ICTP Lecture Notes
SCHOOL ON AUTOMORPHIC FORMS ON GL(n)
31 July - 18 August 2000
Editors
L. G¨ ottsche The Abdus Sa...

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ICTP Lecture Notes

SCHOOL ON AUTOMORPHIC FORMS ON GL(n)

31 July - 18 August 2000

Editors

L. G¨ ottsche The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy G. Harder Mathematisches Institut der Universit¨at Bonn, Bonn, Germany M.S. Raghunathan Tata Institute of Fundamental Research, Mumbai, India

SCHOOL ON AUTOMORPHIC FORMS ON GL(n) – First edition c 2008 by The Abdus Salam International Centre for Theoretical Physics Copyright The Abdus Salam ICTP has the irrevocable and indefinite authorization to reproduce and disseminate these Lecture Notes, in printed and/or computer readable form, from each author. ISBN 92-95003-37-3

Printed in Trieste by The Abdus Salam ICTP Publications & Printing Section

iii PREFACE One of the main missions of the Abdus Salam International Centre for Theoretical Physics in Trieste, Italy, founded in 1964, is to foster the growth of advanced studies and scientific research in developing countries. To this end, the Centre organizes a number of schools and workshops in a variety of physical and mathematical disciplines. Since unpublished material presented at the meetings might prove to be of interest also to scientists who did not take part in the schools and workshops, the Centre has decided to make it available through a publication series entitled ICTP Lecture Notes. It is hoped that this formally structured pedagogical material on advanced topics will be helpful to young students and seasoned researchers alike. The Centre is grateful to all lecturers and editors who kindly authorize ICTP to publish their notes in this series. Comments and suggestions are most welcome and greatly appreciated. Information regarding this series can be obtained from the Publications Office or by e-mail to “pub− [email protected]”. The series is published in-house and is also made available on-line via the ICTP web site: “http://publications.ictp.it”.

Katepalli R. Sreenivasan, Director Abdus Salam Honorary Professor

v CONTENTS M.S. Raghunathan Automorphic Forms in GL(n) I: Decomposition of the Space of Cusp Forms and Some Finiteness Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 T.N. Venkataramana Classical Modular Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 J.W. Cogdell Notes on L-functions for GLn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 D. Prasad and A. Raghuram Representation Theory of GL(n) over Non-Archimedean Local Fields . . 159 G. Harder The Langlands Program (An Overview) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 T. Wedhorn The Local Langlands Correspondence for GL(n) over p-adic Fields . . . . 237

vii Introduction The School on Automorphic Forms on GL(n) took place at the Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste from 31 July to 18 August 2000, under the direction of G. Harder (Universit¨at Bonn) and M.S. Raghunathan (Tata Institute for Fundamental Research, Mumbai). The local organizer was Lothar G¨ottsche (ICTP). The central topics of the school were the theory of automorphic forms on GL(n) and the local theory of representations on GL(n) over p-adic fields. The programme included an introduction to automorphic forms, recent results on L-functions on GL(n), an introduction to the Langlands programme, an introduction to the theory of local representations and an outline of the proof of the local Langlands conjecture for GL(n) over local fields. The school consisted of two weeks of lecture courses followed by one week of conference. This lecture notes volume contains the notes of most of the lecture courses. The electronic version of these lecture notes is available at http://publications.ictp.it/ The school was financially supported by a grant from the European Commission. We are very thankful for this support. I take this opportunity to thank G. Harder and M.S. Raghunathan for organizing this school. I would also like to thank the lecturers of the school and the speakers at the conference for their very interesting lectures.

Lothar G¨ottsche June, 2003

Automorphic Forms in GL(n) I: Decomposition of the Space of Cusp Forms and Some Finiteness Results M.S. Raghunathan∗

School of Mathematics, Tata Institute of Fundamental Research, Mumbai, India

Lectures given at the School on Automorphic Forms on GL(n) Trieste, 31 July - 18 August 2000

LNS0821001

∗

[email protected]

Contents 0 Introduction

5

1 Fundamental Domains

6

2 Automorphic Forms

13

3 Cusp Forms

21

4 Proof of Theorem 2.13

33

References

37

Automorphic Forms in GL(n)

0

5

Introduction

These are notes of lectures given by the author at an Instructional School on “Automorphic Forms in GL(n)” held at the Abdus Salam International Centre for Theoretical Physics, Italy, in August 2000. The author would like to thank the ICTP for their hospitality and their patience over the long delay in making these notes available. In this first chapter we introduce some concepts in the theory of Automorphic Forms and prove some basic results. Although our central focus is on the general linear group GL(n), in this chapter we formulate the results in the more general context of reductive algebraic groups over number fields. This is because the techniques for handling the general case are no different from those needed to handle GL(n). We have also indicated what the notions used mean in the special case of GL(n) (or SL(n)), for readers unfamiliar or uncomfortable with the general theory of algebraic groups. All the results about algebraic groups needed here are to be found in the paper [B-T] of Borel and Tits. In §1 we describe a good fundamental domain for G(k) in G(A) where k is a number field, A is its Ad´ele ring and G is a reductive algebraic group over k. This description follows from a theorem of A. Borel [B] that describes Q a fundamental domain for an arithmetic subgroup Γ in v∈∞ G(kv ): here ∞ is the complete set of inequivalent archimedean valuations of k and for v ∈ ∞, kv is the completion of k at u. Conversely Borel’s theorem can be deduced from this description (of a fundamental domain for G(k) in G(A)). A quick and elegant proof of this (due to R. Godement and A. Weil) is to be found in [G]. In §2 the definition of automorphic forms is given and the central finite dimensionality results are formulated. These concepts and results are due to R. Langlands [L] (see also [H-C]). In §3 we introduce the space of cusp-forms and prove that the representation of G(A) in the space of L2 -cusp-forms is completely reducible, a result due to I. Gelfand and I. Piatetishi-Shapiro. This is needed in §4 where the finite dimensionality theorems formulated in §2 are proved. We need in §3 some well known results from Analysis; these can be found in R. Narasimhan’s book [N].

6

1

M.S. Raghunathan

Fundamental Domains

1.1. Notations Throughout these notes we will adopt the following notations. We denote by k a number field and by V a complete set of mutually inequivalent valuations of k. Let ∞ (resp. Vf ) denote the subset of archimedean (resp. non-archimedean) valuations in V. For v ∈ V, kv denotes the completion of k with respect to v. We denote by Ov for v ∈ Vf the ring of integers in kv and by pv the maximal ideal in Ov . The residue field Ov /pv is denoted Fv and we set qv =| Fv |; also pv = characteristic of Fv . We denote by | |v the absolute value on kv determined by v. We assume the absolute value | |v , v ∈ V chosen so that the following holds: Let Qv denote the closure of Q in kv and Nv : kv → Qv be the norm map. If v is archimedean we set | x |v =| Nv (x) | where | | is the usual absolute value on Qv ≃ R. If v ∈ Vf and Nv (x) = prv · y with r ∈ Z and y a unit in the ring of integers in Qv (≃ Zpv ), | x |v = p−r v . With this definition we have the well known 1.2. Product formula If x ∈ k∗ , | x |v = 1 for all but finitely may v ∈ V Q and v∈S | x |v = 1 where S = {v ∈ V || x |v 6= 1}.

Q 1.3. We denote by A the ring of adeles: A = {x = {xv }v∈V ∈ v∈V kv | xv ∈ Ov for all v 6∈ S, S a finite subset of Vf }: A is a ring under coordinateQ Q wise addition and multiplication. Let ∧ denote the subset v∈∞ kv × v∈Vf Ov ; then ∧ is subring of A. Under the product topology ∧ is a locally compact ring topological ring. A has a natural structure of a locally compact ring such that the inclusion of ∧ (with its product topology) in A is an isomorphism of topological rings on to an open subring. We denote by Af the subset {x ∈ A | xv = 0 for v ∈ ∞}. The group of Ideles I of k is the set of invertible elements in A. It is easy to see that I = {x ∈ A | xv 6= 0 for v ∈ V and | xv |v = 1 for all but a finite number of v ∈ Vf }. We also set If = {x ∈ I | xv = 1 for v ∈ ∞}. The group I is given a topology as Q Q follows: Consider the group v∈∞ kv∗ × v∈Vf Ov∗ where for v ∈ Vf , Ov∗ is the compact group of units in kv∗ ; this group has a natural structure of a locally compact group in the product topology. I is given the unique structure of a Q Q topological group for which the natural inclusion of v∈∞ kv∗ v∈Vf Ov∗× in I is an isomorphism (of topological groups) onto an open subgroup. If x ∈ I Q Q and S = {v ∈ Vf || xv |v 6= 1, we set | x |= v∈V | xv |v = v∈S | xv |v ; then x 7→| x | is a continuous homomorphism of I in R∗ and the kernel of this

Automorphic Forms in GL(n)

7

homomorphism is denoted I1 . Elements of I1 will be called “Ideles of norm 1”. We state without proof the following well known 1.4. Theorem The inclusion of k (resp. k∗ ) in A (resp. I1 ) imbeds k (resp. k∗ ) as a closed discrete subgroup of the (additive) group A (resp. (multiplicative) group I1 ) such that A/k (resp. I1 /k∗ ) is compact. 1.5. Let G denote a connected reductive linear algebraic group over k. (Our main interest is in the case when G = GL(n) the general linear group and we will draw attention to what our terms and definitions mean in this special case). Let S be a maximal k-split torus in G (when G = GL(n), S can be taken to be the group of diagonal matrices in G). Let X ∗ (S) denote the group of characters (all characters of S are defined over k) of S and Φ ⊂ X ∗ (S) the subset of k-roots of G with respect to S: recall that a k-root is a non-trivial character for the adjoint action of S on Lie algebra of G (when G = GL(n), Φ consists of the characters d = diagonal (d1 , · · · , dn ) 7→ di /dj for some pair (i, j), 1 ≤ i, j ≤ n, i 6= j). We fix an ordering on X ∗ (S) and denote by Φ+ (resp. ∆) the set of positive (resp. simple) roots in Φ (when G = GL(n), one usually chooses the ordering given as follows: any character χ on S (= diagonal matrices) is of the form χ(d) =

Y

n (χ)

di i

1≤i≤n

where d = diagonal (d1 , · · · , dn ) and ni (χ) are integers; we set χ > 0 if nr(χ) > 0 where r(χ) = min {i | 1 ≤ i ≤ n, ni (χ) 6= 0}. The set of simple roots for this order are the roots d 7→ di /di+1 , 1 ≤ i < n). We fix once and for all a realisation of G as an algebraic subgroup (over k) of GL(N ) for some N (when G = GL(n), we can take N = n). For v ∈ V then G(kv ) is a 2 closed subset of GL(N, kv ) hence a locally closed subset of M (N, kv ) ≃ kvN and thus acquires a locally compact topology. For v ∈ Vf we denote by Mv the compact open subgroup G(kv ) ∩ GL(N, Ov )(Mv = GL(n, Ov ) when G = GL(n)) of G(kv ). G(A) (resp. G(Af ) the A (resp. Af ) points of G has a natural identification with {g = {gv }v∈V (resp.Vf ) | gv ∈ G(kv ) and the set {v ∈ Vf | gv 6∈ Mv } is finite}. We have a natural inclusion Y

v∈Vf

Mv ֒→ G(Af )

8

M.S. Raghunathan

and G(Af ) is made into a locally compact topological group by stipulating that the above inclusion is an isomorphism of topological groups of the group on the left onto an open subgroup (the topological group G(Af ) obtained in this fashion is independent of the realisation of G as a k-algebraic subgroup Q of GL(N ) for some N ). We set G∞ = v∈∞ G(kv ). We then have a natural identification of G(A) with G∞ × G(Af ). We give to G∞ the product Q topology on v∈∞ G(kv ) and to G(A) the product topology on G∞ ×G(Af ). Let pf (resp. p∞ ) be the projection of G(A) on G(Af ) (resp. G∞ ). For g ∈ G(A), we set gf = pf (g) and g∞ = p∞ (g). Let df (resp. d∞ ) = pf ◦ d (resp. p∞ ◦ d). We have a natural inclusion d : G(k) ֒→ G(A) of the k-points of G in G(A) as a closed discrete subgroup of G(A). In the sequel we identify G(k) with d(G(k)). We will now describe certain k-algebraic (resp. closed) subgroups of G and (resp. G(A)) that will be needed in the sequel. Let Ad denote the adjoint representation of G on its Lie algebra (over k) L(G). Under the adjoint action of S, L(G) decomposes into a direct sum of eigenspaces, the eigencharacters being the roots and the trivial character. For α ∈ Φ, let L(G)(α) be the eigenspace corresponding ` to α and let L(U+ ) = α∈Φ+ L(G)(α). Then there is a unique k-subgroup U of G normalised by S and with L(U) as the Lie subalgebra of L(G) corresponding to U. Also U is normalised by the Z(S) the centraliser of S and P = Z(S) · U is a parabolic subgroup defined over k : P is a minimal kparabolic subgroup of G. We denote by o P the intersection of the kernels of all squares of characters on P defined over k; then P = S·o P and o P = B·U where B is a reductive algebraic k-subgroup of o P which is anisotropic over k (i.e. B does not admit a non-trivial k-split toral subgroup). As S is k-split we may treat it as obtained from a Q-split torus by the base change Q ֒→ kthus in the sequel we will treat S as a Q split torus. We then have for each v ∈ ∞ an inclusion S(Qv ) ֒→ S(kv ) when Qv is the closure of Qv in kv ; and since all the v ∈ ∞ induce on Q the unique archimedean topology with R Q as the completion, we obtain a diagonal inclusion S(R) ֒→ v∈∞ S(kv ). Let Q v∈∞ S(kv ) → S(A) be the inclusion s = {sv }v∈∞ 7→ {sv }v∈V where for v ∈ V\∞, sv = 1. We denote by A the image of the identity component of Q S(R) (a subgroup v∈∞ S(kv )) in G(A). The group A is a closed subgroup of G(A) isomorphic to a product of ℓ = dim S copies of the multiplicative group R+ = {x ∈ R | x > 0}. For a constant c > 0 let Ac = {x ∈ A | α(x) ≤ c for α ∈ ∆}: α is a character (defined over k) on S and as is easily seen on A takes positive real values. With these notations we have

Automorphic Forms in GL(n)

9

1.6. Theorem (R. Godement and A. Weil) There is a compact subset Ω of o P(A) such that Ω ·o P(k) = P(A). There is a maximal compact (open) subgroup K of G(A) and a constant c0 > 0 such that for all c ≥ c0 and compact subsets Ω′ of o P(A) with Ω′ ⊃ Ω, one has K.Ac .Ω′ .G(k) = G(A). 1.7. Corollary Kf .Ω′f .df G(k) = G(Af ) where Ω′f = pf (Ωf ). 1.8. Let Lf be a compact open subgroup of G(Af ) = {g ∈ G(A) | gv = 1 for all v ∈ ∞}. Let ΓLf = γ ∈ G(k) | pf (γ) ∈ Lf }. Now G∞ × Lf has a natural identification with an open subgroup of G(A) and ΓLf is in a natural fashion a closed discrete subgroup of G∞ × Lf . Since Lf is compact, the projection d∞ (ΓLf ) of ΓLf on G∞ is a discrete subgroup of G∞ . The maximal compact subgroup K above decomposes as a direct product K∞ × Kf following the product decomposition G(A) = G∞ × G(Af ) with K∞ (resp. Kf ) a maximal compact subgroup of G∞ (resp. G(Af )). We may also assume that Ω is of the form Ω∞ × Ωf , where Ω∞ (resp. Ωf ) is a compact subset of o P∞ (resp. o PAf )). It follows then that if g ∈ G∞ , there exists γ ∈ G(k) such that gγ∞ ∈ K∞ Ac · Ω∞ and γf ∈ Kf · Ωf . Now Lf being an open compact subgroup and Kf · Ωf being a compact set of G(Af ), we can find finitely many elements θ1 , · · · , θr ∈ G(Af ) such that Kf Ωf ⊂ ∪1≤i≤r θi Lf . In particular we have γf = θi ℓi for some ℓi ∈ Lf and 1 ≤ i ≤ r. Now if γf , γf′ are such that γf = θi ℓi and γf′ = θi ℓ′i with ′ ℓi , ℓ′i ∈ Lf , then γf′ = γf .ℓ−1 i ℓi = γf ζ with ζ ∈ Lf . We thus see that we may assume that the θi = ξif with ξi in G(k). Let Ξ = {ξ1 , · · · , ξr } ⊂ G(k). Then one finds that K∞ Ac · Ω∞ · Ξ∞ · d∞ (ΓLf ) = G∞ . We summarize the above discussion in 1.9. Theorem (A. Borel) Let L be a compact open subgroup of G(Af ) and Γ = d∞ (ΓL ), where ΓL is the subgroup d−1 f (L) ⊂ G(k). Then Γ a discrete subgroup of G∞ . There exists a finite subset Ξ in G(k) a constant c0 > 0 and a compact set Ω∞ ⊂◦ P∞ such that K∞ · Ac · Ω′∞ · Ξ∞ · Γ = G∞ where K∞ is a maximal compact subgroup of G∞ . Further for c′ > c if Sc′ = K∞ · Ac′ · Ω′∞ , the set {γ ∈ Γ | Sc′ ξ ∩ Sc′ ξ ′ 6= φ}

10

M.S. Raghunathan

is finite for ξ, ξ ′ ∈ Ξ∞ . (The last assertion is an additional piece of information that cannot be deduced from Theorem 1.6.) 1.10. Suppose now that the centre of G does not contain a split torus (the centre of GL(n) is a 1-dimensional split torus so the considerations of this paragraph are not applicable to GL(n); they are however applicable to the group SL(n)). In this case by using the Iwasawa decomposition in G∞ it can be shown that the Haar measure of the set Sc′ is finite. Since a left translation invariant Haar measure on G∞ is also right translation invariant, one sees that a Haar measure on G∞ defines a left translation (under G∞ ) invariant finite measure on G∞ /Γ. Y [We will elaborate on this in the case G = SL(n). In this case G∞ = SL(n, kv ). We can identify SL(n, kv ) v∈∞

with SL(n, R) or Y SL(n, C) according to kv ≃ R or C. The compact group K∞ is a product Kv where for v ∈ ∞, Kv = SO(n) or SU (n) according v∈∞

to kv ≃ R or C. The group A can be identified with the group of diagonal matrices in SL(n, R) with positive real diagonal entries - the inclusion of Q in k induces for each v ∈ ∞, an inclusion SL(n, R) in SL(n, kv ) and hence a diagonal inclusion of SL(n, R) in G∞ . For c′ > 0, Ac′ = {d = diagonal (d1 , . . . , dn ) | di > 0 for 1 ≤ i ≤ n, d1 · d2 . . . dn = 1 and di /di+1 ≤ c for 1 ≤ i < n}. The group P consists of upper triangular matrices in SL(n) and let U be the Ysubgroup of upper triangular unipotent matrices in P. The o group ◦ P∞ = P(kv ) and o P(kv ) = B(kv ). U(kv ) where B(kv ) = {d = v∈∞

diagonalY(d1 · · · dn ) | di ∈ kv∗ , | di |v = 1 for 1 ≤ i ≤ n}. One sees then that B∞ = B(kv ) is compact and contained in K∞ and as it centralises A, we v∈∞

see that taking Ω′ to be left translation invariant under B∞ , Sc = K∞ ·Ac′ ·Ω′1 with Ω′1 a suitable compact subset of U∞ . The natural map K∞ × A × U∞ → G∞

is an analytic isomorphism of manifolds. The Haar measure on G∞ pulled back to K∞ ×A×U∞ takes the form ρ2 (a)·dk·da·du where dk (resp. da, du) is a Haar measure on K∞ (resp. A, U) and ρ2 is the homomorphism of A in R+ determined as follows: Inner conjugation by a ∈ A carries the Haar 2 measure duRon U+ ∞ into ρ (a) · du (U∞ is normalised by A). It is also easy to see that A′ ρ2 (a)da < ∞. Since K∞ and Ω′1 are compact, it is immediate c

Automorphic Forms in GL(n)

11

that the Haar measure of Sc′ is finite.] We continue with the assumption that G contains no nontrivial k-split central torus (so that G∞ /Γ has finite Haar measure). We assert now that in this case the measure induced on G(A)/G(k) by the Haar measure on G(A) is finite as well. To see this we denote by G1 the closure of G(k) (imbedded diagonally in G(Af ). Now G1 = G(k) if G∞ is compact; in this case one knows (Godement Criterion) that G(A)/G(k) is compact and the finiteness of the Haar measure follows. Thus we assume that G∞ is not compact (this is the case when G = SL(n) with n ≥ 2). Then one knows that G(Af )/G1 is compact and that G1 is normal in G(Af ) (when G = SL(n) one has in fact G1 = G(Af )-this is seen easily using the fact that SL(n, kv ) is generated by upper and lower triangular unipotent matrices and the fact (by the Chinese remainder theorem) that k is dense in Af . π : G(A) → G∞ · G1 \G(A) ≃ G1 \G(Af ) be the natural map. From the definition of G1 it is clear that π factors through G(A)/G(k). Since G1 \G(Af ) is compact we conclude that in order to show that G(A)/G(k) has finite Haar measure it suffices to show that G∞ · G1 /G(k) has finite Haar measure. Now if L is any compact open subgroup of G(Af ), G∞ · G1 ⊂ G∞ · L · G(k) and the quotient L · G1 /G1 is compact. Thus it suffices to show that G∞ · L · G(k)/G(k) has finite Haar measure for a compact open subgroup L ⊂ G(Af ). Clearly one has a natural identification of G∞ · L · G(k)/G(k) with G∞ · L/(G∞ · L ∩ G(k)); since G∞ /Γ (Γ = projection of ΓL = G∞ · L ∩ G(k) on G∞ ) has finite Haar measure and L is compact G∞ · L · G(k)/G(k) has finite Haar measure proving our contention. We have thus 1.11. Theorem Assume that G has no non trivial k split torus in its centre. Then G(A)/G(k) has finite Haar measure. If G contains no non trivial k-split torus G(A)/G(k) is compact. 1.12. Suppose now that C ⊂ G is the maximal central k split torus. Then G = G/C contains no non trivial central split torus. In the case of main interest to us viz when G = GL(n), C(≃ GL(1)) is of dimension 1 and is the entire centre and G is the group P GL(n). By Theorem 1.10, G(A)/G(k) has finite Haar measure. Now it is known that if π : G → G is the natural map, π(G(A)) is a closed normal subgroup of G(A) and G(A)/π(G(A)) is compact. Also the kernel of π : G(A) → G(A) is evidently C(A). It follows that G(A)/π(G(k)) is compact so that G(A)/C(A) · G(k) is compact as well. Once again we elaborate on this in the special case G = GL(n).

12

M.S. Raghunathan

Here for any field k′ ⊃ k the natural map GL(n)(k′ ) → P GL(n)(k′ ) is surjective - this is an immediate consequence of Hilbert Theorem 90. Using the realisation of P GL(n) as an algebraic k subgroup of GL(n2 ) got from the adjoint representation, one shows that GL(n, Ov ) → P GL(n)(Ov ) is surjective for all v ∈ Vf . It is then immediate that the map GL(n, A) → P GL(n)(A) is surjective. Observe next that C(A)(≃ GL(1, A)) ≃ I in a natural fashion and in the sequel we will treat continuous characters χ : I → C∗ as characters on C(A) under this isomorphism. Let F χ (G(A), G(k)) denote the vector space of all functions f : G(A) → C such that we have f (gc) = χ(c)−1 f (g) for c ∈ C(A) and f (gx) = f (g) for x ∈ G(k) Note that in order that F χ (G(A), G(k)) be non-trivial we need χ to be 1 ∗ trivial on C(k). If χ is unitary i.e., maps C(A) into S = {z ∈ C | z |= 1}, then the function g 7→| f (g) | on G(A) is invariant under C(A) · G(k) and hence defines a function f on G(A)/C(A) · G(k). We will say that f ∈ F χ (G(A), G(k)) is square integrable if f is square integrable on G(A)/C(A)· G(k). Let G′ = [G, G] be the commutator subgroup. Then the connected component S′ of the identity in S ∩ G′ is a maximal split torus in G′ . The natural map C × S′ → S is not in general an isomorphism (it is not in the case G = GL(n) where G′ = SL(n)). However the group A in G(A) breaks up into a direct product (C(A) ∩ A) × (S′ (A) ∩ A) i.e., the natural map (C(A) ∩ A) × (S′ (A) ∩ A) → A is an isomorphism. Let A0 = C(A) ∩ A and A′ = S′ (A) ∩ A so that AY ≃ A0 × A′ . We now define a subgroup A10 in | χ(av ) |v = 1 for every character χ of G A0 : A10 = {a ∈ A0 ⊂ C(A) | v∈∞

defined over k}. Since χ(av ) ∈ R+ for all a ∈ A0 and any character χ on C, and the subgroup of characters on C which are restrictions of characters 1 on G has finite Y index in the group of all characters on C, we see that A0 = {a ∈ A0 | | χ(av ) |v = 1 for all characters χ on C}. The sequence v∈∞

{1} → A10 → A0 → A0 /A10 → {1} splits and has in fact a natural splitting. To see this fix an isomorphism of C with a product GL(1)r of r copies of GL(1). Note that such an isomorphism yields compatible decompositions of A0 and A10 as products of r copies of the same groups. This means that we need to give the ‘natural splitting’ in the case when C = GL(1). Here C(A) = I and A0 is the product of

Automorphic Forms in GL(n) | ∞ | copies of R+ imbedded in

Y

13

kv∗ through the inclusions R ⊂ kv ≃ R

v∈∞ 1 or C(vY ∈ ∞). The group YA0 is then naturally isomorphic to the subgroup {x ∈ kv∗ | xv ∈ R+ , xv = 1} and a natural supplement to A10 in A is v∈∞ v∈∞ Y ∗ kv | xv = x for all v ∈ ∞ with x ∈ R+ }. Thus we have the group {x ∈ v∈∞

obtained a splitting using an isomorphism of C with (GL(1))r . It is easy to see that the splitting is independent of the choice of this isomorphism. Now let G∗ = {g ∈ G(A) | | χ(g) |= 1 for every character χ of G defined over k. Then G∗ ⊃ A10 and if ′ A0 is the natural supplement to A10 in A0 , then ′ A ∩ G∗ = {1} and the natural map 0 ′

A0 × G∗ → G(A)

is an isomorphism (of locally compact groups). The product formula (1.2) shows that G(k) ⊂ G∗ . Using the fact that I1 /k∗ is compact, it is now easy to deduce the following from Theorem 1.10. 1.13. Theorem G∗ /G(k) has finite Haar measure. 1.14. It is easy to see that any compact subgroup of G(A) is contained in G∗ . Consequently there is a c > 0 and a compact subset Ω ⊂ o P(A) ∩ G∗ = P∗ such that for all Ω ⊃ Ω′ , Ω ⊂ o P(A) ∩ G∗ , and c′ ≥ c we have G∗ = K · A10 · A′c′ · Ω′ · G(k) where A′c′ = {a ∈ A1 | α(a) ≤ c′ for all α ∈ △}. Let C(A)1 = {g ∈ C(A) | | χ(g) |= 1 for all characters χ on C}. Then C(k) ⊂ C(A)1 (product formula) and C(A)1 /C(k) is compact. Hence we can find a compact subset C of C(A)1 such that C · C(k) = C(A)1 . Clearly C · Ω′ = Ω0 is a compact subset of P∗ and one has G∗ = K · A′c · Ω0 · G(k).

2

Automorphic Forms

2.1. We continue with the notations introduced in §1. We fix a unitary character χ : C(A) → S 1 on C(A) and assume that χ is trivial on C(k). We introduce now some function spaces. Recall that we defined F χ (G(A), G(k))

14

M.S. Raghunathan

as the vector space of all functions f : G(A) → C satisfying the following condition f (xgγ) = χ(x)−1 f (x) for all g ∈ G(A), x ∈ C(A) and γ ∈ G(k). In the sequel we often write F χ for F χ (G(A), G(k)) (where there is no ambiguity about the group G we are talking about). We set C χ = C χ (G(A), G(k)) = {f ∈ F χ | f

continuous}

Ccχ = Ccχ (G(A), G(k)) = {f ∈ C χ | f has compact support modulo C(A)G(k)}. A function f R: G(A) → C in F χ is square integrable iff f is Borel measurable and k f k22 = G(A)/C(A)G(k) | f (g) |2 dµ(g) < ∞− | f (xgγ) |=| f (g) | for g ∈ G(A), x ∈ C(A) and γ ∈ G(k) so that | f (g) | can be treated as a function on G(A)/C(A)G(k); µ is the Haar measure on this homogeneous space of the locally compact group G(A)/C(A). We denote by Lχ2 = Lχ2 (G(A), G(k)) the set of all square integrable functions in F χ modulo the equivalence of equality almost everywhere with respect to µ. It is a Hilbert space under the inner product Z ′ f (g)f ′ (g)dµ(g). (f, f ) 7→ G(A)/C(A)G(k)

We have (as usual) an inclusion Ccχ ֒→ Lχ2 . A function f : Ω → C, Ω an open set in G(A) is said to be C ∞ if the following conditions hold: (i) for any g0 ∈ Ω, there is a neighbourhood Ω′ of g0 in Ω and a compact open subgroup L ⊂ G(Af ) such that LΩ′ ⊂ Ω and f (xg) = f (g) for all g ∈ Ω′ and x ∈ L. (ii) For any g0 ∈ Ω, there is neighbourhood U of the identity in G∞ such that U g0 ⊂ Ω and the map x 7→ f (xg0 ) is C ∞ on U . Following the decomposition G∞ × G(Af ) ∼ = G(A) as a direct product we may regard any complex-valued function f on G(A) as a function of two variables (one in G∞ and the other in G(Af )). If f is C ∞ , then it is C ∞ in the first variable and locally constant in the second variable; also the partial derivatives of f with respect to the first variable are themselves C ∞ on G(A). We denote by C χ∞ = C χ∞ (G(A), G(k)) (resp. Ccχ∞ = Ccχ∞ (G(A), G(k)) the vector space of C ∞ functions (resp. C ∞ functions with support compact modulo C(A) · G(k)). We have evidently inclusions Ccχ∞ ֒→ Ccχ ֒→ Lχ2 .

Automorphic Forms in GL(n)

15

2.2. In the sequel E will denote any one of the function spaces F χ , C χ , Ccχ , C χ∞ , Ccχ∞ , Lχ2 introduced above and for an open compact subgroup L of G(Af ), E L will denote the subspace of functions in E which are invariant under left translations by L. We observe that a function f in E is determined by its restriction to G∗ (since ′ A0 G∗ = G(A) and ′ A0 ⊂ C(A)). On the other hand G∗ = K · A10 · A′c · Ω′ · C · G(k) with C ⊂ C(A) (cf. 1.14). Further K = K∞ · Kf with K∞ ⊂ G∞ and Kf ⊂ G(Af ) and one may assume that Ω′ = Ω∞ · Ωf with Ω∞ (resp. Ωf ) a compact subset of ◦ P∞ (resp. ◦ P(Af )). Now Kf · Ωf is a compact subset of G(Af ). It follows that if Lf ⊂ G(Af ) is a compact open subgroup of G(Af ), there exists a finite set H′ ⊂ G(Af ) such that Kf · Ωf ⊂ Lf · H′ . It follows from Corollary 1.7 that we have Lf · H′ · df (G(k)) = G(Af ). We introduce an equivalence relation on H′ as follows: for θ, θ ′ ∈ H′ , θ ∼ θ ′ iff ′ there exists ρ ∈ G(k) with θρf θ −1 ∈ Lf (it is easily checked that this is an equivalence relation). Let H ⊂ H′ be a subset containing exactly one element in each equivalence class. We then assert that any f ∈ E Lf is determined by its restriction to G∞ Lf H (and in view of Lf -invariance) by its restriction to G∞ · H. Evidently to prove this it suffices to show that G∞ · Lf · H · C(A) · G(k) = G(A). We have seen that G(A) = ′ A0 · G∗ =

′

A0 · K∞ · A10 · A′c · Ω∞ · Kf · Ωf · C(A)G(k) ⊂ G∞ · Lf · H′ · C(A) · G(k).

Thus it suffices to show that if θ ∼ θ ′ , G∞ · Lf · θ · G(k) ⊃ G∞ · Lf · θ ′ . ′

Since θ ∼ θ ′ , there exists ρ ∈ G(k) with θρf θ −1 = ℓ ∈ Lf so that θ ′ = ℓ−1 θρf . It follows that G∞ Lf θ ′ ρ−1 = G∞ Lf θρf ρ−1 = G∞ ρ−1 ∞ Lf θ = G∞ Lf θ. It follows that Lf · H · df (G(k)) = G(Af ). For θ ∈ H, let rθ (f ) : G∞ → C be the function rθ (f )(g) = f (gθ) = f (θg). Then rθ (f )(gγ∞ ) = rθ (f )(g) for all γ ∈ G(k) such that γf ∈ θ −1 Lf θ. In fact one has rθ (f )(gγ∞ ) = f (gγ∞ θ) = f (gγ∞ θγ −1 ) = f (gθγf−1 θ −1 θ) = f (gθ). −1 Let Γθ = d∞ (G(k) ∩ d−1 f (θ Lf θ)) i.e.,

Γθ = d∞ {γ | γf ∈ θ −1 Lf θ}.

16

M.S. Raghunathan

Let C χ (G∞ /Γθ ) (resp. Ccχ (G∞ /Γθ ), resp. C χ∞ (G∞ /Γθ ), resp. Ccχ∞ (G∞ /Γθ ), resp. Lχ2 (G∞ /Γθ )) be the space of all continuous functions (resp. continuous functions with compact support modulo C∞ (resp. C ∞ functions, resp. C ∞ functions with compact support modulo C∞ , resp. square integrable functions) on G∞ /Γθ satisfying the condition f (gx) = χ(x)−1 f (g) for x ∈ C∞ . Then if E = C χ , (resp. Ccχ , resp. C χ∞ , resp. Ccχ∞ , resp. Lχ2 ), and f ∈ E Lf , rθ (f ) ∈ Cχ (G∞ /Γθ ) (resp. Cχc (G∞ /Γθ ), resp. Cχ∞ (G∞ /Γθ ), χ resp. Cχ∞ c (G∞ /Γθ ), resp. L2 (G/Γθ ). We thus obtain maps a ` rθ : CχLf → θ∈H Cχ (G∞ /Γθ ) θ∈H a ` χL rθ : Cc f → θ∈H Cχc (G∞ /Γθ ) θ∈H a ` rθ : Cχ∞Lf → θ∈H Cχ∞ (G∞ /Γθ ) θ∈H a ` χ∞L rθ : Cc f → θ∈H Cχ∞ c (G∞ /Γθ ) θ∈H a

χLf

rθ : L2

θ∈H

→

`

χ θ∈H L2 (G∞ /Γθ ).

2.3. Proposition The maps above are isomorphisms of topological vector spaces. We record here for future use the following fact proved above. 2.4. Lemma

Lf · H · df (G(k)) = G(Af ).

2.5. Proof of 2.3 The topologies on these spaces are the standard ones. On CχL and Cχ (G∞ /Γθ ) (resp. Cχ∞L and Cχ∞ (G∞ /Γθ )) it is the topology of uniform convergence (resp. together with all derivatives) on compact sets. On CχL and Cχ∞ (resp. Cχc (G∞ /Γθ ) and Cχ∞ c c c (G∞ /Γθ )) it is the inductive limit of the topology of uniform convergence (resp. together with all derivatives) on closed sets of G(A) which are compact modulo C(A)·G(k) and closed subsets of G∞ which are compact modulo C∞ · Γθ respectively. The L2 spaces are of course given the Hilbert space structure. That the maps are continuous injections is easy to see. We need to show that the maps are surjective - the open mapping theorem would then guarantee that the maps are isomorphisms. Let {fθ }θ∈H be a collection of functions on G∞ /Γθ , θ ∈ H belonging to one of the above spaces. Define a function f on G∞ LH by setting f (gℓθ) = fθ (g). We extend the function f to all of

Automorphic Forms in GL(n)

17

G(A) by setting f (g) = f (gγ) where γ ∈ G(k) is an element such that gγ ∈ G∞ LH. We need only check that f is well defined i.e., if gγ ′ ∈ G∞ LH also for some γ ′ ∈ G(k), f (gγ) = f (gγ ′ ). Let gγ = hℓθ and gγ ′ = h′ ℓ′ θ ′ . Then setting γ −1 γ ′ = ζ, we have hℓθζ = h′ ℓ′ θ ′ . It follows that ℓθζf = ℓ′ θ ′ ′ leading to θζf θ −1 = ℓ−1 ℓ′ ∈ L i.e., θ ∼ θ ′ and hence θ = θ ′ ; and when θ = θ ′ , ζf ∈ θ −1 Lθ so that ζ ∈ Γθ , and fθ is Γθ -invariant. This proves that f is well defined. That f has the required properties of continuity, smoothness etc. if the {fθ }θ∈H have them is immediate from the definitions. 2.6. Let U denote the universal enveloping algebra of the real Lie algebra L(G) of G. Then U operates on the space of C ∞ functions on G(A) leaving stable the subspaces C χ∞ (G(A), G(k)) and Ccχ∞ (G(A), G(k)) as well as the subspace of L-invariants in these spaces for any compact open subgroup L of G(Af ). A function f ∈ C χ is K-finite if the C-linear span of {Lk f | k ∈ K} is finite dimensional. The group K acts continuously on this finite dimensional vector space V and hence the image of the profinite group Kf in GL(V ) is finite. Thus if ϕ is K-finite, ϕ is invariant under an open compact subgroup of G(Af ); and the C-span of {Lk ϕ | k ∈ K∞ } is finite dimensional. Conversely if ϕ satisfies these two conditions ϕ is K-finite. Let Z be the centre of U. A C ∞ function ϕ in C χ∞ is Z-finite iff the C-linear span of {zf | z ∈ Z} is finite dimensional. Functions relevant to harmonic analysis on G(A)/G(k) are those that do not grow too rapidly at infinity. To describe the kind of growth at infinity that we need to impose we need some preliminary definitions. We assume, as we may, that the imbedding of G in GL(N ) maps G into SL(N ) (when G = GL(n) we can take N = 2n and the imbedding to be g 0 g 7→ g ∈ GL(n)). 0 tg−1 This is done so that one takes care that the entries of g as well as those of g−1 are handled simultaneously.YWe define for g ∈ G(A) the height of g denoted k g k in the sequel as (max1≤i,j≤N | gvij |v ) where gvij , 1 ≤ v∈V

i, j ≤ N are the entries of the v-adic component gv ∈ G(kv ) of g. Observe that since gv ∈ GL(N, Ov ) for all v ∈ V\S for a finite subset S containing ∞, max1≤i,j≤N | gvij |= 1 for all but a finite number of v and thus the product above over all v ∈ V reduces to a finite product. It is easy to see that g 7→k g k is a continuous function of G(A) in R+ . In the sequel we will say that a right G(k)−invariant function f : G(A) 7→ C has moderate

18

M.S. Raghunathan

growth if there exist constants c, r > 0 such that for all g ∈ G(A) one has | f (g) |≤ c k g kr with this notion of moderate growth we have 2.7. Definition An automorphic form (with central unitary character χ) on G(A) is a C ∞ function f : G(A) → C satisfying the following conditions (i) U f is of moderate growth for all U ∈ U. (ii) f (gx) = χ(x)−1 f (g) for all g ∈ C(A) and all x ∈ C(A). (iii) f (gγ) = f (g) for all g ∈ G(A) and γ ∈ G(k). (iv) f is K-finite (v) f is Z finite. The automorphic forms with central character form a vector space which we denote Aχ 2.8. Remarks (i) If a nonzero automorphic form is to exist, with χ as central character, evidently one must have χ(ρ) = 1 for all ρ ∈ C(k) in view of the condition (iii). Thus we will consider only characters χ on C(A) that are trivial on C(k). Also note that since χ is unitary, the growth condition does not lead to any contradiction. (ii) We have assumed that the group G is reductive. This means that the Lie algebra L(G∞ ) of G∞ is a direct product of an abelian Lie algebra h and a semisimple Lie algebra s. The Lie algebra s has a Cartan - decomposition s = k ⊕ p with the Killing form of s restricted to k (resp. p) negative (resp. positive) definite. Let Xi , 1 ≤ i ≤ r, Yj , 1 ≤ j ≤ s and Zk , 1 ≤ k ≤ t ′ be bases of h, k and p respectively so chosen j , Yj ′ >= −δ Xjj and Xthat < YX 2 2 Yj2 is Xi + Zk − < Zk , Zk′ >= δkk′ . Then the element C = 1≤i≤r

a central element of the enveloping algebra U. Let

1≤p≤t C′ =

X

1≤j≤s Yj2 . Then

1≤j≤s

C + 2C ′ is an element of U and the corresponding differential operator △ on G∞ is an elliptic operator. Let B be the subalgebra of End C χ (G(A), G(k))

Automorphic Forms in GL(n)

19

generated by the centre Z of U and {Lk | k ∈ K} (Lk is the left translation by k). Then the C-span of {T (f ) | T ∈ B} is finite dimensional for the automorphic form f . Now the span of the Lk f, k ∈ K contains U f for all U in the subalgebra of U generated by k. Thus C and C ′ belong to B and hence so does △. It follows that {△n f | n ∈ N} spans a finite dimensional vector space over C. We conclude that there is a monic polynomial P in one variable such that P (△)f = 0. Now P (△) is an elliptic operator on G∞ with analytic coefficients. By the regularity theorem for elliptic operators with analytic coefficients we conclude that f is analytic (in the variable in G∞ ). (iii) The third comment is that if U ∈ U and f is an automorphic form then so is U f . It is clear that it suffices to check this when U ∈ L(G). Now the map L(G∞ ) ⊗ C χ∞ given by (U, f ) 7→ U f is compatible with the action of G(A) on the two sides. G(A) acts on L(G) by the adjoint representations of the factor G∞ and on C χ∞ by left translation; we take the tensor product representation on the left-hand side. Suppose now V is a finite dimensional subspace of C χ∞ which is K as well as Z stable. Then the image of L(G)⊗V in C χ∞ under the above map is finite dimensional Z-stable as well as Kstable. As this space contains U f , for U ∈ L(G∞ ), we see that U f is Kfinite and Z-finite. That U ′ (U f ) = (U ′ U )(f ) has moderate growth is clear from the definitions. (iv) The vector space of all automorphic forms Aχ with central character χ is stable under left translations by elements of G(Af ). This is seen as follows: for g ∈ G(Af ), gKg−1 ∩ K has finite index in K and Lg f is (g−1 Kg ∩ K)finite and hence K-finite as well. Since the action of G(Af ) on the left and the action of U on the space of C ∞ functions commute, we see that (Lg f ) is Z-finite for g ∈ G(Af ). The growth condition is immediate from the fact that for g0 ∈ G(Af ) one has a constant C > 0 such that for all g ∈ G(A), k g0 g k≤ C k g k. 2.9. Lemma (Harish-Chandra: Acta Math. 116 p.118). If f is a K∞ ∞ funcfinite Z-finite C ∞ function on G∞ , there is an α ∈ C∞ c (G∞ ) (= C tions with compact support in G∞ ) such that α ∗ f = f . This lemma shows that the growth condition (i) in (2.6) can be replaced by the weaker condition: (i′ )·f is of moderate growth. In fact fix α ∈ Cc∞ (G) such that α ∗ f = f . Then one has for U ∈ U, U f = U (α ∗ f ) = U α ∗ f ; and

20

M.S. Raghunathan

it is easy to see that for any β ∈ Cc∞ , β ∗ f has moderate growth. In the sequel we will make use of Lemma 2.8 in other contexts too. 2.10. Let χ be a unitary character on C(A)/C(k). Let σ : K → GL(W ) be a finite dimensional representation of the compact subgroup K of G(A) and λ := Z → EndC W ′ a finite dimensional representation of Z. Let Aχ (G(A), G(k), σ, λ) denote the space of complex values C ∞ functions f on G(A) satisfying the following conditions (cf. 2.8). (i) f has moderate growth. (ii) f (gγ) = f (g) for g ∈ G(A), γ ∈ G(k). (iii) f (gh) = χ(h−1 )f (g) for g ∈ G(A), h ∈ C(A). (iv) The K-span of f (in F χ ) is a quotient of W as a K-module. (v) The Z-span of f (in F χ ) is a quotient of W ′ as a Z-module. With this notation our central result is 2.11. Theorem

dim Aχ (G(A), G(k), σ, λ) < ∞.

2.12. The representation σ of K when restricted to the totally disconnected subgroup Kf has a kernel Lf of finite index. Then it is easily seen that Aχ (G(A), G(k), σ, λ) is contained in the space of C ∞ functions f on G(A)/G(k) such that f (gx) = χ(x)−1 f (g) for g ∈ G(A), x ∈ C(A), f (kg) = f (g) for k ∈ Lf , the K∞ -span of f is a quotient of σ∞ as a K∞ -module and the Z-span of f is a quotient of λ. We can now appeal to Proposition 2.4 to conclude that Theorem 2.11 is equivalent to: 2.13. Theorem Let Γ ⊂ G(k) be an arithmetic subgroup. Let Aχ (G∞ , Γ, σ∞ , λ) be the space of C ∞ functions f on G such that (i) f (gγ) = f (g) for all g ∈ G∞ and γ ∈ Γ. (ii) f (gc) = χ(c)f (g) for all g ∈ G∞ and c ∈ C∞ . (iii) The K∞ -span of f is a quotient of σ∞ . (iv) The Z-span of f is a quotient of λ.

Automorphic Forms in GL(n)

21

(v) there is a constant c > 0 and integer r > 0 such that | f (g) |≤ c k g kr for g ∈ G∞ . Then Aχ (G∞ , Γ, σ∞ , λ) is finite dimensional.

3

Cusp Forms

3.1. We need a number of facts from analysis which we collect together in 3.2 - 3.11 below for convenient later use. We refer to R. Narasimhan [N] for proofs of results not given here. We begin with the following 3.2. Proposition Let X be a locally compact space and µ a Borel probability measure on X. Let Cb (X) denote the space of bounded continuous functions on X and for ϕ ∈ Cb (X), let k ϕ k∞ = lim Sup{ϕ(x) | x ∈ X}. Suppose T : L2 (X, µ) → L2 (X, µ) is a linear map such that T (ϕ) ∈ Cb (X) for all ϕ ∈ L2 (X, = µ) and there is a constant C > 0 such that k T (ϕ) k∞ ≤ C k ϕ k2 . Then T is an operator of the Hilbert Schmidt type (and hence compact). Proof. For x ∈ X, the linear form ϕ 7→ T (ϕ)(x) on L2 (= L2 (X, µ)) is bounded. It follows that there exists kx ∈ L2 such that k kx k2 ≤ C and T (ϕ)(x) =< kx , ϕ > for all ϕ ∈ L2 . Let k(x, y) = kx (y) for x, y ∈ X : k(x, y) is defined for all most all y for each x. Clearly for each fixed x, kx (y) is measurable so that k(x, y) is measurable on X × X. Now R R R |2 dµ(x) · dµ(y) = X dµ(x) Y | kx (y) |2 dµ(y) X×X R | k(x, y) ≤ X dµ(x)C 2 = C (since µ(X) = 1) Thus k(x, y) ∈ L2 (X × X, µ × µ) and Z Z k(x, y)ϕ(y)dµ(y). kx (y)ϕ(y)dµ(y) = T ϕ(x) = X

X

Thus T is a Hilbert-Schmidt operator. 3.3. The next result we want to state is the Sobolev inequality. We need to introduce some preliminary notation to state the result. Let M be be a smooth manifold. Let Ω, be a relatively compact subset of M ; the compact closure of Ω is denoted Ω. Let {Ui , | 1 ≤ i ≤ m} be an open covering of

22

M.S. Raghunathan

the closure Ω of Ω by coordinate open sets; let Vi , 1 ≤ i ≤ m, be open subsets of Ui such that the closure V i of Vi is compact and contained in Ui and further ∪1≤i≤r Vi ⊃ Ω. We fix a C ∞ volume form on M and denote the corresponding measure by µ. If x1 , · · · , xn are the coordinates in Ui , we 1 ∂α ∂1 denote by D β the operator ∂x 1 · · · ∂xn for a multi index β = (β1 , · · · , βn ) of n

n

non-negative integers. We introduce on the space of C ∞ functions on Ω (i.e. C ∞ functions defined a neighbourhood of Ω) the following norms k kr,p where r is an integer ≥ 0 and p > 1. For a C ∞ function f on Ω, k f kr,p = Sup {k D β f kpV i | | β |≤ r,

1 ≤ i ≤ m}.

Here for h a C ∞ function on Ω, k h kpV i =

Z

| h |p dµ. Vi

The norm defined above depends on the covering and the shrinking chosen, the coordinates chosen in the covering open sets, and the volume form on M . However the equivalence class of the norm k kr,p depends only on r and p and not on the choices made above. With these definitions we can now state Sobolev’s inequality. 3.4. Theorem There is a constant C, C(p, Ω) > 0 such that for all f, C ∞ on Ω, k f k∞ ≤ C k f k[n/p]∗ ,p where [n/p]∗ is the minimal integer ≥ n/p (n = dim M ). 3.5. Remark In the special case when M = Rn and we take the coordinates to be standard coordinates, the volume form as the standard one, and Ω is a disc of radius ρ centred at a point x0 , we have for any f, C ∞ on Ω, k f k∞ ≤ C k f k[n/p]∗ ,p with C = C(ρ) independent of the point x0 . This is because the measure as well as the vector fields ∂/∂xi are translation invariant. 3.6. Our next result is a very special case of a general theorem about elliptic operators on a compact manifold. We consider a connected real nilpotent Lie group N with a discrete subgroup Φ such that N/Φ is compact. Let L(N ) denote the Lie algebra of N and X1 , · · · , Xn be a basis of L(N ) over R. We consider the Xi to be right translation - invariant vector fields on N , hence they define vector fields on N/Φ which we continue to denote Xi .

Automorphic Forms in GL(n)

23

P Then ∆ = − Xi2 is a non-negative self adjoint elliptic operator on the space of RC ∞ - functions on N/Φ with respect to the inner product : (f, g) 7→ N/Φ f gdn where dn is the Haar measure on N/Φ. One has in fact for X ∈ L(N ), < Xf, g >=< f, −Xg > so that − < X 2 f, f >=< Xf, Xf >≥ 0 and hence < ∆f, f >≥ 0. In fact one has X < ∆f, f >= < Xi f, Xi f > so that ∆f = 0 if and only if Xi f = 0 for all i; and since the Xi , 1 ≤ i ≤ r give a basis for the tangent space at every point this means that f is a constant. Now the general theory of self adjoint elliptic operators applied here shows that L2 (N/Φ), dn) decomposes as an orthogonal direct sum a C⊕ H(λn ) 1≤n 0 such that for any α with | α |≤ 2r and any ϕ ∈ C ∞ (N/Φ), k X α ϕ k2 ≤ C k ∆r ϕ k2 +C ′ k ϕ k2 . Note The norm Sup {k X β ϕ k2 | | β |≤ r} is equivalent to the norm k kr,2 introduced earlier; this is easily seen from the fact that the Xi can be expressed as a linear combination of the ∂/∂xj of the local coordinates with coefficients that are C ∞ . 3.10. Corollary If ϕ ∈ C ∞ (N/Φ) is such that | α |≤ 2r, one has k X β ϕ k2 ≤ C k ∆r ϕ k2

R

N/Φ

ϕdn = 0, then for

for a constant C > 0 (independent of ϕ). 3.11. Corollary Let (N, Φ) be asRin 3.6. Then there is a constant C > 0 such that for ϕ ∈ C ∞ (N/Φ) with N/Φ ϕ(n)dn = 0, we have | ϕ(x) |p ≤ R C N/Φ | ∆r ϕ(x) |p dx for r ≥ [n/p]∗ . This is immediate from Theorem 3.4 and Corollary 3.10. From now on we go back to the notations of §1 and §2. 3.12. A function ϕ ∈ C χ (cf. 2.1) is cuspidal if the following holds: let P′ be a proper parabolic subgroup of G defined over k one U′ its unipotent radical. Then for almost all g ∈ G(A) the function u R7→ ϕ(gu), u ∈ U′ (A) is integrable for the Haar measure on U′ (A)/U′ (k)) and U′ (A)/U′ (k) ϕ(gu)du = 0 for almost all g ∈ G(A). 3.13. Remarks (i) ‘Almost all g’ above means for all g ∈ G(A) outside a set of Haar measure zero. (ii) It is known that every parabolic subgroup defined over k is conjugate to a parabolic subgroup of G defined over k containing a fixed minimal parabolic subgroup P of G defined over k. Using the G(k) invariance χ of elements R of F , it is easily seen that for ϕ to be cuspidal, it′ suffices that U′ (A)/U′ (k) ϕ(gu) vanish for almost all g ∈ G(A) for U the unipotent radical of any of the (finitely many) k-parabolic subgroups P′ containing P.

Automorphic Forms in GL(n)

25

(iii) We observe next that for ϕ to be cuspidal, it sufficient that the integral vanish for maximal parabolic subgroups defined over k and containing ′′ P. In fact if P′ is a k-parabolic subgroup and P is a maximal kparabolic subgroup containing P′ , then the unipotent radical U′ of P′ ′′ ′′ contains the unipotent radical U of P as a normal subgroup. We ′′ ′′ have thus a fibration U′ (A)/U′ (k) → U′ (A)/U (A)U (k) whose fibres ′′ ′′ are U (A)/U (k); our contention now follows from Fubinis theorem which is valid for this fibration. 3.14. Notation If E denotes one of the spaces in E = {F χ , F χ , C χc , C χ∞ and Lχ2 } we denote by o E the subspace {ϕ ∈ E | ϕ cuspidal}. 3.15. If E is one of the spaces in E, then o E is invariant under the left action of G(A) on G(A)/G(k). This is clear from the definition of cuspidal functions. Also if E ∈ E and E is not F χ , for ϕ ∈ E and α ∈ Fc∞ (G(A)), α ∗ ϕ ∈ E as is easily seen from the definition of the convolution operation R ((α ∗ ϕ)(g) = G(A) α(h)ϕ(h−1 g)dh (dh Haar measure on G(A))). It is further easy to see that ϕ 7→ α∗ϕ is a continuous linear operator on E which leaves the subspace o E stable (note that o E is closed in E (for E 6= F χ )). By the definition of Cc∞ (G(A)), there is a compact open subgroup K(α) of G(Af ) such that α is invariant under left translations by K(α). From the definition of α ∗ ϕ, it is immediate that it is K(α)-invariant on the left. Let AR∗ : E → E(E ∈ E, E 6= F χ ) be the averaging over K(α) : (A∗ ϕ)(g) = K(α) ϕ(kg)dk (dk Haar measure on K(α), g ∈ G(A)). Then A∗ is a continuous projection of E on E K(α) and of o E on o E K(α) . We note that for all ϕ ∈ E, α ∗ A∗ (ϕ) = A∗ (α ∗ ϕ)α ∗ ϕ. In particular α ∗ ϕ = A(α ∗ Aϕ) on E as well as o E. We now state the central result of this chapter. 3.16. Theorem operator.

The operator ϕ 7→ α ∗ ϕ of o Lχ2 into itself is a compact

3.17. We will reformulate the theorem for a space of functions on G∞ . In the light of the remarks made at the end of 3.15, it is clear that it suffices to χK(α) show that convolution with α is a compact operator on o L2 . Consider ` now the isomorphism r = θ∈H rθ (defined in 2.2: we take for Lf of 2.2 the ` χK(α) group K(α)) of L2 K(α) on θ∈H Lχ2 (Gα /Γθ ).

26

M.S. Raghunathan

3.18. Claim For θ ∈ H, let o Lχ2 (G∞ R /Γθ ) be the subspace of all those χ functions ϕ in L2 (G∞ , Γθ ) for which U′ ∩Γ ϕ(gu)du = 0 for almost all ∞ g ∈ G∞ and for all U′ , the unipotent radical of proper k parabolic subgroup ` P′ of G. Then r maps o Lχ2 isomorphically onto θ∈H o Lχ2 (G∞ /Γθ ). 3.19. Proof It is known that the group df (U′ (k)) is dense in U′ (Af ) (this is known as the strong approximation property). Let Mθ = U′ (Af ) ∩ θ −1 K(α)θ; then Mθ is an open compact subgroup of U′ (Af ). Consequently U′ (Af ) = Mθ · df (U′ (k)). It follows that U′ (A) = U′∞ · Mθ · U′ (k) and hence the natural map ≃ U′ (A)/U′ (k) (U′∞ · Mθ )/((U′∞ · Mθ ) ∩ U′ (k)) −→

(∗)

R is a homeomorphism. For ϕ ∈o Lχ2 , we have U(A)/U′ (k) ϕ(θgu)du = 0 for θ ∈ H and g ∈ G∞ . In view of the identification (*) described above, we see that Z ϕ(θgu)du = 0. U′∞ ·Mθ /U′∞ ·Mθ ∩U′ (k)

Now if we set u = u∞ · uf with u∞ ∈ U′∞ and uf ∈ U′ (Af ), we have ϕ(θgu) = ϕ(θgu∞ uf ) = ϕ(θuf gu∞ ) = ϕ(θuf θ −1 · θgu∞ ) = ϕ(θgu∞ ) (since by definition of Mθ , uf ∈ Mθ implies that θuf θ −1 ∈ K(α) and ϕ is left K(α)-invariant). Thus integrating over Mθ first we find that Z Z ϕ(θgu)du ϕ(θgu)du = 0= (U′∞ ·Mθ )/((U′∞ ·Mθ )∩U′ (k))

proving that r maps o Lχ2 into

`o

χ θ∈H L2 (G∞

U′∞ /U′∞ ∩Γθ

| Γθ ). We need to show that χK(α)

the map is onto this subspace. To see this let ϕ ∈ L2 be such that rθ (ϕ) is in o Lχ2 (G∞ /Γθ ) for all θ-we know that such a ϕ exists and is unique. We need to show that ϕ ∈o Lχ2 i.e. we have to show that Z ϕ(gu)du = 0 (∗∗) U′ (A)/U′ (k)

for almost all g ∈ G(A) for the unipotent radical U′ of any proper k-parabolic subgroup P′ of G. Now g = g∞ · gf where g∞ ∈ G∞ and gf ∈ G(Af ). By Lemma 2.4, gf = kf ·θ·df (γ) with kf ∈ K(α), θ ∈ H and γ ∈ G(k). Since ϕ is left K(α)-invariant, ϕ(gu) = ϕ(g∞ gf u) = ϕ(g∞ kf θdf (γ)u) = ϕ(g∞ θdf (γ)u) −1 γ γ u) = ϕ(g ′ θγu). (since ϕ is left K(α)-invariant) = ϕ(g∞ θγ∞ ∞ f ∞

Automorphic Forms in GL(n)

27

′′

Set U = γuγ −1 . Then one has R

′ ′ (A)/U′ (k) ϕ(g∞ θγu)du UR ′ u′ )du′ = U′′ (A)/U′′ (k) ϕ(θg∞

R ′ θ · γuγ −1 )du = U′ (A)/U′ (k) ϕ(g∞ R ′ u′ )du′ = (U′′ ·M ′ )/(U′′ ·M ′ ∩U′′ (k)) ϕ(θg∞ ∞

θ

θ

′′

where Mθ′ = (U (Af ) ∩ θ −1 K(α)θ); and the integral in the last integral ′′ ′ )(U′′ · M ′ ∩ U ′′ (k)). It follows on (U∞ · M∞ is an Mθ′ -invariant function ∞ ∞ R ′ u)du and is thus zero since r ϕ ∈ that integral equals U′′ /U′′ ∩Γθ ϕ(θg∞ θ ∞ ∞ o Lχ (G /Γ ). Thus Claim 3.15 is established. ∞ θ 2 3.20. Lemma ForR α ∈ Cc (G(A)), let α∞ ∈ Cc (G∞ ) be the function defined by α∞ (g) = G(Af ) α(gh)dh. Then rθ (α ∗ ϕ) = α∞ ∗ ϕ for all ϕ ∈ χK(α)

L2 . This is immediate from the definition of rθ and the K(α)-invariance of α. The lemma shows that Theorem 3.13 is equivalent to the following:

3.21. Theorem Let Lf be a compact open subgroup of G(A) and set Γ = d∞ (ΓLf ). Let χ be a unitary character on C(A)(C = centre of G) which is trivial on C(k). Let α ∈ Cc (G∞ ). Then ϕ 7→ α ∗ ϕ is a compact operator on o Lχ2 (G∞ /Γ). 3.22. We begin with the observation that there is no loss of generality in assuming that the centre of G is finite. To see this observe first that if G′ is the commutator subgroup of the identity connected component of G, then C∞ · G′∞ = H has finite index in G∞ . Let ξ1 . . . ξr be a complete set of inequivalent representatives for G∞ /H in G∞ . For 1 ≤ i ≤ r and ϕ ∈ Lχ2 , let ϕi : C∞ · G∞ → C be the function defined by ϕi (g) = ϕ(ξi · g). Then ϕi (gγ) = ϕi (g) for g ∈ C∞ ·G∞ ′ = H and γ ∈ H∩Γ. Also ϕi (gc) = χ(c)ϕi (g) for c ∈ C∞ . Now every unipotent element of G∞ is contained in G∞ ′ . From this we see immediately that if ϕ is a cuspidal function so is ϕi restricted to G∞ ′ . Further ϕi is completely determined by its restriction ϕ′i to G∞ ′ . Evidently the map o χ L2 (G∞ /Γ)

−→

Y

r copies

o

′

Lχ2 (G∞ ′ /Γ′ )

28

M.S. Raghunathan

where χ′ = χ |(G′ ∩C)∞ and Γ′ = G∞ ′ ∩ Γ is an isomorphism onto a closed subset. Moreover for 1 ≤ i ≤ r R R (α ∗ ϕ)i (g) = G∞ α(h)ϕ(h−1 ξi g)dh = G∞ α(ξi−1 h)ϕ(h−1 g)dh X R X R −1 −1 ξ −1 g)dh αij (h)ϕj (h−1 = ξ h)ϕ(h ( α(ξ j g )dh j i H H 1≤j≤r

=

X1≤j≤r αij ∗ ϕj

1≤j≤r

where αij ∈ Cc∞ (H) is the function αij (h) = α(ξi−1 ξh h). The natural map π : C∞ × G∞ → H has for kernel the finite group C∞ ∩ G∞ whose order we denote by q. We regard functions on H as functions on C∞ × G∞ by composing with π. With this convention we have Rfor α ∈ Cc∞ (H) and ϕ ∈ R Lχ2 (H/H ∩ Γ), (α ∗ ϕ)(g) = H α(h)ϕ(h−1 g)dh = 1q C∞ ×G∞ α(h)ϕ(h−1 g)dh and setting h = c.x with c ∈ C∞ and x ∈ G∞ we have for g ∈ G∞ (α ∗ ϕ)′ (g) = =

1 q 1 q

R R α(xc)ϕ(c−1 x−1 g)dc ′ dx C G R ∞ R ∞ −1 ′ ′ G′ dx C∞ α(xc)χ(c g)dc = α ∗ ϕ (g) ∞

R where α′ in Cc∞ (G∞ ) is defined by α′ (x) = 1q C∞ α(xc)χ(c)dc. This discussion shows that we need to prove the theorem only in the case when G∞ = G∞ ′ has a finite centre. Now the group Γ admits a torsion free subgroup Γ′ of finite index and Γ′ evidently intersects the centre trivially. We have a natural inclusion Rof o Lχ2 (G∞ ′ (Γ) in o L2 (G∞ ′ /Γ′ ) (where o L2 (G′∞ /Γ′ ) = {ϕ ∈ L2 (G∞ ′ /Γ′ ) | U/U∩Γ′ ϕ(gu)du = 0 for all unipotent radicals of proper k-parabolic subgroups of G∞ ′ ) compatible with convolution by elements of Cc∞ (G∞ ′ ) on both the spaces. We have thus to prove the following 3.23. Theorem Let G be a connected semisimple algebraic k-group and Γ an arithmetic subgroup of G∞ . Let o L2 (G∞ /Γ) be the space of all Γunvariant functions ϕ on G∞ which are square summable on G∞ /Γ and in addition are cuspidal i.e. satisfy the following condition: If U is the R unipotent radical of a proper k-parabolic subgroup P of G, U∞∩Γ ϕ(gu)du = 0 for almost all g ∈ G∞ . Then the operator ϕ 7→ α ∗ ϕ on o L2 (G/Γ) is a compact operator for all α ∈ Cc∞ (G∞ ). Since G∞ is semisimple, G∞ /Γ has finite Haar measure. We now see from Proposition 3.2 that Theorem 3.20 is a consequence of the following:

Automorphic Forms in GL(n)

29

3.24. Proposition Let G∞ , Γ, α be as above. Then there is a constant C = C(α) > 0 such that for ϕ ∈ L2 (G∞ /Γ), | α ∗ ϕ(x) |≤ C k ϕ k2 for all x ∈ G∞ (note that α ∗ ϕ is C ∞ for ϕ ∈ L2 (G∞ /Γ)). 3.25. For c > 0 and a compact subset Ω ⊂o P∞ (notation introduced in §1.5 - 1.9) we define S(c, Ω) to be the subset K∞ ·Ac ·Ω of G∞ . According to Theorem 1.9, there is a finite set Ξ ⊂ G(k), a constant c0 > and a compact set Ω0 ⊂ o P∞ with the following properties: (i) (ξΩ0 ξ −1 )(ξ o P∞ ξ −1 ∩ Γ) = ξ o P∞ ξ −1 . (ii) If c > c0 and Ω ⊃ Ω0 is a compact subset of o P , then S(c, Ω)ΞΓ = G∞ . (iii) If c, Ω are as above the set {γ ∈ Γ | S(c, Ω)Ξγ ∩ S(c, Ω)Ξ 6= φ is finite. Condition (iii) implies in particular the following. There is a constant M = M (c, Ω) > 0 such that for ξ ∈ Ξ and all ϕ ∈ L2 (G∞ /Γ), we have Z | ϕ(g) |2 dg ≤ M k ϕ k22 (∗) S(c,Ω)ξ

3.26. We have seen that o P is a semi-direct product B · U and we have correspondingly a semi-direct product decomposition o P∞ = B∞ · U∞ . The group B∞ is reductive and K∞ ∩ B∞ is a maximal compact subgroup of B∞ . We then have an Iwasawa decomposition of B∞ as (K∞ ∩ B∞ ) · F where F is a connected solvable closed Lie subgroup of B∞ . Moreover as B∞ commutes with A, we see that for ξ ∈ Ξ the map K∞ × F × A × U∞ → G∞

(∗)

given by (k, f, a, u) 7→ k.f.a.u.ξ is an analytic diffeomorphism. Suppose now that P′ is a proper maximal k-parabolic subgroup of G containing P. Then P′ is the semidirect product Z(S′ ) · U′ where Z(S′ ) is the centraliser of a suitable 1-dimensional sub-torus of S and U′ is the unipotent radical of P′ . We have correspondingly a semidirect product decomposition Z(S′ )∞ ·U′∞ of P′∞ . Now U′∞ is a normal subgroup of U∞ and (hence) U∞ is the semidirect ′′ product (Z(S′ )∞ ∩ U∞ ) · U′∞ . Let Z(S′ )∞ ∩ U∞ = U∞ . Then we have a further refinement of the product decomposition (*) above: ′′

Φξ : K∞ × F × a × U∞ × U′∞ → G

30

M.S. Raghunathan ′′

′′

given by (k, f, a, u , u′ ) 7→ k, f, a, u , u′ , ξ. It is convenient to denote by ′′ ′′ H1 , H2 , H3 , H4 and H5 the groups K∞ , F, A, U∞ and U∞ respectively. We assume as we may say that the compact set Ω0 (resp. Ω) in 3.22 is chosen to be of the form (K∞ ∩ o P∞ ).Ω02 .Ω04 .Ω05 (resp. (K∞ ∩ o P∞ ).Ω02 .Ω04 .Ω05 with Ω0i (resp. Ωi ) a compact subset of Hi , 1 = 2, 4, 5 with Ω0i contained in the interior of Ωi . We also assume Ω05 so chosen that ξ −1 Ω05 .ξ.(ξ −1 H5 ξ ∩ Γ) = ξ −1 H5 ξ. The Lie algebra L(G∞ ) of G∞ is identified with the Lie algebra of right translation invariant vector fields on G∞ . The Lie algebras L(Hi ) of the Hi , 1 ≤ i ≤ 5 are identified with subalgebras of L(G∞ ) and hence their elements will also be regarded as right translation invariant vector fields on G∞ . On the other hand an element X of L(Hi ) determines a right translation invariant vector field on the group Hi . We denote this vector field by X ′ (to distinguish it from the vector field X on G∞ ). For e on G∞ as follows: X e is the X ∈ L(Hi ), 1 ≤ i ≤ 5 we define a vector field X image under the analytic diffeomorphism Φξ of the vector field on the product Q 1≤i≤5 Hi whose component in Hi is X while all the other components are zero. If g ∈ G∞ is such that g = h1 .h2 .h3 .h4 .h5 .ξ with hi ∈ H then we have - as is easily checked - for Xi ∈ L(Hj ), Y e (∗) X(g) = (Ad( hi )(X))(g) i<j

Q

Here when j = 1, i<j hi is the identity element while for j > 1, the product is taken in the order of increasing i : h1 h2 · · · hj−1 . 3.27. We now fix a basis B = {Xi }1≤i≤p ∪ {Yi }1≤i≤q ∪ {Zi }1≤i≤r of L(G∞ ) with the following properties: (i) Xi and Yj are eigenvectors for Ad(S), the corresponding character on S being denoted ηi and ζj respectively: Adt(Xi ) = ηi (t)Xi and Adt(Yi ) = ζi (t)Yi . (ii) {Zi | 1 ≤ i ≤ r} ∩ L(Hi ) is a basis for L(Hi ) for 1 ≤ i ≤ 3. (iii) {Xi | 1 ≤ i ≤ p} (resp. {Yi | 1 ≤ i ≤ q}) is a basis for L(H5 ) (resp. L(H4 )). From our choice of ordering on the character group of S, ηi , 1 ≤ i ≤ p and Q Q ζi , 1 ≤ i ≤ q, are positive roots so that ηi = α∈∆ αmiα , ζi = α∈∆ αni iα with miα , niα ≥ 0. For a multi-index β = (β1 , · · · , βp ) (resp. γ = (γ1 , · · · , γ1 ), resp. δ = (δ1 , · · · δr ), (β1 , γj , δk non-negative integers) we set

Automorphic Forms in GL(n)

Xβ = Yγ = Zδ =

X1β1 X2β2 Y1γ1 Y2γ2 Z δ1 Z2δ2

··· ··· ···

With this notation we have

β e 1β Xp p , X γq e 1γ Yq , Y Zrδr , Ze1δ

= = =

31

e 1β1 X e 1β2 X 2 1γ Ye1 1 Ye11γ2 Ze1δ1 · Ze1δ2 1

2

··· ··· ···

1β

Xp p 1γ Yeq q , er1δr . Z

3.28. Lemma Let g = h1 , h2 , h3 , h4 , h5 , ξ ∈ G∞ with hi ∈ Hi . Then for multi-indices β, γ, δ as above we have, setting m =| β + | γ | + | δ |, X e′ δ Ye ′ γ X e ′ β = η β (h3 )ζ γ (h3 ) Z Cβ ′ ,γ ′ ,δ′ (h1 , h2 , h3 , h4 , h−1 3 ) |β ′ |+|γ ′ |+|δ′ |≤m

with Cβ ′ γ ′ δ′ , C ∞ -functions on H1 × H2 × H4 (note that H3 normalises H4 ). Q Q Here η β = 1≤i≤p ηiβi and ζ γ = 1≤i≤q = ζiγi . Proof. One argues by induction on | β | + | γ | + | δ |. When | β | + | γ | + | δ |= 1, this follows from (*) of 3.23 and the fact that the Xi and Yj are eigenvectors with eigencharacter ηi and ζj respectively. Suppose that | β | + | γ | + | δ |> 1; then we can find β ′ , γ ′ , δ′ and with | β ′ | + e′ δ Ye ′ β X e ′ β = Te′ Ze′ δ Ye ′ γ X e ′ β ′ with | γ ′ | + | δ′ |=| β | + | γ | + | δ | −1 and Z ′δ e Ye ′ γ ′ X e ′β′ T ∈ B. Using the induction hypothesis we have an expansion for Z e′ δ Ye ′ δ X e ′ β now in terms of the X, Y, Z of the desired kind. The result for Z follows from the following observations: Te′ has the desired kind of expansion; secondly for a character χ on H3 treated as a function on G∞ , (Te′ , χ)(h3 ) = 0 ·

·

unless T ∈ LH3 and if T ∈ LH3 , (Te′ χ)(h3 ) = χ(T ) χ (h3 ) where χ is the tangent map L(H3 ) → R induced by χ. This proves the lemma.

3.29. We fix an element g0 = h01 , h02 , h03 , h05 , ξ in G∞ with hi ∈ Hi , 1 ≤ i ≤ 5. We take P′ to be the parabolic subgroup defined as follows: let α0 ∈ ∆ be such that α0 (h0 ) ≤ α(h′3 ) for all α ∈ ∆; then P′ is the proper maximal k-parabolic subgroup determined by α0 . The Lie algebra L(H5 ) is the sum of the eigenspace in L(G∞ ) for AdH3 corresponding to the eigencharacters Q (i.e. k-roots) ψ of the form α∈∆ αmα with mα integers ≥ 0 and mα0 > 0- in Q other words in the notation of Lemma 3.25, ηi = α∈∆ αmi α with miα0 > 0. Let (c′0 , Ω0 ) and (c, Ω) be as in 3.22. Then there is neighbourhood V of the identity in A such that for g0 ξ −1 = h01 , h02 , h03 , h04 , h05 ∈ S(c0 , Ω0 ) and x ∈ V, h01 , h02 , h03 x, h04 , h05 ∈ S(c, Ω). Fix such a neighbourhood V of 1 in

32

M.S. Raghunathan

A once and for all (for a fixed (c, Ω)). Let ∆ denote the elliptic operator P P ′2 2 e′ e ′2 1≤i≤p Xi on H5 and let ∆ = 1≤i≤p Xi . Then we have the L -Sobolev inequality (Theorem 3.4) if 2p > r/2, Z 0 2 e ′ p ϕ(h0 .h0 .h3 .h0 .h0 .ξ) |2 dh5 . . . |∆ (I) | ϕ(g ) | ≤ C 1 2 4 5 Ω5

where ϕ ∈ Cc∞ (G∞ ). Treating the right-hand side as a function on H1 ×H2 × H3 × H4 and applying the L1 Sobolev inequality combined with Schwarz’s inequality we have Z X e ′ δ Ye ′ γ ∆ e ′ p ϕ |2 |X | ϕ(g0 ) |2 ≤ C ′ K∞ ×Ω2 ×h03 V ×Ω4 ×Ω5 |δ|+|γ|≤q+4

dh1 , dh2 , dh3 , dh4 , dh5 e ′p = Now ∆

X

1≤i1 ,i2 ,···ip ≤p

e′ X e′ e′ ai1 · · · io X i1 i2 · · · Xip . It is now immediate from

Lemma 3.25 that we have Z ′′ 0 2 | ϕ(g ) | ≤ C E

(II)

X

| ηi1 ηi2 · · · ηip (h3 ) |2 | Z δ Y γ X β ϕ |2

1≤i1 ,i2 ,···ip ≤p |β|+|γ|+|δ|≤2p+q+r

dh1 , dh2 , dh3 , dh4 , dh5

(III)

where E = K∞ ×Ω2 ×h03 V ×Ω4 ×Ω5 . Now for h′3 , h3 V one has (ηi1 ηi2 · · · ηip )2 (h′3 ) ≤ λp · αp0 (h′3 ) for a suitable constant λp > 0. This is because we have Q for any i, with 1 ≤ i ≤ r, ηi = α∈∆ αmiα with miα ≥ 0, miα0 ≥ 1 integers; also α(h′3 ) ≤ c for α ∈ ∆ so that (ηi1 ηi2 · · · ηip · · · ηip )(h′3 ) ≤ cN α0 (h′3 )p P P where N = ( 1≤r≤p α∈∆ mir α − p). Next observe that if p is sufficiently Q Q large we have α(h3 )p < C1 δ2 (h3 ) where δ2 = 1≤i≤p ηi 1≤j≤q ζj . This is seen as follows: since α0 (h03 ) ≤ α(h03 ) for all α ∈ ∆, we see that there is a constant b > 0 such that α0 (h3 ) ≤ bα(h3 ) for all α ∈ ∆ and h3 ∈ Q h03 V . Now δ2 (h3 ) = α∈∆ α(h3 )να with να integers ≥ 1 so that δ2 (h3 ) ≥ P Q −1 να ≥ C −1 α (h )ν where ν = 0 3 1 α∈∆ να . Now if p ≥ ν, we α∈∆ (b α0 (h3 )) have α0 (h3 )ν ≤ α0 (h3 )p · cν−p ≤ cν−p C1 δ2 (h3 ). We see from (III) above now that we have Z X ′′ 2 | (Z δ Y γ X β )ϕ |2 dg (IV) | ϕ(g0 ) | ≤ C C1 S(c,Ω)·ξ |β|+|γ|+|δ|≤2p+q+r

Automorphic Forms in GL(n)

33

Note that E ⊂ S(c, Ω) and dh1 .dh2 .dh3 .dh4 .dh5 .δ2 (h3 ) = dg. Proposition 3.21 now follows for the inequality IV above: We have for α ∈ A∞ c (G∞ ) and D in the enveloping algebra of L(G∞ ), D(α ∗ ϕ) = Dα ∗ ϕ so that by (IV) R P ′′ (α ∗ ϕ)(g0 ) |2 ≤ C C1 S(c,Ω)ξ |β|+|γ|+|δ|≤2p+q+r | (Z δ Y γ X β α) ∗ ϕ |2 dg ≤ C(α) k ϕ k22 P ′′ where C(α) = C C1 M (α) with M (α) = |β|+|γ|+|δ|≤2p+q+r k Z δ Y γ X β α k01 . Note that we have for α ∈ Cc∞ (G∞ ) and ϕ ∈ L2 (G∞ /Γ), k α ∗ ϕ k2 ≤k α k2 . This proves the proposition and hence the Theorem.

4

Proof of Theorem 2.13 We begin by proving the following:

4.1. Theorem Let o Aχ (G∞ , Γ, σ∞ , λ) = {f ∈ Aχ (G∞ , Γ, σ∞ , λ) | f cuspidal }. Then o Aχ (G∞ , Γ, σ∞ , λ) is finite dimensional. 4.2. We will first show that any ϕ ∈o Aχ (G∞ , Γ, σ∞ , λ) decreases to zero rapidly at ∞ so that in particular ϕ ∈ L2 (G∞ /Γ) (and hence in o Lχ (G /Γ)). By Lemma 3.25 combined with (I) of 3.26, we have for p′ ≥ p ∞ 2 and g0 = h01 .h02 .h03 .h04 .h05 .ξ, 2

o

| ϕ(g ) | ≤

Z

X

| (ηi1 ηi1 ηi2 · · · ηip (h03 ) |2

Ω5 1≤ .i ···i ≤p′ i 1 p′

X

| cβγδ Z δ Y γ X β ϕ(g0 ξ −1 )h5 ξ) |2 dh5

|β|+|α|+|δ|≤2p′

Let I be the set of p′ -tuples i1 , . . . , ip′ with 1 ≤ i1 ≤ i2 · · · ≤ ip′ ≤ p′ and for I ∈ I, let ηI ηi1 ηi2 · · · ηip . Since ϕ and its derivatives have moderate growth there is constant B > 0 and an integer r > 0 such that | (Z δ Y γ X β ϕ(g0 ) |2 ≤ B k g0 kr for | α | + | β | + | γ |≤ 2p′ . Thus, we have any p′ ≥ p | ϕ(g0 ) |2 ≤ C ≤ C

′′

′′′

R

Ω5

X (ηI (h03 ))2

I∈I ′ α0 (h03 )p

k

g0

kr

X

|δ|+|γ|+|β|≤2p′

| (Z δ Y γ X β ϕ)(g0 ξ −1 h5 ξ) |2 dh5

34

M.S. Raghunathan

Now k g0 kr ≤ b′ Supα∈∆ α(h03 )−s for some integer s for g0 ∈ S(c0 , Ω0 ). It follows that we have ′′

′

′′

′

| ϕ(g0 ) |2 ≤ b α0 (h03 )p · α0 (h03 )−s = b α(h03 )p −s . Since p′ is at our choice we see that | ϕ(g0 ) |2 ≤ is bounded in S(c0 , Ω0 ). Hence ϕ ∈o L2 (G∞ /Γ). 4.3. Lemma (Godement) Let X be a locally compact space and µ a probability measure. Let V ⊂ L2 (X, µ) be a closed subspace. Suppose that every ϕ ∈ V is essentially bounded. Then dim H < ∞. Proof V is a closed subspace of L∞ as well since L∞ -convergence implies L2 convergence. It follows also that the identity map of V is a continuous homomorphism of V with L∞ topology on V with the L2 topology. By the open mapping theorem there is a constant c > 0 such that k ϕ k∞ < c k ϕ k2 for all ϕ ∈ V . Suppose ϕ1 , · · · , ϕn is an orthonormal set in V , and a = (a1 , · · · , an ) ∈ Cn ; then we have for almost all x ∈ X, |

n n X X X 1 k2 = C( | ai ϕi |2 ) 2 ai ϕi (x) |≤ C k i=1

i=1

It follows that if D is a dense countable subset of Cn , there is a set Y of measure zero in X such that we have for all x ∈ X\Y and all a ∈ D, n n X X 1 | ai |2 ) 2 ai ϕi (x) |≤ c( | i=1

i=1

Since D is dense in Cn , the inequality holds for all a ∈ Cn . Taking ai = fi (x), we conclude that n X | fi (x) |2 ≤ c2 i=1

Integrating over X, we have n ≤ c2 . Thus dim V < ∞. 4.4. Proof of 4.1. We have seen that we may assume that G has no central split torus so that G∞ /Γ has finite measure and further that χi trivial. We claim that o A(G∞ , Γ, σ∞ , λ) is a closed subspace of o L2 (G∞ /Γ). In fact if ϕn ∈ o A(G∞ , Γ, σ∞ , λ) converges to ϕ in o L2 , then ϕn → ϕ in

Automorphic Forms in GL(n)

35

the sense of distributions. All the ϕn are in the kernel of a fixed elliptic operator with C ∞ coefficients on G∞ /Γ hence the distribution ϕ is also in this kernel. But any distribution in the kernel of an elliptic operator with C ∞ coefficients is a C ∞ function. Thus ϕ is a C ∞ function in o L2 . Since it is the limit of the ϕn as a distribution, the K∞ -span of ϕ is a quotient of σ∞ and the Z-span a quotient of λ. Further as ϕ is k-finite and Z finite there is an α ∈ Zc∞ (G) such that α ∗ ϕ = ϕ and one has thus k ϕ k∞ ≤k α ∗ ϕ k∞ ≤ C k α k2G∞ k ϕ kG∞ /Γ . It follows that ϕ has moderate growth so that ϕ ∈ o A((G∞ , Γ, σ∞ , λ). Lemma 4.3 now clearly implies Theorem 4.1. 4.5. Clearly Theorem 4.1 is equivalent to the assertion that o (A)χ (G(A), G(k), σ, λ) is finite dimensional. To prove Theorem 2, we will argue by induction on the k-rank of [G, G]. Let {Pi }icI be the collection of all the maximal parabolic subgroups of G containing P. Let Ui be the cusp radical of Pi and Gi ⊃ S. Let supplement to Ui , Pi . Let ri : C ∞ (G(A)/G(k)) → C ∞ (Gi (A)/Gi (k)) be the map defined as follows: Z ϕ(xu)du. γi (ϕi )(x) = Ui (A)/Ui (k)

We assert that for a suitable χi and λi and σi∞ , λi (ϕ) ∈ Aχ (Gi∞ (A), Gi∞ (k), λi , σi∞ ) if ϕ ∈ Aχ (G(A), G(k), λ, σ∞ ). Assume that this is true. Observe that k-rank ([Gi , Gi ]) = k rank G − 1. Thus by induction hypothesis Aχi (Gi (A), Gi (k), σ, λ∞ ) is finite dimensional for all i ∈ I. On the other hand if ϕ is in ∩i∈I ker λi , ϕ is a cuspidal automorphic form in o Aχ (G(A), G(k), σ, λ ); and this last space is finite dimensional by Theo∞ rem 4.1. This completes the proof of Theorem 2 assuming the following: 4.6. Lemma Given χ, σ, λ there exists for every i ∈ I, χi , σi , λi , χi = χ (note that C ⊂ Gi (σi )), representation of K∞ ∩ Gi and λi : Zi → EndWi′ , (Zi the centre of the enveloping algebra Ui of L(Gi )) such that ri (Aχ (G(Af ), G(k), σ, λ)) ⊂ Aχi (Gi (Af ), Gi (k), σi , λi ) Proof Let U− be the unipotent algebraic k-subgroup of G normalised by XS and whose Lie algebra is spanned by {L(G)(β) | β a root, β = − mα α, mαi < 0}. Let P− i be the normaliser of Ui . One then has α∈∆

36

M.S. Raghunathan

− Pi ∩ P− i = Gi and L(G) = L(Ui ) ⊗ L(Gi ) ⊗ L(Ui ). Correspondingly the enveloping algebra U of L(G) can be written as

U(L(U− i )) · Ui · U(L(Ui )) − − where Ui is the enveloping algebra of L(Gi ). Now U(L(U− i )) = L(Ui )(U(L(Ui ))⊕ C and similarly U(L(Ui )) = U(L(Ui ))L(Ui ) ⊕ C. Consequently we have

U

− = L(U− i )U(L(Ui )) · Ui · U(L(Ui ))L(Ui ) − ⊕ L(Ui )U(L(U− i )) · Ui ⊕ Ui · U(L(Ui ))L(Ui ) ⊕ Ui

Let hi denote the projection of U on Ui following this direct sum decomposition. Then according to a theorem of Harish-Chandra hi restrict to U maps Z injectively into Zi and Zi is finitely generated as a Z-module. − Also the component of z ∈ CU in the direct factors L(U− i )(U(L(Ui )) · Ui and Ui U(L(Ui ))L(Ui ) are trivial. From this it is easy to see that ri (zϕ) = hi (z)ri (ϕ) for ϕ ∈ (Aχ (G(A), G(k), σ, λ). It follows that if we set Wi′ = W ⊗Z Zi , Wi′ is a finite dimensional vector space and we have a homomorphism λi : Zi → End Wi′ . Evidently Ui (ri (ϕ)) is a quotient of Wi′ as a Zi -module. Also, if Ki = K ∩ Gi (Af ), it is clear that the Ki (K-span of ϕ) and thus is a quotient of the Ki -module W (obtained by restriction of the K-module structure to K0 ). Lemma 4.5 is immediate from this.

Automorphic Forms in GL(n)

37

References [B] A. Borel, Introduction aux groupes arithm´etiques, Hermann (1969), Paris. [B-T] A. Borel and J. Tits, Groupes r´eductifs, Publ. Math. de ℓ’IHES, 27 (1965), 55–150. [G] R. Godement, Domaines fondamentaux de groupes arithm´etiques eminaire Bourbaki, (1962/63), Fas 3, No.257, Paris. [H-C] Harish-Chandra, Automorphic forms on semisimple Lie groups, LN 62, Springer-Verlag (1968). [L] R.P. Langlands, On the functional equations satisfied by Eisenstein series, LN 544, Springer-Verlag (1976). [N] R. Narashimhan, Analysis on real and complex manifolds, Advanced Studies in Pure Mathematics, Vol. 1, North Holland, Amsterdam, (1968).

Classical Modular Forms T.N. Venkataramana∗

School of Mathematics, Tata Institute of Fundamental Research, Colaba, Mumbai, India

Lectures given at the School on Automorphic Forms on GL(n) Trieste, 31 July - 18 August 2000

LNS0821002

∗

[email protected]

Contents 1 Introduction

43

2 A Fundamental Domain for SL(2, Z)

43

3 Modular Forms; Definition and Examples

49

4 Modular Forms and Representation Theory

54

5 Modular Forms and Hecke Operators

62

6 L-functions of Modular Forms

70

Classical Modular Forms

1

43

Introduction

The simplest kind of automorphic forms (apart from Gr¨ossencharacters, which will also be discussed in this conference) are the “elliptic modular forms”. We will study modular forms and their connection with automorphic forms on GL(2), in the sense of representation theory. Modular forms arise in many contexts in number theory, e.g. in questions involving representations of integers by quadratic forms, and in expressing elliptic curves over Q as quotients Jacobians of modular curves (this was the crucial step in the proof of Fermat’s Last Theorem by Andrew Wiles), etc. The simplest modular forms are those on the modular group SL(2, Z) and we will first define modular forms on SL(2, Z). To begin with, in section 2, we will describe a fundamental domain for the action of SL(2, Z) on the upper half plane h. The fundamental domain will be seen to parametrise isomorphism classes of elliptic curves. In section 3, we will define modular forms for SL(2, Z) and construct some modular forms, by using the functions E4 and E6 which we encounter already in the section on elliptic curves. In section 4, a representation theoretic interpretation of modular forms will be given, which will enable us to think of them as automorphic forms on GL(2, R). In section 5, we will give an adelic interpretation of modular forms. This will enable us to think of Hecke operators as convolution operators in the Hecke algebra; using this, we show the commutativity of the Hecke operators. We will also prove a special case of the Multiplicity One theorem.

2

A Fundamental Domain for SL(2, Z)

Notation 2.1 Denote by h the “Poincar´e upper half-plane” i.e. the space of complex numbers whose imaginary part is positive: h = {z ∈ C; z = x + iy, x, y ∈ R, y > 0}. If z ∈ C, denote respectively by Re(z) and Im(z) the real and imaginary parts of z. On the upper half plane h, the group GL(2, R)+ of real2 × 2 matrices a b with positive determinant operates as follows: let g = ∈ GL(2, R)+ , c d and let z ∈ h. Set g(z) = (az + b)/(cz + d). Notice that if cz + d = 0

44

T.N. Venkataramana

and c 6= 0 then, z = −d/c is real, which is impossible since z has positive imaginary part. Thus, the formula for g(z) makes sense. Observe that Im(g(z)) = Im(z)(det(g))/ | cz + d |2 .

(1)

The equation (1) shows that the map (g, z) 7→ g(z) takes GL(2, R)+ × h into h. One checks immediately that this map gives an action of GL(2, R)+ on the upper half plane h. Note also that | cz + d |2 = c2 y 2 + (cx + d)2 .

(2)

Therefore, | cz + d |2 ≥ y 2 or 1 according as | c |= 0 or nonzero. Therefore, Im(γ(z)) ≤ y/ min{1, y 2 } ∀γ ∈ Γ0 ⊂ SL(2, Z),

(3)

where min{1, y 2 } denotes the minimum of 1 and y 2 and Γ0 ⊂ SL(2, Z) is the group generated by the elements T = ( 10 11 ) and S = ( −10 01 ). Later we will see that Γ0 is actually SL(2, Z). The element T acts on the upper half plane h by translation by 1: 1 1 T (z) = (z) = z + 1 ∀z ∈ h. (4) 0 1 Similarly, the element S acts by inversion: 0 1 S(z) = (z) = −1/z ∀z ∈ h. −1 0

(5)

Consider the set F = {z ∈ h; −1/2 < Re(z) ≤ 1/2,

| z |≥ 1, and 0 ≤ Re(x)} if | z |= 1.

Theorem 2.2 Given z ∈ h there is a unique point z0 ∈ F and an element γ ∈ SL(2, Z) such that γ(z) = z0 . Moreover, given γ ∈ SL(2, Z), we have γ(F ) ∩ F = φ unless γ lies in a finite set ( of elements of SL(2, Z) which fix the point ω = 1/2 + i31/2 /2 ∈ h or i ∈ h). [one then says that F is a fundamental domain for the action of SL(2, Z) on the upper half plane h]. Proof We will first show that any point z on the upper half plane can be translated by an element of the subgroup Γ0 of SL(2, Z) (generated by 0 1 )) into a point in the “fundamental domain” F . T ( 10 11 ) and S = ( −1 0

Classical Modular Forms

45

Now, given a real number x, there exists an integer k such that −1/2 < x + k ≤ 1/2. Therefore, equation (4) shows that given z ∈ h there exists an integer k such that the real part x′ of T k (z) satisfies the inequalities −1/2 < x′ ≤ 1/2. Let y denote the imaginary part of z and denote by Sz the set Sz = {γ(z); γ ∈ Γ0 , Im(γ(z)) ≥ y , −1/2 < Re(γ(z)) ≤ 1/2} . We will first show that Sz is nonempty and finite. Let k be as in the previous paragraph. Then −1/2 < Re(T k (z)) ≤ 1/2 and Im(T k (z) = Im(z); therefore, T k (z) lies in Sz and Sz is nonempty. Now, equation (3) shows that the imaginary parts of elements of the set Sz are all bounded from above by y/ min 1, y 2 . By definition, the imaginary parts of points on Sz are bounded from below by y. The definition of Sz shows that Sz is a relatively compact subset of h. We get from (3) that | cz + d |2 ≤ 1; now (2) shows that | c |≤ 1/y 2 . Suppose γ ∈ γ0 = ( ac db ) is such that γ(z) ∈ Sz then, c is bounded by 1/y 2 and is in a finite set. The fact that cz + d is bounded now shows that d also lies in a finite set. Since Sz is relatively compact in h, it follows that γ(z) = (az + b)/(cz + d) is bounded for all γ(z) ∈ Sz ; therefore, az + b is bounded as well, and hence a and b run through a finite set. We have therefore proved that Sz is finite. Let y0 be the supremum of the imaginary parts of the elements of the finite set Sz ; let S1 = {z ′ ∈ Sz ; Im(z ′ ) = y0 } and let z0 ∈ S1 be an element whose real part is maximal among elements of S1 . We claim that z0 ∈ F . First observe that if z ′ ∈ Sz then S(z ′ ) = −1/z ′ has imaginary part y0 / | z |2 = Im(z ′ )/ | z ′ |2 ≤ y0 whence | z ′ |2 ≥ 1. If | z0 |> 1, then it is immediate from the definitions of F and Sz that z0 ∈ F . Suppose that | z0 |= 1. Then, S(z0 ) = −1/z0 also has absolute value 1, its imaginary part is y0 and its real part is the negative of Re(z0 ); hence S(z0 ) ∈ S1 . The maximality of the real part of z0 among elements of S1 now implies that Re(z0 ) ≥ 0. Therefore, z0 ∈ F . We have proved that every element z0 may be translated by an element of Γ0 into a point in the fundamental domain F . Suppose now that z ∈ γ −1 (F )∩F for some γ ∈ SL(2, Z). Write γ = ( ac db ) with a, b, c, d ∈ Z and ad − bc = 1. Suppose that Im(γ(z)) ≥ Im(z) = y (otherwise, replace z by γ(z)). Then, by (3) one gets (cx + d)2 + c2 y 2 ≤ 1.

(6)

Since z ∈ F , we have x2 + y 2 ≥ 1 and 0 ≤ x ≤ 1/2. Therefore y 2 ≥ 3/4 and (1 ≥)c2 y 2 ≥ c2 4/3.

(7)

46

T.N. Venkataramana

This shows that c2 ≤ 1 since c is an integer. Suppose c = 0. Then, ad = 1, a, d ∈ Z and we may assume (by multiplying by the matrix −Id [minus identity] if necessary) that d = 1. Hence γ = ( 10 1b ). Then, γ(z) = z + b ∈ D which means that 0 ≤ x + b ≤ 1/2 and 0 ≤ x ≤ 1/2. Thus, −1/2 ≤ b ≤ 1/2, i.e. b = 0 and γ is the identity matrix. The other possibility is c2 = 1, and by multiplying by the matrix −Id (minus identity) we may assume that c = 1. Suppose first that d = 0. Then, bc = −1 whence b = −1. Now, (7) shows that x2 + y 2 ≤ 1. Moreover, γ(z) = az + b/z = a + bz/ | z |2 = a − z whence its real part is a − x which lies between 0 and 1/2. Since 0 ≤ x ≤ 1, it follows that 0 ≤ a ≤ 1. If a = 0 then γ = ( 01 −10 ) and lies in the isotropy of the point i ∈ h. If a = 1 then, γ = ( 11 −10 ) which lies in the isotropy of the point ω = 1/2 + i31/2 /2. We now examine the remaining case of c = 1 and d 6= 0. From (6) we get (x + d)2 + y 2 ≤ 1. If d ≥ 1 then the inequality 0 ≤ x ≤ 1/2 shows that 1 ≤ d ≤ x + d which contradicts the inequality (x + d)2 + y 2 ≤ 1, which is impossible. Thus, d ≤ −1; then the inequality 0 ≤ x ≤ 1/2 implies that x + d ≤ 1/2 + (−1) = −1/2 whence (x + d)2 ≥ 1/4. Since y 2 ≥ 3/4 the inequality (x+d)2 +y 2 ≤ 1 implies that equalities hold everywhere: y 2 = 3/4, x = 1/2 and d = −1. Thus, z = ω and z −1 = z 2 . Since 1 = ad−bc = −a−b (d = −1 and c = 1), and γ(z) = (az+b)/(z−1) = (az+b)/z 2 = −(az+b)z = a+(−a−b)z = a+z ∈ D, the real part of γ(z) is a + x = a + 1/2 and is between 0 and 1/2, i.e. −1 −1/2 ≤ a ≤ 0 i.e. a = 0 and b = −1. Therefore, γ = ( 01 −1 ) lies in the isotropy of ω. This completes the proof of Theorem (2.2). Corollary 2.3 The group SL(2, Z) is generated by the matrices T = ( 10 11 ) 0 1 ). and S = ( −1 0 Proof In the proof of Theorem (2.2), a point on the upper half plane is brought into the fundamental domain F by applying only the transformations generated by S and T . The fact that the points on the fundamental domain are inequivalent under the action of SL(2, Z) now implies that SL(2, Z) is generated by S and T .

Classical Modular Forms

47

(The Corollary can also be proved directly by observing that ST −1 S −1 = ( 11 01 ). Now, the usual row-column reduction of matrices with integral entries implies that T and ST S −1 generate SL(2, Z)). Notation 2.4 Elliptic Functions. We recall briefly some facts on elliptic functions (for a reference to this subsection, see Ahlfors’ book on Complex Analysis). Given a point τ on the upper half plane h, the space Γτ = Z ⊕ Zτ of integral linear combinations of 1 and τ forms a discrete subgroup of C with compact quotient. The quotient Eτ = C/Γτ may be realised as the curve in P2 (C) whose intersection with the complement of the plane at infinity is given by y 2 = 4x3 − g2 x − g3

(8)

The curve Eτ = C/Γτ is called an “elliptic curve”. The map of C/Γτ to P2 is given by z 7→ (℘′ (z), ℘(z), 1) for z ∈ C. Recall the definition of ℘: if z ∈ C and does not lie in the lattice Γτ , then write ′

X ℘(z) = 1/z + (1/(z + w)2 − 1/w2 ), 2

P′ is the sum over all the non-zero points w in the lattice Γτ . The where derivative ℘′ (z) of ℘(z) is then given by X ℘′ (z) = 1/(z + w)3 , where the sum is over all the points of the lattice Γτ . One has the equation (cf. equation (8)) ℘′ (z)2 = 4℘(z)3 − g2 (τ )℘(z) − g3 (τ ).

(9)

a b ∈ SL(2, Z) and τ ∈ h, then the elliptic curve Eγ(τ ) is c d isomorphic as an algebraic group (which is also a projective variety) to the elliptic curve Eτ . The explicit isomorphism on C is given by z 7→ z/(cτ + d). It is also possible to show that if Eτ and Eτ ′ are isomorphic elliptic curves, then τ ′ is a translate of τ by an element of SL(2, Z). Thus the fundamental domain F which was constructed in Theorem (2.2) parametrises isomorphism classes of elliptic curves. If γ =

48

T.N. Venkataramana In equation (9), recall that the coefficients g2 and g3 are given by ′

g2 (τ ) = 60G4 (τ ) = 60 and

X

(mτ + n)−4

′

X g3 (τ ) = 140G6 (τ ) = 140 (mτ + n)−6

P′ is the sum over all the pairs of integers (m, n) such that not both where m and n are zero. The discriminant of the cubic equation in (9) is given by 1/(16)∆(τ ) where ∆(τ ) = g23 − 27g32 . (10) It is well known and easily proved that ℘′ (z) has a simple zero at all the 2-division points 1/2,τ /2 and (1+τ )/2 and that ℘(1/2),℘(τ /2) and ℘((1+ τ )/2) are all distinct. Thus equation (9) transforms to ℘′ (z)2 = 4(℘(z) − ℘(1/2))(℘(z) − ℘(τ /2))(℘(z) − ℘((1 + τ )/2))

(11)

Thus the discriminant of the (nonsingular) cubic in equation (9) is non-zero and so we obtain that ∆(τ ) 6= 0 (12) for all τ ∈ h. Notation 2.5 On the upper half plane h, there is a measure denoted y −2 dxdy, as z = x + iy varies in h. This measure is easily seen to be invariant under the action of elements of the group GL(2, R)+ of nonsingular matrices with positive determinant. Lemma 2.6 With respect to this measure, the fundamental domain F has finite volume. Proof We compute the volume of F . Note that if z = x + iy lies in F , then, −1/2 ≤ x ≤ 1/2 and 1/(1 − x2 )1/2 ≤ y < ∞. Thus the volume of F is the integral Z ∞ Z 1/2 Z 2 dy/y 2 ) dx( dxdy/y = F

−1/2

(1−x2 )1/2

which is easily seen to be π/3. In particular, F has finite volume.

Classical Modular Forms

49

Notation 2.7 Let S denote the inverse image of the fundamental domain F ⊂ h under the quotient map GL(2, R) → GL(2, R)/O(2)Z = h. Then, we have proved that GL(2, Z)S = GL(2, R). The set S is called a Siegel Fundamental Domain.

3

Modular Forms; Definition and Examples

Notation 3.1 Given z ∈ h (h is the upper half plane) and an element g = ( ac db ), write j(g, z) = cz + d. Note that if j(g, z) = 0, then by comparing the real parts and imaginary parts we get c = 0 and d = 0 which is impossible since ad − bc 6= 0. Thus, j(g, z) is never zero. Definition 3.2 A function f : h → C is weakly modular of weight w if the following two conditions hold. (1) f is holomorphic on the upper half plane. (2) for all γ ∈ SL(2, Z), with γ = ( ac db ), we have the equation f ((az + b)/(cz + d)) = (cz + d)w f (z).

(13)

Given g = ( ac db ) and a function f on the upper half plane h, define g−1 ∗ f (z) = (cz + d)−w f (g(z)) ∀z ∈ h. Then, it is easily checked that the map (g, f ) → g−1 ∗ f defines an action of GL(2, R) on the space of functions on h. Thus, the condition (2) above is that the function f there is invariant under this action by SL(2, Z). Now 0 1 ) and by Corollary (2.3), SL(2, Z) is generated by the matrices S = ( −1 0 −1 T = ( 10 11 ). Thus condition (2) is equivalent to saying that γ ∗ f = f for γ = S, T . This amounts to saying that f (−1/z) = z w f (z)

(14)

f (z + 1) = f (z).

(15)

and Note that the invariance of f under the action of −1 where 1 is the identity matrix in SL(2, Z) implies that f is zero of w is odd: f (z) = (−1)w f (z). Therefore, we assume from now on (while considering modular forms for the group SL(2, Z) ) that w = 2k where k is an integer.

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T.N. Venkataramana

Definition 3.3 The map exp : h → D ∗ given by z 7→ e2πiz = q is easily seen to be a covering map of the upper half plane h onto the set D ∗ of non-zero complex numbers of modulus less than one. The covering transformations are generated by T (z) = z + 1. A weakly modular function f is invariant under T and therefore yields a holomorphic map f ∗ : D ∗ → C given by f ∗ (q) = f (z) for all z ∈ h. We say that a weakly modular function of weight w is a modular function of weight w if f ∗ extends to a holomorphic function of D (the set of complex numbers of modulus less than one) i.e. f ∗ extends to 0 ∈ D. Let f be a weakly modular function on h. Then, f is a modular function if and only if the function f ∗ has the “Fourier expansion” (or the “qexpansion”) X f ∗ (q) = an q n , (16) n≥0

where an are complex numbers and the summation is over all non-negative integers n. Observe that a weakly modular function is modular if and only if it is bounded in the fundamental domain F . We will say that a modular form is a cusp form if the constant term of its q-expansion is zero: i.e. a0 = 0 in the notation of equation (15). Notation 3.4 Examples of modular forms. First we note that if f and g are modular forms of weights w and w′ then, the product function f g is a modular form of weight ww′ . We will first prove that for the modular group SL(2, Z), there are no non-constant “weight zero” modular forms. First note that if f is a weight zero modular form, then the function f ∗ extends to 0 and hence is bounded in a disc of radius r < 1. Its inverse image under exp : F → D ∗ is precisely the set A = {z = x + iy ∈ F ; y > − log r} and f is bounded on the set A. The complement of the set A in the fundamental domain F is compact, and f is bounded there as well, whence f is bounded on all of the fundamental domain F as well as at “infinity”. By the maximum principle, f is constant. We will now show that there are no modular forms of weight two on SL(2, Z). Suppose f is one and let F (z) be its integral from z0 to z for some fixed z0 ∈ h. The modularity of f shows that γ 7→ F (γ(z0 )) gives a homomorphism from SL(2, Z) to C. But, SL(2, Z) is generated by the finite order elements S and ST whence, this homomorphism is identically zero. This and the modularity of f shows that the integral F is invariant under SL(2, Z). It is easy to show that F ∗ is holomorphic at 0 (integrate both

Classical Modular Forms

51

sides of equation (15)), and use the invariance of F under T ). Hence F is a modular form of weight zero. By the foregoing paragraph, f is a constant, i.e. f = 0. Fix an even positive integer 2k, with k ≥ 2. We will construct a modular form of degree k as follows. Let τ ∈ h and write (compare the definition of g2 and g4 in section (2.4)) X ′ G2k (τ ) = (mτ + n)−2k , (17) P′ is the sum over all the pairs of integers (m, n) not both of which are where zero. Then, G2k is easily shown to be a weakly modular function of weight 2k on the upper-half plane. If τ is varying in the fundamental domain and its imaginary part tends to infinity, then it is clear from the formula for G2k P′ −2k n = 2ζ(2k) where that G2k (τ ) tends to X ζ(s) = n−s

is the Riemann zeta function (the sum is over all the positive integers n and in the sum, the real part of s exceeds 1). Consequently, G2k is a modular form of weight 2k. We will now outline a derivation of the q-expansion of G2k . Start with the partial fraction expansion X πcot(πz) = z −1 + (z + n)−1 + (z − n)−1 (18)

where the sum is over all positive integers n. This series converges uniformly on compact subsets of the complement of Z in C. Write q = e2πiz (where i ∈ h and i2 = −1). Then one has the q-expansion X πcot(πz) = πi(q + 1)/(q − 1) = −πi − 2πi qn (19) n≥1

Differentiate 2k- times, the right-hand sides of equations (17) and (18) with respect to z. We then get the equality X X n2k−1 q n (20) (z + n)−2k = ((2k − 1)!)−1 (2πi)2k n∈Z

n≥1

Fix m and in equation (19) take for z the complex number mτ . Then sum over all m. We obtain by equations (16) and (18), the q-expansion X σ2k−1 q n (21) G2k (τ ) = 2ζ(2k) + ((2k − 1)!)−1 (2πi)2k n≥1

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T.N. Venkataramana

where for an integer r and n ≥ 1, σr (n) is defined to be the sum d runs over the positive divisors of n. By using the power series expansion X (1 + x)−2 = nxn−1

P

dr where

n≥1

and equation (17) one has the power series identity XX X πcot(πz) = z −1 + 2 n2j z 2j−1 = z −1 + 2 ζ(2j)z 2j−1 n≥1 j≥1

(22)

j≥1

P By comparing the power series expansions cos(x) = m≥0 ((2m)!)−1 x2m P and sin(x) = m≥0 ((2m + 1)!)−1 x2m+1 with the right-hand side of equation (21) one obtains ζ(2) = π 2 /6, ζ(4) = π 4 /90 andζ(6) = π 6 /(33 .5.7).

(23)

Using (20) and (22) we get g2 = 60G4 = (4/3)π 4 + 160π 4 (q + · · · )

(24)

where the expression q + · · · is a power series in q with integral coefficients with the coefficient of q being 1. Similarly, we get (again from (20) and (22)) g3 = 140G6 = (8/27)π 6 − 25 .7π 6 /3(q + · · · ) Therefore, we get, after some calculation, that for all z ∈ h, X ∆(z) = g2 (z)3 − 27g3 (z)2 = 211 π 12 (q + τ (n)q n )

(25)

(26)

n≥2

where the τ (n) are integers. We recall that ∆(z) is never zero on the upperhalf plane (section (2.4)). The equation (26) shows that the coefficient of q in q-expansion of ∆ is non-zero, (and that its constant term is zero). Lemma 3.5 There are no modular forms of negative weight. Proof Suppose that f is a modular form of weight −l with l > 0. Form the product g = f 12 ∆l . Since f and ∆ are modular forms, so is the product. Since its weight is zero, g is a constant (see the beginning of this subsection). But, (26) shows that the q-expansion of g has no constant term. Hence g = 0 whence, f = 0.

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Lemma 3.6 Suppose that f is a cusp form. Then ∆ divides f i.e., there is a modular form g such that f = ∆g. In particular, the weight of f is at least 12. Proof Consider the quotient g = f /∆. Since ∆ has no zero in h, it follows that g is holomorphic in h. Clearly, g is weakly modular of weight =weight of f -12. Now the q-expansion of f (and also ∆), has no constant term; and the coefficient of q in the q-expansion of ∆ is non-zero. Therefore, g∗ extends to a holomorphic function in a neighbourhood of 0. That is, g is a modular function. Since the weight of g is non-negative (by Lemma (3.5)), it follows that the weight of f must be at least that of ∆, namely, 12. Corollary 3.7 The space of cusp forms of weight 12 (for SL(2, Z)), is one dimensional. Proof If f is a cusp form of weight 12, then f /∆ is a modular form of weight zero, hence is a constant. That is, the space of cusp forms of weight 12 is spanned by ∆. Theorem 3.8 The space of modular forms of weight 2k with k ≥ 0 is n spanned by the modular forms Gm 4 G6 with 4m + 6n = 2k. Proof Argue by induction on k. We have already excluded the possibilities k < 0 and k = 0 and k = 1. Suppose that k ≥ 2 and that f is modular of weight 2k. First observe that any integer k ≥ 2 may be written as 2m + 3n for non-negative integers m and n. Now, the q-expansion of G4 and G6 have non-zero constant term. n Hence h = f − λGm 4 G6 ) for a suitable constant λ, has no constant term in its q-expansion, and is a cusp form. Now, Lemma (3.6) shows that g = h/∆ is a modular form of weight 2k−12. By induction, g is a linear combination of the modular forms Ga4 Gb6 with k − 6 = 2a + 3b whence, h is a sum of monomials of the form Gp4 Gq6 with 2p + 3q = k (recall that ∆ is (60G4 )3 − 27(140G6 )2 ). Therefore, so is f . Notation 3.9 Define E2k (z) = G2k /2ζ(2k). Then, it follows from the Fourier expansion of G2 and G6 that the modular forms E4 and E6 have inP tegral Fourier coefficients. One sometimes writes ∆(z) = q + n≥2 τ (n)q n . Then, ∆ has integral Fourier coefficients as well. We now consider the Zmodule spanned by E4m E6n with 4m + 6n = 2k. We get an integral lattice

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T.N. Venkataramana

of modular forms of weight 2k. We will see later that this integral lattice is stable under the Hecke operators.

4

Modular Forms and Representation Theory

Notation 4.1 We will begin with some calculations on the Lie the group GL(2, R). Write, 0 1 1 0 1 0 0 X= , Y = , Z= , and H = 0 0 0 0 0 1 −1

algebra g of 1 . 0

(27)

The complexified Lie algebra of GL(2, R) is M2 (C) the space of 2 × 2 matrices with complex entries; the Lie algebra structure is given by (a, b) 7→ [a, b] = ab − ba; M2 (C) is spanned by X, Y, Z and A. Write A = −iH (where i ∈ h is the unique element whose square is -1). Then, A acts semisimply (under the adjoint action) on g with real eigenvalues. Write g = CE + ⊕ CE − ⊕ CZ ⊕ CA

(28)

where E − = X + iY − (i/2)A − (i/2)Z and E + = X − iY − (i/2)A + (i/2)Z. (29) Then E − and E + are eigenvectors for A with eigenvalues −2 and 2 respectively. Of course, on A and Z, A acts by 0. Thus, the complex Lie algebra spanned by E + , E − and A is isomorphic to sl2 (C). Definition 4.2 Fix the subgroup K∞ = O(2) of GL(2, R). This is the group generated by cosθ sinθ SO(2) = Rθ = :θ∈R (30) −sinθ cosθ and ι=

−1 0 . 0 1

(31)

Then, O(2) is a maximal compact subgroup of GL(2, R). Suppose that (π, V ) is a module for g as well as for O(2) such that the module structures are compatible. That is, suppose that v ∈ V and ξ ∈ g, and σ ∈ O(2). Then, π(σ)π(ξ)(v) = π(σ(ξ))(v)

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where σ(ξ) is the inner conjugation action of O(2) on the Lie algebra g. One then says that (π, V ) is a (g, K∞ )-module. If, as a K∞ -module, (π, V ) is a direct sum of irreducible representations of K∞ with each irreducible representation occurring only finitely many times, then one says that the (g, K∞ )module is admissible. One then sees at once that a (g, K∞ )-submodule (or a quotient module) of an admissible module is also admissible. One says that a vector v ∈ V generates (π, V ) as a (g, O(2))-module, if the smallest submodule of V containing v is all of V . Notation 4.3 The Tensor Algebra. Given a (complex) vector space V , denote by T n (V ) = V ⊗n the n-th tensor power of V . This is of course, spanned by the pure tensors, i.e. vectors of the form v1 ⊗ · · · ⊗ vn with vi ∈ V . By definition, T 0 (V ) = C and T 1 (V ) = V . Denote by T (V ) = ⊕T n (V ) where the direct sum is over all the non-negative integers n. Given non-negative integers m and n, there exists a linear map T m (V ) ⊗ T n (V ) → T m+n (V ) which on pure tensors is the map (v1 ⊗ · · · ⊗ vm ) ⊗ (w1 ⊗ · · · ⊗ wn ) 7→ (v1 ⊗ · · · ⊗ vm ⊗ w1 ⊗ · · · wn ). This extends by linearity to all of T m (V ) ⊗ T n (V ) and thence to all of the direct sum T (V ). Under this “multiplication”, T (V ) becomes an associative algebra, and is called the tensor algebra of the vector space V . The subspace T 0 (V ) = C acts simply by scalar multiplication. Notation 4.4 The Universal Enveloping Algebra. Given now a Lie algebra g, let u(g) denote the quotient of the tensor algebra T (g) of g, by the two sided ideal generated by the elements x⊗y −y ⊗x−[x, y], as x and y vary over the elements of the Lie algebra g. Here, ⊗ denotes the multiplication in the tensor algebra T (g) and the bracket [x, y] denotes the Lie bracket in g. The algebra u(g) is called the universal enveloping algebra of g. Note that g is a subspace of u(g), with [x, y] = xy − yx. Here, x, y are the images of x, y ∈ g = T 1 (g) under the quotient map T (g) → u(g). Suppose that u is some algebra over C and f : g → u a linear map such that f ([x, y]) = f (x)f (y) − f (y)f (x) for all elements x, y ∈ g. Here f (x)f (y) refers to the product of the two elements in the algebra u. Then, there exists

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a unique algebra map F : u(g) → u which extends f . This is why u(g) is called the universal enveloping algebra of g. In particular, if V is a module over g, we have a map f : g → End(V ) of Lie algebras, and we have f ([x, y]) = [f (x), f (y)] where the bracket structure on End(V ) is simply the commutator in the algebra End(V ). Therefore, by the last paragraph, we get a unique extension F : u(g) → End(V ) , with F an algebra map. In other words, the g module V is naturally a module over u(g) as well. Theorem 4.5 The Poincar´ e-Birkhoff-Witt Theorem: Let a, b and c be subalgebras of a Lie algebra g such that g = a ⊕ b ⊕ c. Then, one has the decomposition u(g) = u(a) ⊗ u(b) ⊗ u(c) Proof We must prove that every element of u(g) lies in the subspace of the right-hand side of the above equation. Argue by an induction on the degree of an element ξ ∈ T n (g). If we have an element yx for example, with y ∈ c and x ∈ c, then, we may write it as xy − [x, y]. Now xy is in the above subspace, and since g is by assumption a direct sum of a, b and c, the element [x, y] also lies in the relevant subspace. We omit the details, since this would be rather technical, and the reader can easily supply the details. We now return to the group G = GL(2, R) and its Lie algebra g. We will now prove the basic fact from representation theory which we will use. Theorem 4.6 Let (π, V ) be a (g, K∞ )-module. Suppose that v ∈ V has the following properties: (1) v generates V . (2) The connected component SO(2) of O(2) acts on v by the character determined by Rθ (v) = e2πiθm v, for some positive integer m (i.e. v is an eigenvector for A with eigenvalue m). (3) E − (v) = 0 and Z(v) = 0. Then the (π, V ) is admissible and irreducible. Proof Let u(g) denote the universal enveloping algebra of the Lie-algebra g. One has the decomposition (the Poincar´e-Birkhoff-Witt Theorem) u(g) = u(g).[E − ] + u(g).[Z] ⊕ C[E + ] ⊗ C[A]

(32)

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57

where C[ξ] denotes the algebra generated by the operator ξ. Therefore, if (as in (31)) −1 0 ι= 0 1 then by assumptions (1) and (2) of the Theorem, V = C[E + ](v) ⊕ ιC[E + ](v).

(33)

On E + the element A acts by the eigenvalue 2. Therefore, for an integer p ≥ 0, the element (E + )p (v) is an eigenvector for A with eigenvalue (2p + m), and ι(E + )p (v) is an eigenvector with eigenvalue (−2p − m) (note that under the conjugation action of ι, the element A goes to −A, hence ι takes an r-eigenspace for A into the −r-eigenspace). Since all these weights are different, equation (33) shows that V is admissible as an SO(2) module (A generates the complexified Lie algebra of SO(2)). In fact, equation (33) shows that the multiplicity of an irreducible representation of SO(2) in V is at most one, i.e. V is admissible. Suppose that W ⊂ V is a submodule. In the last paragraph, we saw that the action of A on V is completely reducible; hence the same holds for W . Suppose that w is a weight vector in W of weight j, say. By replacing w by ι(w) if necessary, we may assume that j > 0. The last paragraph shows that j = 2p + m for some p ≥ 0 and also that (E + )p (v) = w (up to scalar multiples). We may assume that p is the smallest non-negative integer such that W contains the eigenvector (E + )p (v) = w with eigenvalue 2p + m. The minimality of p implies that E − (w) = 0. Let W ′ be the submodule of W generated by the vector w. To prove the irreducibility, it is enough to show that W ′ = V . We may assume then that W = W ′ . Since v generates V and Z annihilates v, it follows that Z acts by zero on all of V . Therefore, the vector w satisfies all the properties that v does in the assumptions of the Theorem (except that in (2) the eigencharacter is 2p + m). Therefore, cf. equation (33), we have W = C[E + ](w) ⊕ ιC[E + ](w) = C[E + ](E + )p (v) ⊕ ιC[E + ](E + )p (v).

(34)

Now the equations (33) and (34) show that the codimension of W in V is finite: dim(V /W ) < ∞. Hence V /W also satisfies the assumptions of the Theorem (with v ∈ V replaced by its image v ∈ V /W ), but is finite dimensional. This is impossible by the finite dimensional representation theory of sl(2, C): a lowest weight vector (i.e. one killed by E − of sl(2))

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cannot have positive weight for A. But v has exactly this property in V /W . This shows that V /W = 0 i.e. W = V . Proposition 4.7 Given m > 0, there is a unique irreducible (g, K∞ )module ρm which satisfies the properties of 4.6. Proof The uniqueness follows easily from the above proof of Theorem (4.6). Let χm denote the one dimensional complex vector space on which the group SO(2) acts by the character Rθ 7→ e2πimθ (where Rθ is, as in (30), the rotation by θ in 2-space). Consider the space u(g) ⊗ χm . This is a representation for SO(2) (as well as for the universal enveloping algebra u(g)). Let ρm be the O(2)-module induced from this SO(2)-module. Then, ρm satisfies the properties of Theorem (4.6) and is therefore irreducible. Moreover, it is clear that any module V of the type considered in 4.6 is a quotient of ρm . By irreducibility, V = ρm . Remark 4.8 The modules ρ2k are called the discrete series representations of weight 2k of (g, O(2)). This means the following. Suppose there exists an irreducible unitary representation of the group GL(2, R), call it ρ. Suppose that this occurs discretely (i.e. is a closed subspace of ) in a space of functions L2 (G, ω) which transform according to the unitary character ω of the centre Z of GL(2, R) and which are square summable with respect to the Haar measure on the quotient GL(2, R)/Z. Given such a unitary module ρ, consider the space of vectors whose translates under the compact group O(2) form a finite dimensional vector space. This is the Harish-Chandra module of the unitary representation ρ and is a (g, O(2))-module. The representations ρ2k are the Harish-Chandra modules of discrete series representations of even weight. We will show in the next section, that these are closely related to modular forms of weight 2k. There are also the discrete series representations of odd weight, which we will not discuss, since we are dealing with the group SL(2, Z) and it has no modular forms of odd weight. We are now in a position to state the precise relationship of modular forms with representation theory.

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Notation 4.9 Let f be a modular form of weight 2k with k > 0. We will now construct a function on the group G+ = GL(2, R)+ as follows. Set Ff (g) = j(g, i)−2k f (g(i))det(g)k where i ∈ h is the point whose isotropy is the group SO(2) as in equation (30). As before, j(g, i) = cz + d, where g = ( ac db ). By using the modularity of f and the equation j(gh, z) = j(g, h(z))j(h, z) for the “automorphy factor” j(g, z), it is easy to see that Ff is invariant under left translation by elements of SL(2, Z) and also under the centre Z∞ of GL(2, R). We will now check that the (g, O(2))-module generated by Ff is isomorphic to ρ2k , with ρ2k as in 4.7. Note that Ff is contained in the space C ∞ (Z∞ GL(2, Z)\GL(2, R), the space of smooth functions on the relevant space and that the latter is a (g, O(2))-module under right translation by elements of GL(2, R). Moreover, for all y > 0 and x ∈ R we have y x = y k f (x + iy) (35) Ff 0 1 The function g 7→ f (g(i)) is right invariant under the action of SO(2) since i is the isotropy of SO(2). Using the fact that j(Rθ , i) = e−iθ (where Rθ is as in (30)) one checks that j(gRθ , i) = j(g, i)(e−2iθ ). Therefore, it follows that Ff (gRθ ) = Ff (g)e2ikθ . (36) 0 1 ) (A This equation implies that under the action of the element A = i( −1 0 generates the Lie algebra of SO(2)), Ff is an eigenvector with eigenvalue 2k. Compute the action of E − (E − as in (29)) on Ff . Using the invariance of of Ff under Z∞ and that it is an eigenvector of A with eigenvalue 4k, one sees that E − (Ff ) = (X + iY )Ff − ikFf .

Now use equation (35) to conclude that E − Ff = y 2k (∂f /∂z). Since f is holomorphic, one obtains that E − Ff = 0. The (g, O(2)) module generated by Ff satisfies the conditions of 4.6.

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Notation 4.10 Growth Properties of Ff . Consider now the growth properties of Ff . We have the quotient map GL(2, R)+ → h(⊃ F ) where F is the fundamental domain constructed in section 2. Let S be the pre-image of F under this quotient map. Then, we have the inclusion y x 2 : y > 3/4 and − 1/2 < x < 1/2 S ⊂ Z∞ O(2) 0 1 and the latter is a “Siegel set”. Now, y x Ff = y k f (x + iy). 0 1 From the modularity property of f , it follows that f is “bounded at infinity”, which means that there exists a constant C > 0 such that on the fundamental domain F of SL(2, Z), the function z 7→ f (z) = f (x + iy) is bounded by C: | f (z) |≤ C ∀z ∈ F. Therefore, on the Siegel set S, we have y x Ff ≤ Cy k , 0 1 i.e. Ff has moderate growth on the Siegel Set. Suppose now that f is a cusp form. Then, the Fourier expansion at infinity of f is of the form f (z) =

n=∞ X

an exp(2πinz)

n=1

where a(n) are the Fourier coefficients. The function X

an q n−1

is clearly bounded in a neighbourhood of infinity in the fundamental domain F and the complement of a neighbourhood of infinity being compact, it is bounded on all of F too, by a constant C.

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This shows the existence of a constant C > 0 such that for all z ∈ F , one has f (x + iy) ≤ Cexp(−y). The Haar measure on the group GL(2, R) is the product of the Haar measure on the group O(2) and the invariant measure dxdy/y 2 on the upper half plane h = GL(2, R)/O(2) constructed at the end of section 2. This is an easy exercise. Thus, the square of the absolute value of Ff integrated on S = π −1 (F ) (where π is the quotient map GL(2, R) → h ) is simply the integral of the square of the absolute value of f (z) on the domain F . The above estimate for f shows that this integral over F is finite: ! Z Z 1/2

∞

dx

−1/2

(dy/y 2 )exp(−y)

3/4. Suppose that f is a cusp form for SL(2, Z) and Ff be as in section (4.5). Given g ∈ GL(2, A) = GL(2, R) × GL(2, Af ), write g = (g∞ , gf ) accordingly. Define the function Φf on GL(2, Q)\GL(2, A) as follows. Set Φf (g∞ , gf ) = Ff (g∞ ) if gf ∈ K0 and extend to G(A) by demanding that Φf be GL(2, Q)-invariant. The SL(2, Z)-invariance of Ff implies that Φf is well defined. Now, 4.12 shows that Φf is an automorphic form on GL(2, A). By 4.13, Ff is rapidly decreasing on S0 ; moreover, Ff is a cuspidal automorphic form in the sense that for all g ∈ G(A), the following holds. Z Φf (ng)dn = 0 (43) U (Q)\U (A)

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where U is the group of unipotent upper triangular matrices in GL(2) with ones on the diagonal and dn is the Haar measure on U (A). To prove this, we note that the vanishing of the integral is unaltered by changing g on the (1) right by an element of Z∞ O(2) × K0 , since Φf is an eigenvector for the right translation action by K∞ Z∞ × K0 . We may hence assume that b an element of G(Af ) g = (g∞ , gf ). Now, up to elements of K0 = GL(2, Z) is upper triangular; (2) left by an element of B(Q) since G(Q) normalises U (A) and U (Q), and preserves the Haar measure on U (A). Note that (the Iwasawa decomposition) the double coset B(Q)\G(Af )/K0 Z(A) is a singleton. Hence, we may assume that g = g∞ = ( y0 x1 ). Then the above integral is the same as Z f (x + n + iy)dn U (Z)\U (R)

which is nothing but the zero-th Fourier coefficient of f , and by the cuspidality of f , this is zero. On the Siegel domain, the modular function f satisfies an estimate of the form |f (x + iy)| < Cexp(−y) where C is some constant. This implies that on the Siegel set S, the function Φf satisfies an estimate of the form Φf (g) = O(|g|−N ) for some positive integer N . This can be shown to imply that the function Φf is square summable on the quotient Z(A)GL(2, Q)\GL(2, A) with respect to the Haar measure. Further, one has the L2 -metric < , > on the space of cuspidal automorphic forms which translates to the “Petersson” metric Z f (z)g(z)y 2k (y −2 dxdy) < f, g >= F

for cusp forms f and g of weight k. As before, F is the fundamental domain for SL(2, Z). Notation 5.5 From now on, we will fix our attention on cusp forms. We have the natural inclusion of GL(2, Q) in GL(2, Af ). Let p > 0 be a prime and let gp = ( p0 10 ) be thought of as an element in GL(2, Af ) under the

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foregoing inclusion. Let Xp denote the characteristic function of the double coset of K0 through the element gp . If χp denotes the characteristic function Xp , and f is a cuspidal modular form of weight 2k, then Φ′ = Φf ∗ R(χp ) (where R(φ) denotes the right convolution by the function φ) is a smooth function on the quotient GL(2, Q)Z(A)\GL(2, A) whose “infinite” component is still ρ2k (since χp commutes with the right action of GL(2, R) on the above quotient). Since χp is K0 invariant, it follows that Φ′ is also right K0 invariant. Therefore, it corresponds to a modular form g, i.e. Φ′ = Φg . It is easy to show that Φ′ is cuspidal (the space of cusp forms is stable under right convolutions). Therefore, g is a cusp form of weight 2k as well. Denote g = T (p)(f ). Then, T (p) is called the Hecke operator corresponding to the prime p. By noting that convolution by χp is self-adjoint for the L2 metric on cuspidal automorphic functions on GL(2) one immediately sees that the operators T (p) are self-adjoint for the Petersson metric on the space of cusp forms of weight 2k. The commutativity of the unramified Hecke algebra implies that the operators T (p) (as p varies) commute as well. Definition 5.6 Now a commuting family of self-adjoint operators on a finite dimensional complex vector space can be simultaneously diagonalised. Consequently, there exists a basis of cusp forms of weight 2k which are simultaneous eigenfunctions for all the Hecke operators T (p); these are called Hecke eigenforms. If f is a Hecke eigenform for SL(2, Z) and has constant term 1, then it is called a normalised Hecke eigenform. Theorem 5.7 The Iwasawa Decomposition: Any matrix in GL(2, Af ) may be written as a product bk with b ∈ B(Af ) (the group of upper triangular b matrices), and k ∈ K0 = GL(2, Z).

Proof This is an easy application of the elementary divisors theorem. By identifying B\G with the projective line P1 , we see that the Iwasawa decomb on P1 (Af ). position amounts to the transitivity of the action of GL(2, Z) But, any element of P1 (Af ) may be written as a vector (x, y) ∈ A2f where for every prime p, the p-th components (xp , yp ) are not both zero. By changing (x, y) by an element of A∗f if necessary, (x, y) may be asb 2 . Further, x, y may be assumed to be coprime, in the sense sumed to be in Z

that for every prime p, the p-adic components xp , yp of x, y are coprime. Now, by writing everything in the notation of row vectors, we want to solve

Classical Modular Forms b the equation for g ∈ GL(2, Z)

67

(x, y) = (0, 1)g

(note that the isotropy at (0, 1) is precisely B(Af )). This amounts to finding b 2 to a basis of Z b 2 , which can be done precisely because x, y are (z, t) ∈ Z coprime. Notation 5.8 This implies that for a prime p, we have Xp = ∪( p0 x1 )K0 ∪ ( 10 0p )K0 b where the union is a disjoint union, and 0 ≤ x ≤ p − 1. Here K0 = GL(2, Z), as before. Notation 5.9 We will now state without proof the computation of T (p) for a prime p. Note that by strong approximation, the K0 invariant function Φf on the quotient Z(A)GL(2, Q)\GL(2, A) is completely determined by its values on elements of the form ( y0 x1 ) with y > 0, in the quotient. We compute (using the description of Xp in the previous section) Φf ∗ R(χp )( y0 x1 ) and find that this is equal, to p2k−1 Φg ( y0 x1 ) where g(z) = (1/p)

X

f ((z + m)/p) + p2k−1 f (pz) = T (p)(f )(z).

0≤m≤p−1

The Fourier coefficients of g at infinity are given by g(m) = a(mp) if m is coprime to p and g(m) = a(mp) + p2k−1 a(m/p) if p divides m.

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Notation 5.10 In particular, if f is an eigenfunction for all the T (p) with eigenvalue λ(p) say, the equation T (p)f = λ(p)f implies, by comparing the p-th Fourier coefficients, that a(p) = λ(p)a(1) for each p. Remark 5.11 In particular, we get from the last two paragraphs, that if f is an eigenform whose first Fourier coefficient a(1) is zero, then, a(m) = 0 for all positive m. This easily follows from induction and the formula a(mp) = λ(p)a(p) if m and p are coprime, and a(m)λ(p) = a(mp) + p2k−1 a(m/p) if p divides m. Hence f = 0. Thus, we have proved that every Hecke eigenform is a nonzero multiple of a normalised Hecke eigenform. Theorem 5.12 (The Multiplicity 1 Theorem): Let f1 and f2 be two normalised Hecke eigenforms for the action of the Hecke operators T (p) with the same eigenvalues λ(p) for every prime p. Then, f1 = f2 . Proof We will prove this by showing that the Fourier coefficients of f1 and f2 are the same. This will imply, by the Fourier expansion for modular forms, that f1 = f2 . Write f = f1 − f2 . Now, the first Fourier coefficient of f is zero, since f1 and f2 are normalised. Further, f is also a Hecke eigenform, since f1 and f2 are so, and with the same eigenvalues. Therefore, by the previous remark, f = 0. Recall that we have identified [representations π of GL(2, A) whose infinite component π∞ is ρ2k and whose finite component πf contains a non-zero GL(2, Af ) invariant vector], with [normalised eigenforms f of weight 2k for the group GL(2, Z)]. Therefore, we have proved that the multiplicity of such a π in the space of cusp forms on GL(2, A) is one. Remark 5.13 Later, Cogdell will prove that the multiplicity of a cuspidal automorphic representation of GL(n) is always 1. This is the famous multiplicity 1 theorem due to Jacquet-Langlands for GL(2) and to PiatetskiiShapiro and Shalika in general. What we have proved is therefore a very special case when n = 2, the infinite component is ρ2k , and the representation is unramified at all the local places.

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Definition 5.14 We now define inductively, the operators T (n) as follows. If m and n are coprime, we define T (mn) = T (m)T (n). This reduces us to defining T (pm ) where p is a prime. Define recursively the operators T (pm ) by the formula T (p)T (pm ) = T (pm+1 ) + pT (pm−1 ). This now implies (by the similarity of the recursive formulae for T (n) P and the Fourier coefficients a(n)), that if f = a(n)q n is a normalised Hecke eigenform, then T (n)f = a(n)f for all n. These T (n)’s are the classical Hecke operators. By construction, they commute, one has T (mn) = T (m)T (n) if m, n are coprime, and they are self-adjoint for the Petersson inner product on modular forms of weight 2k. Theorem 5.15 If f is a normalised Hecke eigenform for SL(2, Z), then, all its Fourier coefficients are algebraic integers. Proof We consider the action of the Hecke operators T (n) on the space 0 of cups forms. Note that the space of cusp forms contains the (adM2k ditive) subgroup L of those cusp forms whose Fourier coefficients are rational integers. This subgroup is stable under the action of the operators T (n). To see this, first suppose that n = p is a prime. By the formula aT (p)f (m) = a(pm) + p2k−1 a(m/p) if p divides m and aT (p)f (m) = a(pm) otherwise, we see that T (p) stabilises the subgroup L. Since the T (p) generate T (n), the operator T (n) also stabilises L. By induction on k, we see that the space M2k of modular forms of weight 2k has a basis whose Fourier coefficients are integral: M2k = CE2k ⊕ M2k−12 ∆, and ∆ and E2k have integral Fourier coefficients. This shows that the subgroup L of the last paragraph contains a basis of the space of cusp 0 . forms M2k Since the operator T (n) is self adjoint with respect to a suitable metric 0 , it follows that the eigenvalues of T (n) are all real and are on the space M2k all algebraic integers. Now the Fourier coefficients a(n) of the eigenform f are nothing but the eigenvalue λ(n) of T (n) corresponding to the eigenvector f , by the last paragraph of the previous section. Consequently, the Fourier coefficients of f are all real algebraic integers.

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6

L-functions of Modular Forms

In this section, we define the L-function of a cusp form for SL(2, Z) and prove that it has analytic continuation to the entire plane and has a nice functional equation. Later, Cogdell will prove analogous statements for cuspidal automorphic representations for GL(n). To begin with, we prove an estimate –due to Hecke– for the n-th Fourier coefficient of a cusp form of weight 2k. Pn=∞ Lemma 6.1 (Hecke) Let f = n=1 a(n)q n be a cusp form of weight 2k. Then, there exists a constant C > 0 such that a(n) ≤ Cnk/2

∀n ≥ 1.

Proof Consider the function φ defined and continuous on the upper half plane h, given by φ(z) = y k/2 |f (z)|. If γ = ( ac db ) ∈ SL(2, Z) and z ′ = γ(z) = x′ + iy ′ , then recall that y ′ = y/(|cz + d|2 ). Therefore, we obtain from the modularity property of f , that φ(γ(z)) = φ(z), i.e. φ is invariant under SL(2, Z). Hence, φ is determined by its restriction to the fundamental domain F . As z tends to infinity in F , the cuspidality condition of f shows that f (z) = O(exp(−2πy)). Therefore, φ is bounded on F and hence on all of the upper half plane h. Thus, there exists a constant C1 such that |f (z)| ≤ C1 y −k/2

∀z ∈ h.

Now consider the n-th Fourier coefficient a(n) of f . Clearly, 2πny

a(n) = e

Z

1

f (x + iy)e−2iπnx dx.

0

By applying the foregoing estimate for f to this equation, we obtain |a(n)| ≤ C1 e2πny y k/2 for all y > 0. Take y = 1/n. We then get a(n) ≤ (C1 e2π )nk/2 .

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Remark 6.2 Deligne has proved that for all primes p, |a(p)| = O(p((k−1)/2)) , by linking these estimates with the “Weil Conjectures” for the number of rational points of algebraic varieties over finite fields. P Definition 6.3 If f (z) = a(n)q n is a cusp form of weight 2k, define for a complex variable s, the Dirichlet Series L(f, s) by the formula L(f, s) =

∞ X

a(n)/ns .

n=1

From Lemma (6.1), it follows that the series converges and is holomorphic in the region Re(s) > 1 + (k/2). The function L(f, s) is called the L-function of the cusp form f . Notation 6.4 an integral expression for L(f, s)). We will now write an integral formula for the L-function of f . First consider the integral Z ∞ f (iy)y s (dy/y). 0

Since f (iy) = O(exp(−2πy)) for all y > 0 (cf. the proof of Lemma (6.1)), it follows that if Re(s) > 0, then the integral converges. Let σ be the real part of s. Then, for each n ≥ 1 the integral Z ∞ |a(n)|e−2πny y σ (dy/y) 0

converges, and is equal to (|a(n)|/nσ )(2π)−σ Γ(σ) where Γ is the classical Γ-function: Z ∞ e−t tz (dt/t). Γ(z) = 0

From the Hecke estimate a(n) = O(nk/2 ) of Lemma (6.1), it follows that the infinite sum of these integrals also converges, provided σ > k/2 + 1. Thus, by the Dominated Convergence Theorem (to justify the interchange of sum and integral), we obtain the equation Z ∞ ∞ X a(n)n−s . f (iy)y s (dy/y) = (2π)−s Γ(s) 0

n=1

We finally obtain the integral expression: Z ∞ f (iy)y s (dy/y) = (2π)−s Γ(s)L(f, s). 0

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Remark 6.5 Notice that the integral expression says that the function L(f, s) can be analytically continued to the region Re(s) > 0, since the left-hand side is analytic there, and the function (2π)−s Γ(s) has no zero’s on the complex plane. Notation 6.6 The functional Equation. The L-Function and the integral expression could have been defined for any holomorphic function P f (z) = a(n)q n (q = e2πiz ), provided a(n) satisfy a Hecke estimate. We will now prove a functional equation for L(f, s), by using the modularity property, especially that f (−1/z) = (−1)k z 2k f (z). R∞ Consider now the integral I(s) = 0 f (iy)y s (dy/y), which converges for Re(s) > 0. We write the integral as a sum of the integral from 1 to ∞ and the integral from from 0 to 1. By making a change of variable y 7→ 1/y we get, Z ∞ Z 1 f (−1/(iy))y −s (dy/y). f (iy)y s (dy/y) = 1

0

Note that as f (1/(iy)) is bounded on the interval [1, ∞], and Re(s) > 0, the integral on the right side converges. Therefore, we get, using the functional equation f (−1/iy) = (−1)k y 2k f (iy), that Z

1

s

k

f (iy)y (dy/y) = (−1)

0

We then get I(s) =

Z

∞

Z

∞

f (iy)y 2k−s (dy/y).

1

f (iy)(y s + (−1)k y 2k−s )(dy/y).

1

This holds for all s, with Re(s) > 0. We now make the change of variable s 7→ 2k − s, for s in the region 0 < Re(s) < 2k. Then the above expression for I(s) shows that I(2k − s) = (−1)k I(s). This is the functional equation for L(f, s): (2π)−s Γ(s)L(f, s) = (−1)k (2π)2k−s Γ(2k − s)L(f, 2k − s) for all s in the region 0 < Re(s) < 2k. The left side of this equation is analytic in the region Re(s) > 0 and the right side is analytic in the region Re(s) < 2k. Using the functional equation (and the fact the (2π)−s Γ(s) never vanishes on the complex plane), we now see that L(f, s) has an analytic continuation over the entire complex plane.

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73

Notation 6.7 Euler Factors. In the last few sections, we derived the functional equation and analytic continuation of a cusp form for SL(2, Z). We now derive an Euler product, for the L-function of a normalised Hecke eigenform f . P n Let then f = ∞ n=1 a(n)q be a normalised Hecke eigenform of weight 2k. Fix a prime p and consider the infinite sum Lp (f, s) =

∞ X

a(pm )/pms

m=0

where Re(s) > k/2 + 1. It converges, by the Hecke estimate a(n) = O(nk/2 ). We now use the relations a(pm )a(p) = a(pm+1 ) + p2k−1 a(pm−1 ). Multiplying these equations by 1/p(m+1)s and then summing over all m ≥ 0 we get a(p)Lp (f, s) = (Lp (f, s) − 1)ps + (p2k−1 /p2s )Lp (f, s). That is, Lp (f, s)−1 = 1 − a(p)/ps + p2k−1 /p2s . We now use the fact that a(mn) = a(m)a(n) if m, n are coprime, since f is a normalised Hecke eigenform. Form the product over all primes p of these Lp (f, s). We then get (by using the Dominated Convergence Theorem Q P to justify interchanges) the equation Lp (f, s) = a(n)/ns = L(f, s). Thus we have the infinite product expansion (the product being over all primes p) Y 1/(1 − a(p)/ps + p2k−1 /p2s ). L(f, s) = p

From now on we consider the function L∗ (f, s) = (2π)−s Γ(s)L(f, s) and refer to this as the L-function of f . We have thus proved the following Theorem. P n Theorem 6.8 Let f = ∞ of n=1 a(n)q be a normalised Hecke eigenform P weight 2k for SL(2, Z). Then, the L-function L∗ (f, s) = (2π)−s Γ(s) a(n)/ns converges for Re(s) > k/2, has an analytic continuation to the entire complex plane and satisfies the functional equation L∗ (f, s) = (−1)k L∗ (f, 2k − s). Moreover, in the region Re(s) > k/2, one has the Euler product Y L∗ (f, s) = (2π)−s Γ(s) 1/(1 − a(p)/ps + p2k−1 /p2s ). p

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T.N. Venkataramana

Remark 6.9 Recall that to each Hecke eigenform f of weight 2k, there corresponds an irreducible cuspidal automorphic representation π(f ) = π = π∞ ⊗p πp of the (restricted direct product) group GL(2, A) = GL(2, R) × Q p GL(2, Qp ) such that π∞ is the discrete series representation ρ2k and each πp is an irreducible unramified representation of GL(2, Qp ). In the lectures of Cogdell, you will see that each cuspidal automorphic representation π of GL(n, A) has an L-function attached to it –denoted L(π, s)– which satisfies a functional equation, has an analytic continuation to the entire plane, and has an Euler product comprising of terms which are monic polynomials in p−s of degree n. It turns out that for the representation π(f ) = π attached to the Hecke eigenform f of weight 2k, the L-function is nothing but L(π, s) = L∗ (f, s + (k − 1/2)), which can easily be seen to satisfy the equation L(π, s) = (−1)k L(π, 1 − s). Moreover, the local factors are of the form L(πp , s)−1 = (1 − (a(p)/pk−1/2 )/ps + 1/p2s , a monic polynomial in in p−s of degree two.

Notes on L-functions for GLn J.W. Cogdell

∗

Oklahoma State University, Stillwater, USA

Lectures given at the School on Automorphic Forms on GL(n) Trieste, 31 July - 18 August 2000

LNS0821003

∗

[email protected]

Abstract The theory of L-functions of automorphic forms (or modular forms) via integral representations has its origin in the paper of Riemann on the ζ-function. However the theory was really developed in the classical context of L-functions of modular forms for congruence subgroups of SL2 (Z) by Hecke and his school. Much of our current theory is a direct outgrowth of Hecke’s. L-functions of automorphic representations were first developed by Jacquet and Langlands for GL2 . Their approach followed Hecke combined with the local-global techniques of Tate’s thesis. The theory for GLn was then developed along the same lines in a long series of papers by various combinations of Jacquet, Piatetski-Shapiro, and Shalika. In addition to associating an L-function to an automorphic form, Hecke also gave a criterion for a Dirichlet series to come from a modular form, the so-called Converse Theorem of Hecke. In the context of automorphic representations, the Converse Theorem for GL2 was developed by Jacquet and Langlands, extended and significantly strengthened to GL3 by Jacquet, Piatetski-Shapiro, and Shalika, and then extended to GLn . What we have attempted to present here is a synopsis of this work and in doing so present the paradigm for the analysis of automorphic L-functions via integral representations. We begin with the Fourier expansion of a cusp form and results on Whittaker models since these are essential for defining Eulerian integrals. We then develop integral representations for L-functions for GLn × GLm which have nice analytic properties (meromorphic continuation, finite order of growth, functional equations) and have Eulerian factorization into products of local integrals. We next turn to the local theory of L-functions for GLn , in both the archimedean and non-archimedean local contexts, which comes out of the Euler factors of the global integrals. We finally combine the global Eulerian integrals with the definition and analysis of the local L-functions to define the global L-function of an automorphic representation and derive their major analytic properties. We next turn to the various Converse Theorems for GLn and their applications to Langlands liftings.

Contents Introduction

79

1 Fourier Expansions and Multiplicity One Theorems 80 1.1 Fourier Expansions . . . . . . . . . . . . . . . . . . . . . . . . 81 1.2 Whittaker Models and the Multiplicity One Theorem . . . . . 86 1.3 Kirillov Models and the Strong Multiplicity One Theorem . . 89 2 Eulerian Integrals for GLn 2.1 Eulerian Integrals for GL2 . . . . . . . 2.2 Eulerian Integrals for GLn × GLm with 2.2.1 The projection operator . . . . 2.2.2 The global integrals . . . . . . 2.3 Eulerian Integrals for GLn × GLn . . . 2.3.1 The mirabolic Eisenstein series 2.3.2 The global integrals . . . . . .

. . . . m> 0 after the unfolding. As we have seen in Lecture 1, if Wϕ ∈ W(π, ψ) corresponds to a decomposable vector ϕ ∈ Vπ ≃ ⊗′ Vπv then the Whittaker function factors into a product of local Whittaker functions Wϕ (g) =

Y

Wϕv (gv ).

v

Since the character χ and the adelic absolute value factor into local components and the domain of integration A× also factors we find that our global integral naturally factors into a product of local integrals Z

A×

Wϕ

a

1

×

s−1/2

d a=

χ(a)|a|

YZ v

kv×

W ϕv

av

1

χv (av )|av |s−1/2 d× av ,

with the infinite product still convergent for Re(s) >> 0, or I(s; ϕ, χ) =

Y

Ψv (s; Wϕv , χv )

v

with the obvious definition of the local integrals Ψv (s; Wϕv , χv ) =

Z

kv×

W ϕv

av 1

χv (av )|av |s−1/2 d× av .

Thus each of our global integrals is Eulerian. In this way, to π and χ we have associated a family of global Eulerian integrals with nice analytic properties as well as for each place v a family of local integrals convergent for Re(s) >> 0.

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2.2

J.W. Cogdell

Eulerian Integrals for GLn × GLm with m < n

Now let (π, Vπ ) be a cuspidal representation of GLn (A) and (π ′ , Vπ′ ) a cuspidal representation of GLm (A) with m < n. Take ϕ ∈ Vπ and ϕ′ ∈ Vπ′ . At first blush, a natural analogue of the integrals we considered for GL2 with GL1 twists would be Z h ϕ ϕ′ (h)| det(h)|s−(n−m)/2 dh. I n−m GLm (k)\ GLm (A) This family of integrals would have all the nice analytic properties as before (entire functions of finite order satisfying a functional equation), but they would not be Eulerian except in the case m = n − 1, which proceeds exactly as in the GL2 case. The problem is that the restriction of the form ϕ to GLm is too brutal to allow a nice unfolding when the Fourier expansion of ϕ is inserted. Instead we will introduce projection operators from cusp forms on GLn (A) to cuspidal functions on on Pm+1 (A) which are given by part of the unipotent integration through which the Whittaker function is defined. 2.2.1

The projection operator

In GLn , let Y n,m be the unipotent radical of the standard parabolic subgroup attached to the partition (m + 1, 1, . . . , 1). If ψ is our standard additive character of k\A, then ψ defines a character of Yn,m (A) trivial on Yn,m (k) since Yn,m ⊂ Nn . The group Yn,m is normalized by GLm+1 ⊂ GLn and the mirabolic subgroup Pm+1 ⊂ GLm+1 is the stabilizer in GLm+1 of the character ψ. Definition If ϕ(g) is a cusp form on GLn (A) define the projection operator Pnm from cusp forms on GLn (A) to cuspidal functions on Pm+1 (A) by “ ”Z n−m−1 p − 2 Pnm ϕ(p) = | det(p)| ϕ y ψ −1 (y) dy I n−m−1 Yn,m (k)\ Yn,m (A) for p ∈ Pm+1 (A). As the integration is over a compact domain, the integral is absolutely convergent. We first analyze the behavior on Pm+1 (A). Lemma The function Pnm ϕ(p) is a cuspidal function on Pm+1 (A).

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95

Proof: Let us let ϕ′ (p) denote the non-normalized projection, i.e., for p ∈ Pm+1 (A) set “ ” ϕ′ (p) = | det(p)|

n−m−1 2

Pnm ϕ(p).

It suffices to show this function is cuspidal. Since ϕ(g) was a smooth function on GLn (A), ϕ′ (p) will remain smooth on Pm+1 (A). To see that ϕ′ (p) is automorphic, let γ ∈ Pm+1 (k). Then Z γ 0 p 0 ′ ϕ y ϕ (γp) = ψ −1 (y) dy. 0 1 0 1 Yn,m (k)\ Y n,m (A) Since γ ∈ Pm+1 (k) and Pm+1 normalizes Yn,m and stabilizes ψ we may make −1 γ 0 γ 0 the change of variable y 7→ y in this integral to obtain 0 1 0 1 Z γ 0 p 0 ′ ϕ y ϕ (γp) = ψ −1 (y) dy. 0 1 0 1 Yn,m (k)\ Y n,m (A) Since ϕ(g) is automorphic on GLn (A) it is left invariant under GLn (k) and we find that ϕ′ (γp) = ϕ′ (p) so that ϕ′ is indeed automorphic on Pm+1 (A). We next need to see that ϕ′ is cuspidal on Pm+1 (A). To this end, let U ⊂ Pm+1 be the standard unipotent subgroup associated to the partition (n1 , . . . , nr ) of m + 1. Then we must compute the integral Z ϕ′ (up) du. U(k)\ U(A)

Inserting the definition of ϕ′ we find Z ϕ′ (up) du U(k)\ U(A)

=

Z

U(k)\ U(A)

Z

u 0 p 0 ϕ y ψ −1 (y) dy du. 0 1 0 1 Yn,m (k)\ Yn,m (A)

The group U′ = U ⋉ Y n,m is the standard unipotent subgroup of GLn associated to the partition (n1 , . . . , nr , 1, . . . , 1) of n. We may decompose this group in a second manner. If we let U′′ be the standard unipotent subgroup e n−m−1 of GLn associated to the partition (n1 , . . . , nr , n−m−1) of n and let N be the subgroup of GLn obtained by embedding Nn−m−1 into GLn by I 0 n 7→ m+1 0 n

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e n−m−1 ⋉ U′′ . If we extend the character ψ of Ym,n to U′ by then U′ = N e n−m−1 ⋉ U′′ , ψ making it trivial on U, then in the decomposition U′ = N e n−m−1 component and there it is the standard is dependent only on the N character ψ on Nn−m−1 . Hence we may rearrange the integration to give Z ϕ′ (up) du U(k)\ U(A)

=

Z

Nn−m−1 (k)\ Nn−m−1 (A)

Z

′′

ϕ u U′′ (k)\ U′′ (A)

1 0 0 n

p 0 du′′ ψ −1 (n) dy. 0 1

But since ϕ is cuspidal on GLn and U′′ is a standard unipotent subgroup of GLn then Z p 0 ′′ 1 0 du′′ ≡ 0 ϕ u 0 n 0 1 U′′ (k)\ U′′ (A) from which it follows that Z

ϕ′ (up) du ≡ 0

U(k)\ U(A)

so that ϕ′ is a cuspidal function on Pm+1 (A).

2

From Lecture 1, we know that cuspidal functions on Pm+1 (A) have a Fourier expansion summed over Nm (k)\ GLm (A). Applying this expansion to our projected cusp form on GLn (A) we are led to the following result. h n Lemma Let ϕ be a cusp form on GLn (A). Then for h ∈ GLm (A), Pm ϕ 1 has the Fourier expansion “ ” n−m−1 X h γ 0 h − 2 Pnm ϕ = | det(h)| Wϕ 1 0 In−m In−m γ∈Nm (k)\ GLm (k)

with convergence absolute and uniform on compact subsets. Proof: Once again let ′

“

ϕ (p) = | det(p)|

n−m−1 2

”

Pnm ϕ(p)

with p ∈ Pm+1 (A). Since we have verified that ϕ′ (p) is a cuspidal function on Pm+1 (A) we know that it has a Fourier expansion of the form X γ 0 ′ ϕ (p) = W ϕ′ p 0 1 γ∈Nm (k)\ GLm (k)

L-functions for GLn where Wϕ′ (p) =

Z

ϕ′ (np)ψ −1 (n) dn.

Nm+1 (k)\ Nm+1 (A) To obtain our expansion for Pnm ϕ we need terms of ϕ rather than ϕ′ .

We have Wϕ′ (p) = =

Z

Z

97

to express the right-hand side in

ϕ′ (n′ p)ψ −1 (n′ ) dn′ Nm+1 (k)\ Nm+1 (A)

Nm+1 (k)\ Nm+1 (A)

′ np 0 ϕ y g · 0 1 Y n,m (k)\ Y n,m (A)

Z

· ψ −1 (y) dy ψ −1 (n′ ) dn′ . It is elementary to see that the maximal unipotent subgroup Nn of GLn can be factored as Nn = Nm+1 ⋉ Y n,m and if we write n = n′ y with n′ ∈ Nm+1 and y ∈ Yn,m then ψ(n) = ψ(n′ )ψ(y). Hence the above integral may be written as Z p 0 p 0 −1 ′ ϕ n ψ (n) dn = Wϕ Wϕ (p) = . 0 In−m−1 0 In−m−1 Nn (k)\ Nn (A) Substituting this expression into the above we find that h n Pm ϕ 1 ” “ n−m−1 X γ 0 h − 2 Wϕ = | det(h)| 0 In−m In−m γ∈Nm (k)\ GLm (k)

and the convergence is absolute and uniform for h in compact subsets of GLm (A). 2 2.2.2

The global integrals

We now have the prerequisites for writing down a family of Eulerian integrals for cusp forms ϕ on GLn twisted by automorphic forms on GLm for m < n. Let ϕ ∈ Vπ be a cusp form on GLn (A) and ϕ′ ∈ Vπ′ a cusp form on GLm (A). (Actually, we could take ϕ′ to be an arbitrary automorphic form on GLm (A).) Consider the integrals Z h 0 n ′ Pm ϕ I(s; ϕ, ϕ ) = ϕ′ (h)| det(h)|s−1/2 dh. 0 1 GLm (k)\ GLm (A)

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J.W. Cogdell

The integral I(s; ϕ, ϕ′ ) is absolutely convergent for all values of the complex parameter s, uniformly in compact subsets, since the cusp forms are rapidly decreasing. Hence it is entire and bounded in any vertical strip as before. Let us now investigate the Eulerian properties of these integrals. We first replace Pnm ϕ by its Fourier expansion. Z h 0 n ′ ϕ′ (h)| det(h)|s−1/2 dh Pm ϕ I(s; ϕ, ϕ ) = 0 In−m GLm (k)\ GLm (A) Z X γ 0 h 0 Wϕ · = 0 In−m 0 In−m GLm (k)\ GLm (A) γ∈Nm (k)\ GLm (k)

· ϕ′ (h)| det(h)|s−(n−m)/2 dh. Since ϕ′ (h) is automorphic on GLm (A) and | det(γ)| = 1 for γ ∈ GLm (k) we may interchange the order of summation and integration for Re(s) >> 0 and then recombine to obtain Z h 0 ′ Wϕ I(s; ϕ, ϕ ) = ϕ′ (h)| det(h)|s−(n−m)/2 dh. 0 I n−m Nm (k)\ GLm (A) This integral is absolutely convergent for Re(s) >> 0 by the gauge estimates of [31, Section 13] and this justifies the interchange. Let us now integrate first over Nm (k)\ Nm (A). Recall that for n ∈ Nm (A) ⊂ Nn (A) we have Wϕ (ng) = ψ(n)Wϕ (g). Hence we have I(s; ϕ, ϕ′ ) = Z

Nm (A)\ GLm (A)

=

= =

Z Z

Z

Wϕ Nm (k)\ Nm (A)

Wϕ Nm (A)\ GLm (A)

Wϕ

Nm (A)\ GLm (A) Ψ(s; Wϕ , Wϕ′ ′ )

n 0 0 In−m

h 0 0 In−m

·

· ϕ′ (nh) dn | det(h)|s−(n−m)/2 dh Z 0 ψ(n)ϕ′ (nh) dn

h 0 In−m

Nm (k)\ Nm (A)

· | det(h)|s−(n−m)/2 dh 0 Wϕ′ ′ (h)| det(h)|s−(n−m)/2 dh

h 0 In−m

where Wϕ′ ′ (h) is the ψ −1 -Whittaker function on GLm (A) associated to ϕ′ , i.e., Z ϕ′ (nh)ψ(n) dn,

Wϕ′ ′ (h) =

Nm (k)\ Nm (A)

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99

and we retain absolute convergence for Re(s) >> 0. From this point, the fact that the integrals are Eulerian is a consequence of the uniqueness of the Whittaker model for GLn . Take ϕ a smooth cusp form in a cuspidal representation π of GLn (A). Assume in addition that ϕ is factorizable, i.e., in the decomposition π = ⊗′ πv of π into a restricted tensor product of local representations, ϕ = ⊗ϕv is a pure tensor. Then as we have seen there is a choice of local Whittaker models so that Q Wϕ (g) = Wϕv (gv ). Similarly for decomposable ϕ′ we have the factorizaQ tion Wϕ′ ′ (h) = Wϕ′ ′v (hv ). If we substitute these factorizations into our integral expression, then Q since the domain of integration factors Nm (A)\ GLm (A) = Nm (kv )\ GLm (kv ) we see that our integral factors into a product of local integrals YZ hv 0 ′ W ϕv Wϕ′ ′v (hv ) · Ψ(s; Wϕ , Wϕ′ ) = 0 I n−m Nm (kv )\ GLm (kv ) v

·| det(hv )|vs−(n−m)/2 dhv . If we denote the local integrals by Z ′ Ψv (s; Wϕv , Wϕ′v ) =

W ϕv

Nm (kv )\ GLm (kv )

hv 0 Wϕ′ ′v (hv ) · 0 In−m

·| det(hv )|vs−(n−m)/2 dhv , which converges for Re(s) >> 0 by the gauge estimate of [31, Prop. 2.3.6], we see that we now have a family of Eulerian integrals. Now let us return to the question of a functional equation. As in the case of GL2 , the functional equation is essentially a consequence of the existence of the outer automorphism g 7→ ι(g) = gι = tg−1 of GLn . If we define the action of this automorphism on automorphic forms by setting ϕ(g) e = e n = ι ◦ Pn ◦ ι then our integrals naturally satisfy ϕ(gι ) = ϕ(wn gι ) and let P m m the functional equation e − s; ϕ, I(s; ϕ, ϕ′ ) = I(1 e ϕ e′ )

where e ϕ, ϕ′ ) = I(s;

Z

GLm (k)\ GLm (A)

h n e Pm ϕ

We have established the following result.

1

ϕ′ (h)| det(h)|s−1/2 dh.

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Theorem 2.1 Let ϕ ∈ Vπ be a cusp form on GLn (A) and ϕ′ ∈ Vπ′ a cusp form on GLm (A) with m < n. Then the family of integrals I(s; ϕ, ϕ′ ) define entire functions of s, bounded in vertical strips, and satisfy the functional equation e − s; ϕ, I(s; ϕ, ϕ′ ) = I(1 e ϕ e′ ).

Moreover the integrals are Eulerian and if ϕ and ϕ′ are factorizable, we have Y I(s; ϕ, ϕ′ ) = Ψv (s; Wϕv , Wϕ′ ′v ) v

with convergence absolute and uniform for Re(s) >> 0. The integrals occurring on the right-hand side of our functional equation are again Eulerian. One can unfold the definitions to find first that e − s; ϕ, e − s; ρ(wn,m )W fϕ , W f′ ′ ) I(1 e ϕ e′ ) = Ψ(1 ϕ

where the unfolded global integral is Z Z h ′ e Ψ(s; W, W ) = W x In−m−1

dx W ′ (h)| det(h)|s−(n−m)/2 dh 1

with the h integral over Nm (A)\ GLm (A) and the x integral over Mn−m−1,m (A), the space of (n − m − 1) × m matrices, ρ denoting right translation, and 1 . Im with wn−m = . . wn,m the Weyl element wn,m = wn−m 1 the standard long Weyl element in GLn−m . Also, for W ∈ W(π, ψ) we set f (g) = W (wn gι ) ∈ W(e W π , ψ −1 ). The extra unipotent integration is the reme W, W ′ ) is absolutely convergent for Re(s) >> 0. en . As before, Ψ(s; nant of P m ′ e Wϕ , W ′ ′ ) will factor For ϕ and ϕ factorizable as before, these integrals Ψ(s; ϕ as well. Hence we have Y e Wϕ , W ′ ′ ) = e v (s; Wϕv , W ′ ′ ) Ψ(s; Ψ ϕ ϕv v

where e v (s; Wv , Wv′ ) = Ψ

Z Z

hv Wv xv In−m−1

dxv Wv′ (hv )| det(hv )|s−(n−m)/2 dhv 1

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where now the hv integral is over Nm (kv )\ GLm (kv ) and the xv integral is over the matrix space Mn−m−1,m (kv ). Thus, coming back to our functional equation, we find that the right-hand side is Eulerian and factors as Y e v (1−s; ρ(wn,m )W fϕv , W f ′ ′ ). e e fϕ , W f′ ′ ) = Ψ I(1−s; ϕ, e ϕ e′ ) = Ψ(1−s; ρ(wn,m )W ϕv ϕ v

2.3

Eulerian Integrals for GLn × GLn

The paradigm for integral representations of L-functions for GLn × GLn is not Hecke but rather the classical papers of Rankin [52] and Selberg [54]. These were first interpreted in the framework of automorphic representations by Jacquet for GL2 × GL2 [28] and then Jacquet and Shalika in general [36]. Let (π, Vπ ) and (π ′ , Vπ′ ) be two cuspidal representations of GLn (A). Let ϕ ∈ Vπ and ϕ′ ∈ Vπ′ be two cusp forms. The analogue of the construction above would be simply Z ϕ(g)ϕ′ (g)| det(g)|s dg. GLn (k)\ GLn (A)

This integral is essentially the L2 -inner product of ϕ and ϕ′ and is not suitable for defining an L-function, although it will occur as a residue of our integral at a pole. Instead, following Rankin and Selberg, we use an integral representation that involves a third function: an Eisenstein series on GLn (A). This family of Eisenstein series is constructed using the mirabolic subgroup once again. 2.3.1

The mirabolic Eisenstein series

To construct our Eisenstein series we return to the observation that Pn \ GLn ≃ kn − {0}. If we let S(An ) denote the Schwartz–Bruhat functions on An , then each Φ ∈ S defines a smooth function on GLn (A), left invariant by Pn (A), by g 7→ Φ((0, . . . , 0, 1)g) = Φ(en g). Let η be a unitary idele class character. (For our application η will be determined by the central characters of π and π ′ .) Consider the function Z Φ(aen g)|a|ns η(a) d× a. F (g, Φ; s, η) = | det(g)|s A×

If we let P′n = Zn Pn be the parabolicof GL n associated to the partition h y (n − 1, 1) then one checks that for p′ = ∈ P′n (A) with h ∈ GLn−1 (A) 0 d

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and d ∈ A× we have, F (p′ g, Φ; s, η) = | det(h)|s |d|−(n−1)s η(d)−1 F (g, Φ; s, η) = δPs ′ (p′ )η −1 (d)F (g, Φ; s, η), n

with the integral absolutely convergent for Re(s) > 1/n, so that if we extend η to a character of P′n by η(p′ ) = η(d) in the above notation we have that F (g, Φ; s, η) is a smooth section of the normalized induced representation GLn (A) s−1/2 s−1/2 IndP′ (A) (δP′ η). Since the inducing character δP′ η of P′n (A) is invarin n n ant under P′n (k) we may form Eisenstein series from this family of sections by X E(g, Φ; s, η) = F (γg, Φ; s, η). γ∈P′n (k)\ GLn (k)

If we replace F in this sum by its definition we can rewrite this Eisenstein series as Z X s Φ(aξg)|a|ns η(a) d× a E(g, Φ; s, η) = | det(g)| k × \A× ξ∈k n −{0}

= | det(g)|s

Z

k × \A×

Θ′Φ (a, g)|a|ns η(a) d× a

and this first expression is convergent absolutely for Re(s) > 1 [36]. The second expression essentially gives the Eisenstein series as the Mellin transform of the Theta series X ΘΦ (a, g) = Φ(aξg), ξ∈k n

where in the above we have written X Θ′Φ (a, g) = Φ(aξg) = ΘΦ (a, g) − Φ(0). ξ∈k n −{0}

This allows us to obtain the analytic properties of the Eisenstein series from the Poisson summation formula for ΘΦ , namely X X ΘΦ (a, g) = Φ(aξg) = Φa,g (ξ) ξ∈k n

=

X

ξ∈k n

d Φ a,g (ξ) =

ξ∈k n −n

= |a|

X

b −1 ξ tg−1 ) |a|−n | det(g)|−1 Φ(a

ξ∈k n | det(g)|−1 ΘΦˆ (a−1 ,t g−1 )

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ˆ on S(An ) is defined by where the Fourier transform Φ Z ˆ Φ(y)ψ(y tx) dy. Φ(x) = A×

This allows us to write the Eisenstein series as Z s E(g, Φ, s, η) = | det(g)| Θ′Φ (a, g)|a|ns η(a) d× a |a|≥1 Z s−1 + | det(g)| Θ′Φˆ (a,t g−1 )|a|n(1−s) η −1 (a) d× a + δ(s) |a|≥1

where δ(s) =

( 0

if η is ramified

s −cΦ(0) | det(g)| s+iσ

+

s−1 ˆ | det(g)| cΦ(0) s−1+iσ

if η(a) = |a|inσ with σ ∈ R

with c a non-zero constant. From this we easily derive the basic properties of our Eisenstein series [36, Section 4]. Proposition 2.1 The Eisenstein series E(g, Φ; s, η) has a meromorphic continuation to all of C with at most simple poles at s = −iσ, 1 − iσ when η is unramified of the form η(a) = |a|inσ . As a function of g it is smooth of moderate growth and as a function of s it is bounded in vertical strips (away from the possible poles), uniformly for g in compact sets. Moreover, we have the functional equation ˆ 1 − s, η −1 ) E(g, Φ; s, η) = E(gι , Φ; where gι = tg−1 . Note that under the center the Eisenstein series transforms by the central character η −1 . 2.3.2

The global integrals

Now let us return to our Eulerian integrals. Let π and π ′ be our irreducible cuspidal representations. Let their central characters be ω and ω ′ . Set η = ωω ′ . Then for each pair of cusp forms ϕ ∈ Vπ and ϕ′ ∈ Vπ′ and each Schwartz-Bruhat function Φ ∈ S(An ) set Z ′ ϕ(g)ϕ′ (g)E(g, Φ; s, η) dg. I(s; ϕ, ϕ , Φ) = Zn (A) GLn (k)\ GLn (A)

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Since the two cusp forms are rapidly decreasing on Zn (A) GLn (k)\ GLn (A) and the Eisenstein series is only of moderate growth, we see that the integral converges absolutely for all s away from the poles of the Eisenstein series and is hence meromorphic. It will be bounded in vertical strips away from the poles and satisfies the functional equation ˆ I(s; ϕ, ϕ′ , Φ) = I(1 − s; ϕ, e ϕ e′ , Φ),

coming from the functional equation of the Eisenstein series, where we still have ϕ(g) e = ϕ(gι ) = ϕ(wn gι ) ∈ Vπe and similarly for ϕ e′ . These integrals will be entire unless we have η(a) = ω(a)ω ′ (a) = |a|inσ is unramified. In that case, the residue at s = −iσ will be Z ′ ϕ(g)ϕ′ (g)| det(g)|−iσ dg Res I(s; ϕ, ϕ , Φ) = −cΦ(0) s=−iσ

Zn (A) GLn (A)\ GLn (A)

and at s = 1 − iσ we can write the residue as Z ′ ˆ Res I(s; ϕ, ϕ , Φ) = cΦ(0) s=1−iσ

Zn (A) GLn (k)\ GLn (A)

ϕ(g) e ϕ e′ (g)| det(g)|iσ dg.

Therefore these residues define GLn (A) invariant pairings between π and π ′ ⊗ | det |−iσ or equivalently between π e and π e′ ⊗ | det |iσ . Hence a residues ′ iσ can be non-zero only if π ≃ π e ⊗ | det | and in this case we can find ϕ, ϕ′ , and Φ such that indeed the residue does not vanish. We have yet to check that our integrals are Eulerian. To this end we take the integral, replace the Eisenstein series by its definition, and unfold: Z ′ ϕ(g)ϕ′ (g)E(g, Φ; s, η) dg I(s; ϕ, ϕ , Φ) = Zn (A) GLn (k)\ GLn (A) Z ϕ(g)ϕ′ (g)F (g, Φ; s, η) dg = Zn (A) P′n (k)\ GLn (A) Z Z ϕ(g)ϕ′ (g)| det(g)|s Φ(aen g)|a|ns η(a) da dg = × Z (A) Pn (k)\ GLn (A) A Z n ϕ(g)ϕ′ (g)Φ(en g)| det(g)|s dg. = Pn (k)\ GLn (A)

We next replace ϕ by its Fourier expansion in the form X ϕ(g) = Wϕ (γg) γ∈Nn (k)\ Pn (k)

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and unfold to find Z ′ Wϕ (g)ϕ′ (g)Φ(en g)| det(g)|s dg I(s; ϕ, ϕ , Φ) = Nn (k)\ GLn (A) Z Z ϕ′ (ng)ψ(n) dn Φ(en g)| det(g)|s dg Wϕ (g) = Nn (k)\ Nn (A) N (A)\ GLn (A) Z n Wϕ (g)Wϕ′ ′ (g)Φ(en g)| det(g)|s dg = =

Nn (A)\ GLn (A) Ψ(s; Wϕ , Wϕ′ ′ , Φ).

This expression converges for Re(s) >> 0 by the gauge estimates as before. To continue, we assume that ϕ, ϕ′ and Φ are decomposable tensors under the isomorphisms π ≃ ⊗′ πv , π ′ ≃ ⊗′ πv′ , and S(An ) ≃ ⊗′ S(kvn ) so that we Q Q Q have Wϕ (g) = v Wϕv (gv ), Wϕ′ ′ (g) = v Wϕ′ ′v (gv ) and Φ(g) = v Φv (gv ). Then, since the domain of integration also naturally factors we can decompose this last integral into an Euler product and now write Y Ψ(s; Wϕ , Wϕ′ ′ , Φ) = Ψv (s; Wϕv , Wϕ′ ′v , Φv ), v

where Ψv (s; Wϕv , Wϕ′ ′v , Φv )

=

Z

Nn (kv )\ GLn (kv )

Wϕv (gv )Wϕ′ ′v (gv )Φv (en gv )| det(gv )|s dgv ,

still with convergence for Re(s) >> 0 by the local gauge estimates. Once again we see that the Euler factorization is a direct consequence of the uniqueness of the Whittaker models. Theorem 2.2 Let ϕ ∈ Vπ and ϕ′ ∈ Vπ′ cusp forms on GLn (A) and let Φ ∈ S(An ). Then the family of integrals I(s; ϕ, ϕ′ , Φ) define meromorphic functions of s, bounded in vertical strips away from the poles. The only possible poles are simple and occur iff π ≃ π e′ ⊗ | det |iσ with σ real and are then at s = −iσ and s = 1 − iσ with residues as above. They satisfy the functional equation fϕ , W f ′ ′ , Φ). ˆ I(s; ϕ, ϕ′ , Φ) = I(1 − s; W ϕ

Moreover, for ϕ, ϕ′ , and Φ factorizable we have that the integrals are Eulerian and we have Y I(s; ϕ, ϕ′ , Φ) = Ψv (s; Wϕv , Wϕ′ ′v , Φv ) v

with convergence absolute and uniform for Re(s) >> 0.

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We remark in passing that the right-hand side of the functional equation also unfolds as Z fϕ (g)W f ′ ′ (g)Φ(e ˆ n g)| det(g)|1−s dg ˆ = W I(1 − s; ϕ, e ϕ e′ , Φ) ϕ Nn (A)\ GLn (A)

=

Y v

fϕ , W f ′ ′ , Φ) ˆ Ψv (1 − s; W ϕ

with convergence for Re(s) > 0. To make the notation more convenient for what follows, in the case m < n for any 0 ≤ j ≤ n−m−1 let us set Z Z h dx W x Ij Ψj (s : W, W ′ ) = Nm (k)\ GLm (k) Mj,m (k) In−m−j W ′ (h)| det(h)|s−(n−m)/2 dh,

e W, W ′ ) = Ψn−m−1 (s; W, W ′ ), so that Ψ(s; W, W ′ ) = Ψ0 (s; W, W ′ ) and Ψ(s; which is still absolutely convergent for Re(s) >> 0. We need to understand what type of functions of s these local integrals are. To this end, we need to understand the local Whittaker functions. So let W ∈ W(π, ψ). Since W is smooth, there is a compact open subgroup K, of finite index in the maximal compact subgroup Kn = GLn (o), so that W (gk) = W (g) for all k ∈ K. If we let {ki } be a set of coset representatives of GLn (o)/K, using that W transforms on the left under Nn (k) via ψ and the Iwasawa decomposition on GLn (k) we see that W (g) is completely determined by the values of W (aki ) = Wi (a) for a ∈ An (k), the maximal split (diagonal) torus of GLn (k). So it suffices to understand a general Whittaker function on the torus. Let αi , i = 1, . . . , n − 1, denote the standard simple

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a1 roots of GLn , so that if a =

..

.

∈ An (k) then αi (a) = ai /ai+1 .

an By a finite function on An (k) we mean a continuous function whose translates span a finite dimensional vector space [30, 31, Section 2.2]. (For the field k× itself the finite functions are spanned by products of characters and powers of the valuation map.) The fundamental result on the asymptotics of Whittaker functions is then the following [31, Prop. 2.2]. Proposition 3.1 Let π be a generic representation of GLn (k). Then there is a finite set of finite functions X(π) = {χi } on An (k), depending only on π, so that for every W ∈ W(π, ψ) there are Schwartz–Bruhat functions φi ∈ S(kn−1 ) such that for all a ∈ An (k) with an = 1 we have X W (a) = χi (a)φi (α1 (a), . . . , αn−1 (a)). X(π)

The finite set of finite functions X(π) which occur in the asymptotics near 0 of the Whittaker functions come from analyzing the Jacquet module W(π, ψ)/hπ(n)W − W |n ∈ Nn i which is naturally an An (k)–module. Note that due to the Schwartz-Bruhat functions, the Whittaker functions vanish whenever any simple root αi (a) becomes large. The gauge estimates alluded to in Section 2 are a consequence of this expansion and the one in Proposition 3.6. Several nice consequences follow from inserting these formulas for W and W ′ into the local integrals Ψj (s; W, W ′ ) or Ψ(s; W, W ′ , Φ) [31, 33]. Proposition 3.2 The local integrals Ψj (s; W, W ′ ) or Ψ(s; W, W ′ , Φ) satisfy the following properties. 1. Each integral converges for Re(s) >> 0. For π and π ′ unitary, as we have assumed, they converge absolutely for Re(s) ≥ 1. For π and π ′ tempered, we have absolute convergence for Re(s) > 0. 2. Each integral defines a rational function in q −s and hence meromorphically extends to all of C. 3. Each such rational function can be written with a common denominator which depends only on the finite functions X(π) and X(π ′ ) and hence only on π and π ′ .

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In deriving these when m < n − 1 note that one has that h 6= 0 W x Ij In−m−j−1 implies that x lies in a compact set independent of h ∈ GLm (k) [33]. Let Ij (π, π ′ ) denote the complex linear span of the local integrals Ψj (s; W , ′ W ) if m < n and I(π, π ′ ) the complex linear span of the Ψ(s; W, W ′ , Φ) if m = n. These are then all subspaces of C(q −s ) which have “bounded denominators” in the sense of (3). In fact, these subspaces have more structure – they are modules for C[q s , q −s ] ⊂ C(q −s ). To see this, note that for any h ∈ GLm (k) we have h ′ ′ W, π (h)W = | det(h)|−s−j+(n−m)/2 Ψj (s; W, W ′ ) Ψj s; π In−m and Ψ(s; π(h)W, π ′ (h)W ′ , ρ(h)Φ) = | det(h)|−s Ψ(s; W, W ′ , Φ). So by varying h and multiplying by scalars, we see that each Ij (π, π ′ ) and I(π, π ′ ) is closed under multiplication by C[q s , q −s ]. Since we have bounded denominators, we can conclude: Proposition 3.3 Each Ij (π, π ′ ) and I(π, π ′ ) is a fractional C[q s , q −s ]–ideal of C(q −s ). Note that C[q s , q −s ] is a principal ideal domain, so that each fractional ideal Ij (π, π ′ ) has a single generator, which we call Qj,π,π′ (q −s ), as does I(π, π ′ ), which we call Qπ,π′ (q −s ). However, we can say more. In the case m < n recall that from what we have said about the Kirillov model that when we restrict Whittaker functions in W(π, ψ) to the embedded GLm (k) ⊂ Pn (k) we get all functions of compact support on GLm (k) transforming by ψ. Using this freedom for our choice of W ∈ W(π, ψ) one can show that in fact the constant function 1 lies in Ij (π, π ′ ). In the case m = n one can reduce to a sum of integrals over Pn (k) and then use the freedom one has in the Kirillov model, plus the complete freedom in the choice of Φ to show that once again 1 ∈ I(π, π ′ ). The consequence of this is that our generator can be taken to be of the form Qj,π,π′ (q −s ) = Pj,π,π′ (q s , q −s )−1 for m < n or Qπ,π′ (q −s ) = Pπ,π′ (q s , q −s )−1 for appropriate polynomials in C[q s , q −s ].

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Moreover, since q s and q −s are units in C[q s , q −s ] we can always normalize our generator to be of the form Pj,π,π′ (q −s )−1 or Pπ,π′ (q −s )−1 where the polynomial P (X) satisfies P (0) = 1. Finally, in the case m < n one can show by a rather elementary although somewhat involved manipulation of the integrals that all of the ideals Ij (π, π ′ ) are the same [33, Section 2.7]. We will write this ideal as I(π, π ′ ) and its generator as Pπ,π′ (q −s )−1 . This gives us the definition of our local L-function. Definition Let π and π ′ be as above. Then L(s, π × π ′ ) = Pπ,π′ (q −s )−1 is the normalized generator of the fractional ideal I(π, π ′ ) formed by the family of local integrals. If π ′ = 1 is the trivial representation of GL1 (k) then we write L(s, π) = L(s, π × 1). One can easily show that the ideal I(π, π ′ ) is independent of the character ψ used in defining the Whittaker models, so that L(s, π × π ′ ) is independent of the choice of ψ. So it is not included in the notation. Also, note that for π ′ = χ an automorphic representation (character) of GL1 (A) we have the identity L(s, π × χ) = L(s, π ⊗ χ) where π ⊗ χ is the representation of GLn (A) on Vπ given by π ⊗ χ(g)ξ = χ(det(g))π(g)ξ. We summarize the above in the following Theorem. Theorem 3.1 Let π and π ′ be as above. The family of local integrals form a C[q s , q −s ]–fractional ideal I(π, π ′ ) in C(q −s ) with generator the local Lfunction L(s, π × π ′ ). Another useful way of thinking of the local L-function is the following. L(s, π × π ′ ) is the minimal (in terms of degree) function of the form P (q −s )−1 , with P (X) a polynomial satisfying P (0) = 1, such that the raΨ(s; W, W ′ ) Ψ(s; W, W ′ , Φ) tios (resp. ) are entire for all W ∈ W(π, ψ) and L(s, π × π ′ ) L(s, π × π ′ ) W ′ ∈ W(π ′ , ψ −1 ), and if necessary Φ ∈ S(kn ). That is, L(s, π × π ′ ) is the standard Euler factor determined by the poles of the functions in I(π, π ′ ). One should note that since the L-factor is a generator of the ideal I(π, π ′ ), then in particular it lies in I(π, π ′ ). Since this ideal is spanned by our local integrals, we have the following useful Corollary. Corollary There is a finite collection of Wi ∈ W(π, ψ), Wi′ ∈ W(π ′ , ψ −1 ), and if necessary Φi ∈ S(kn ) such that X X L(s, π × π ′ ) = Ψ(s; Wi , Wi′ ) or L(s, π × π ′ ) = Ψ(s; Wi , Wi′ , Φi ). i

i

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111

For future reference, let us set e(s; W, W ′ ) = e˜(s; W, W ′ ) =

Ψ(s; W, W ′ ) , L(s, π × π ′ ) e W, W ′ ) Ψ(s; , L(s, π × π ′ )

and e(s; W, W ′ , Φ) =

ej (s; W, W ′ ) =

Ψj (s; W, W ′ ) , L(s, π × π ′ )

Ψ(s; W, W ′ , Φ) . L(s, π × π ′ )

Then all of these functions are Laurent polynomials in q ±s , i.e., elements of C[q s , q −s ]. As such they are entire and bounded in vertical strips. As above, P there are choices of Wi , Wi′ , and if necessary Φi such that e(s; Wi , Wi′ ) ≡ 1 P or e(s; Wi , Wi′ , Φi ) ≡ 1. In particular we have the following result. Corollary The functions e(s; W, W ′ ) and e(s; W, W ′ , Φ) are entire functions, bounded in vertical strips, and for each s0 ∈ C there is a choice of W , W ′ , and if necessary Φ such that e(s0 ; W, W ′ ) 6= 0 or e(s0 ; W, W ′ , Φ) 6= 0.

3.1.2

The local functional equation

Either by analogy with Tate’s thesis or from the corresponding global statement, we would expect our local integrals to satisfy a local functional equation. From the functional equations for our global integrals, we would expect e − s; ρ(wn,m )W f, W f ′ ) when these to relate the integrals Ψ(s; W, W ′ ) and Ψ(1 f, W f ′ , Φ) ˆ when m = n. This will m < n and Ψ(s; W, W ′ , Φ) and Ψ(1 − s; W indeed be the case. These functional equations will come from interpreting the local integrals as families (in s) of quasi-invariant bilinear forms on W(π, ψ) × W(π ′ , ψ −1 ) or trilinear forms on W(π, ψ) × W(π ′ , ψ −1 ) × S(kn ) depending on the case. First, consider the case when m < n. In this case we have seen that h Ψ s; π

In−m

W, π ′ (h)W ′

= | det(h)|−s+(n−m)/2 Ψ(s; W, W ′ )

f, W f ′ ) has the same quasi-invariance and one checks that Ψ(1 − s; ρ(wn,m )W as a bilinear form on W(π, ψ) × W(π ′ , ψ −1 ). In addition, if we let Yn,m

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denote the unipotent radical of the standard parabolic subgroup associated to the partition (m + 1, 1, . . . , 1) as before then we have the quasi-invariance Ψ(s; π(y)W, W ′ ) = ψ(y)Ψ(s; W, W ′ ) e − s; ρ(wn,m )W f, W f ′ ) satisfies the for all y ∈ Yn,m . One again checks that Ψ(1 same quasi-invariance as a bilinear form on W(π, ψ) × W(π ′ , ψ −1 ). For n = m we have seen that Ψ(s; π(h)W, π ′ (h)W ′ , ρ(h)Φ) = | det(h)|−s Ψ(s; W, W ′ , Φ) f, W f ′ , Φ) ˆ satisfies the same quasiand it is elementary to check that Ψ(1−s; W ′ invariance as a trilinear form on W(π, ψ) × W(π , ψ −1 ) × S(kn ). Our local functional equations will now follow from the following result [33, Propositions 2.10 and 2.11]. Proposition 3.4 (i) If m < n, then except for a finite number of exceptional values of q −s there is a unique bilinear form Bs on W(π, ψ) × W(π ′ , ψ −1 ) satisfying h ′ ′ Bs π W, π (h)W = | det(h)|−s+(n−m)/2 Bs (W, W ′ ) In−m and

Bs (π(y)W, W ′ ) = ψ(y)Bs (W, W ′ )

for all h ∈ GLm (k) and y ∈ Yn,m (k). (ii) If n = m, then except for a finite number of exceptional values of −s q there is a unique trilinear form Ts on W(π, ψ) × W(π ′ , ψ −1 ) × S(kn ) satisfying Ts (π(h)W, π ′ (h)W ′ , ρ(h)Φ) = | det(h)|−s Ts (W, W ′ , Φ) for all h ∈ GLn (k). Let us say a few words about the proof of this proposition, because it is another application of the analysis of the restriction of representations of GLn to the mirabolic subgroup Pn [33, Sections 2.10 and 2.11]. In the case where m < n the local integrals involve the restriction of the Whittaker functions in W(π, ψ) to GLm (k) ⊂ Pn , that is, the Kirillov model K(π, ψ) of π. In the case m = n one notes that S0 (kn ) = {Φ ∈ S(kn ) | Φ(0) = 0}, which has co-dimension one in S(kn ), is isomorphic to the compactly induced

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−1/2

n (δPn ) so that by Frobenius reciprocity a GLn (k) representation indPn (k) quasi-invariant trilinear form on W(π, ψ) × W(π ′ , ψ −1 ) × S0 (kn ) reduces to a Pn (k)-quasi-invariant bilinear form on K(π, ψ) × K(π ′ , ψ −1 ). So in both cases we are naturally working in the restriction to Pn (k). The restrictions of irreducible representations of GLn (k) to Pn (k) are no longer irreducible, but do have composition series of finite length. One of the tools for analyzing the restrictions of representations of GLn to Pn , or analyzing the irreducible representations of Pn , are the derivatives of Bernstein and Zelevinsky [2,11]. These derivatives π (n−r) are naturally representations of GLr (k) for r ≤ n. π (0) = π and since π is generic the highest derivative π (n) corresponds to the irreducible common submodule (τ, Vτ ) of all Kirillov models, and is hence the non-zero irreducible representation of GL0 (k). The poles of our local integrals can be interpreted as giving quasi-invariant pairings between derivatives of π and π ′ [11]. The s for which such pairings exist for all but the highest derivatives are the exceptional s of the proposition. There is always a unique pairing between the highest derivatives π (n) and π ′(m) , which are necessarily non-zero since they correspond to the common irreducible subspace (τ, Vτ ) of any Kirillov model, and this is the unique Bs or Ts of the proposition. As a consequence of this Proposition, we can define the local γ-factor which gives the local functional equation for our integrals.

Theorem 3.2 There is a rational function γ(s, π × π ′ , ψ) ∈ C(q −s ) such that we have e − s; ρ(wn,m )W f, W f ′ ) = ω ′ (−1)n−1 γ(s, π × π ′ , ψ)Ψ(s; W, W ′ ) Ψ(1

if m < n

or

f, W f ′ , Φ) ˆ = ω ′ (−1)n−1 γ(s, π × π ′ , ψ)Ψ(s; W, W ′ , Φ) Ψ(1 − s; W

if m = n

for all W ∈ W(π, ψ), W ′ ∈ W(π ′ , ψ −1 ), and if necessary all Φ ∈ S(kn ). Again, if π ′ = 1 is the trivial representation of GL1 (k) we write γ(s, π, ψ) = γ(s, π × 1, ψ). The fact that γ(s, π × π ′ , ψ) is rational follows from the fact that it is a ratio of local integrals. An equally important local factor, which occurs in the current formulations of the local Langlands correspondence [23, 26], is the local ε-factor.

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Definition The local factor ε(s, π × π ′ , ψ) is defined as the ratio ε(s, π × π ′ , ψ) =

γ(s, π × π ′ , ψ)L(s, π × π ′ ) . L(1 − s, π e×π e′ )

With the local ε-factor the local functional equation can be written in the form e − s; ρ(wn,m )W f, W f′) Ψ(1 Ψ(s; W, W ′ ) ′ n−1 ′ = ω (−1) ε(s, π ×π , ψ) L(1 − s, π e×π e′ ) L(s, π × π ′ )

if m < n

or

f, W f ′ , Φ) ˆ Ψ(s; W, W ′ , Φ) Ψ(1 − s; W ′ n−1 ′ = ω (−1) ε(s, π × π , ψ) L(1 − s, π e×π e′ ) L(s, π × π ′ )

if m = n .

This can also be expressed in terms of the e(s; W, W ′ ), etc.. In fact, since we know we can choose a finite set of Wi , Wi′ , and if necessary Φi so that X Ψ(s, ; Wi , W ′ )

=

X Ψ(s; Wi , W ′ , Φi )

=

i

i

or

L(s, π × π ′ )

i

i

L(s, π × π ′ )

X

e(s; Wi , Wi′ ) = 1

i

X

e(s; Wi , Wi′ , Φi ) = 1

i

we see that we can write either ε(s, π × π ′ , ψ) = ω ′ (−1)n−1

X i

or ε(s, π × π ′ , ψ) = ω ′ (−1)n−1

fi , W f′) e˜(1 − s; ρ(wn,m )W i

X i

π ′ , ψ)

C[q s , q −s ].

and hence ε(s, π × ∈ functional equation twice we get

fi , W f′ , Φ ˆ i) e(1 − s; W i

On the other hand, applying the

ε(s, π × π ′ , ψ)ε(1 − s, π e×π e′ , ψ −1 ) = 1

so that ε(s, π × π ′ , ψ) is a unit in C[q s , q −s ]. This can be restated as: Proposition 3.5 ε(s, π × π ′ , ψ) is a monomial function of the form cq −f s .

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Let us make a few remarks on the meaning of the number f occurring in the ε–factor in the case of a single representation. Assume that ψ is unramified. In this case write ε(s, π, ψ) = ε(0, π, ψ)q −f (π)s . In [34] it is shown that f (π) is a non-negative integer, f (π) = 0 iff π is unramified, that in general the space of vectors in Vπ which is fixed by the compact open subgroup f (π) ∗ K1 (p ) = g ∈ GLn (o) g ≡ 0 ···

∗ .. . ∗ 0 1

f (π) (mod p )

has dimension exactly 1, and that if t < f (π) then the dimension of the space of fixed vectors for K1 (pt ) is 0. Depending on the context, either the integer f (π) or the ideal f(π) = pf (π) is called the conductor of π. Note that the analytically defined ε-factor carries structural information about π. 3.1.3

The unramified calculation

Let us now turn to the calculation of the local L-functions. The first case to consider is that where both π and π ′ are unramified. Since they are assumed generic, they are both full induced representations from unramified n characters of the Borel subgroup [69]. So let us write π ≃ IndGL Bn (µ1 ⊗ ′ ′ ′ m · · · ⊗ µn ) and π ′ ≃ IndGL Bm (µ1 ⊗ · · · ⊗ µm ) with the µi and µj unramified characters of k× . The Satake parameterization of unramified representations associates to each of these representations the semi-simple conjugacy classes [Aπ ] ∈ GLn (C) and [Aπ′ ] ∈ GLm (C) given by

Aπ =

µ1 (̟) ..

. µn (̟)

Aπ′ =

µ′1 (̟)

..

.

. ′ µm (̟)

(Recall that ̟ is a uniformizing parameter for k, that is, a generator of p.) In the Whittaker models there will be unique normalized K = GL(o)– fixed Whittaker functions, W◦ ∈ W(π, ψ) and W◦′ ∈ W(π ′ , ψ −1 ), normalized by W◦ (e) = W◦′ (e) = 1. Let us concentrate on W◦ for the moment. Since this function is right Kn –invariant and transforms on the left by ψ under Nn we have that its values are completely determined by its values on diagonal

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matrices of the form

̟ j1

̟J =

..

. ̟ jn

for J = (j1 , . . . , jn ) ∈ Zn . There is an explicit formula for W◦ (̟ J ) in terms of the Satake parameter Aπ due to Shintani [63] for GLn and generalized to arbitrary reductive groups by Casselman and Shalika [4]. Let T + (n) be the set of n–tuples J = (j1 , . . . , jn ) ∈ Zn with j1 ≥ · · · ≥ jn . Let ρJ be the rational representation of GLn (C) with dominant weight ΛJ defined by t1 j .. ΛJ = t11 · · · tjnn . . tn

Then the formula of Shintani says that ( 0 W◦ (̟ J ) = 1/2 δBn (̟ J ) tr(ρJ (Aπ ))

if J ∈ / T + (n) if J ∈ T + (n)

under the assumption that ψ is unramified. This is proved by analyzing the recursion relations coming from the action of the unramified Hecke algebra on W◦ . We have a similar formula for W◦′ (̟ J ) for J ∈ Zm . If we use these formulas in our local integrals, we find [36, I, Prop. 2.3] J X ̟ ′ Ψ(s; W◦ , W◦ ) = W◦ W◦′ (̟ J ) I n−m + J∈T (m), jm ≥0

−1 (̟ J ) · | det(̟ J )|s−(n−m)/2 δB m X = tr(ρ(J,0) (Aπ )) tr(ρJ (Aπ′ ))q −|J|s J∈T + (m), jm ≥0

=

X

tr(ρ(J,0) (Aπ ) ⊗ ρJ (Aπ′ ))q −|J|s

J∈T + (m), jm ≥0

where we let |J| = j1 +· · ·+jm and we embed Zm ֒→ Zn by J = (j1 , · · · , jm ) 7→ (J, 0) = (j1 , · · · , jm , 0, · · · , 0). We now use the fact from invariant theory that X tr(ρ(J,0) (Aπ ) ⊗ ρJ (Aπ′ )) = tr(S r (Aπ ⊗ Aπ′ )), J∈T + (m), jm ≥0, |J|=r

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where S r (A) is the r th -symmetric power of the matrix A, and ∞ X

tr(S r (A))z r = det(I − Az)−1

r=0

for any matrix A. Then we quickly arrive at Ψ(s; W◦ , W◦′ ) = det(I − q −s Aπ ⊗ Aπ′ )−1 =

Y (1 − µi (̟)µ′j (̟)q −s )−1 i,j

a standard Euler factor of degree mn. Since the L-function cancels all poles of the local integrals, we know at least that det(I − q −s Aπ ⊗ Aπ′ ) divides L(s, π × π ′ )−1 . Either of the methods discussed below for the general calculation of local factors then shows that in fact these are equal. There is a similar calculation when n = m and Φ = Φ◦ is the characteristic function of the lattice on ⊂ kn . Also, since π unramified implies that f◦ as its normalized unramified its contragredient π e is also unramified, with W Whittaker function, then from the functional equation we can conclude that in this situation we have ε(s, π × π ′ , ψ) ≡ 1. Theorem 3.3 If π, π ′ , and ψ are all unramified, then ( Ψ(s; W◦ , W◦′ ) −1 ′ −s L(s, π × π ) = det(I − q Aπ ⊗ Aπ′ ) = Ψ(s; W◦ , W◦′ , Φ◦ )

m 0 if π and π ′ are tempered. The meromorphic continuation and “bounded denominator” statement in the case of a non-archimedean local field is now replaced by the following. Define M(π × π ′ ) to be the space of all meromorphic functions φ(s) with the property that if P (s) is a polynomial function such that P (s)L(s, π × π ′ ) is holomorphic in a vertical strip S[a, b] = {s a ≤ Re(s) ≤ b} then P (s)φ(s) is bounded in S[a, b]. Note in particular that if φ ∈ M(π × π ′ ) then the quotient φ(s)L(s, π × π ′ )−1 is entire. Theorem 3.5 The integrals Ψj (s; W, W ′ ) or Ψ(s; W, W ′ , Φ) extend to meromorphic functions of s which lie in M(π × π ′ ). In particular, the ratios ej (s; W, W ′ ) =

Ψj (s; W, W ′ ) L(s, π × π ′ )

or

e(s; W, W ′ , Φ) =

Ψ(s; W, W ′ , Φ) L(s, π × π ′ )

are entire and in fact are bounded in vertical strips. This statement has more content than just the continuation and “bounded denominator” statements in the non-archimedean case. Since it prescribes the “denominator” to be the L factor L(s, π × π ′ )−1 it is bound up with the actual computation of the poles of the local integrals. In fact, a significant part of the paper of Jacquet and Shalika [38] is taken up with the simultaneous proof of this and the local functional equations:

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Theorem 3.6 We have the local functional equations f, W f ′ ) = ω ′ (−1)n−1 γ(s, π × π ′ , ψ)Ψj (s; W, W ′ ) Ψn−m−j−1 (1 − s; ρ(wn,m )W

or

f, W f ′ , Φ) ˆ = ω ′ (−1)n−1 γ(s, π × π ′ , ψ)Ψ(s; W, W ′ , Φ). Ψ(1 − s; W

The one fact that we are missing is the statement of “minimality” of the L-factor. That is, we know that L(s, π × π ′ ) is a standard archimedean Euler factor (i.e., a product of Γ-functions of the standard type) and has the property that the poles of all the local integrals are contained in the poles of the L-factor, even with multiplicity. But we have not established that the L-factor cannot have extraneous poles. In particular, we do know that we can achieve the local L-function as a finite linear combination of local integrals. Towards this end, Jacquet and Shalika enlarge the allowable space of local integrals. Let Λ and Λ′ be the Whittaker functionals on Vπ and Vπ′ associated ˆ = Λ ⊗ Λ′ with the Whittaker models W(π, ψ) and W(π ′ , ψ −1 ). Then Λ defines a continuous linear functional on the algebraic tensor product Vπ ⊗ Vπ′ which extends continuously to the topological tensor product Vπ⊗π′ = ˆ π′ , viewed as representations of GLn (k) × GLm (k). Vπ ⊗V Before proceeding, let us make a few remarks on smooth representations. If (π, Vπ ) is the space of smooth vectors in an irreducible admissible unitary representation, then the underlying Harish-Chandra module is the space of Kn -finite vectors Vπ,K . Vπ then corresponds to the (Casselman-Wallach) canonical completion of Vπ,K [66]. The category of Harish-Chandra modules is appropriate for the algebraic theory of representations, but it is useful to work in the category of smooth admissible representations for automorphic forms. If in our context we take the underlying Harish-Chandra modules Vπ,K and Vπ′ ,K then their algebraic tensor product is an admissible HarishChandra module for GLn (k) × GLm (k). The associated smooth admissible representation is the canonical completion of this tensor product, which is in fact Vπ⊗π′ , the topological tensor product of the smooth representations π and π ′ . It is also the space of smooth vectors in the unitary tensor product of the unitary representations associated to π and π ′ . So this completion is a natural place to work in the category of smooth admissible representations. Now let ˆ W(π ⊗ π ′ , ψ) = {W (g, h) = Λ(π(g) ⊗ π ′ (h)ξ)|ξ ∈ Vπ⊗π′ }.

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Then W(π⊗π ′ , ψ) contains the algebraic tensor product W(π, ψ)⊗W(π ′ , ψ −1 ) and is again equal to the topological tensor product. Now we can extend all our local integrals to the space W(π ⊗ π ′ , ψ) by setting Z Z h , h dx | det(h)|s−(n−m)/2 dh Ψj (s; W ) = W x Ij In−m−j and Ψ(s; W, Φ) =

Z

W (g, g)Φ(en g)| det(g)|s dh

for W ∈ W(π ⊗ π ′ , ψ). Since the local integrals are continuous with respect to the topology on the topological tensor product, all of the above facts remain true, in particular the convergence statements, the local functional equations, and the fact that these integrals extend to functions in M(π ×π ′ ). At this point, let Ij (π, π ′ ) = {Ψj (s; W )|W ∈ W(π ⊗ π ′ )} and let I(π, π ′ ) be the span of the local integrals {Ψ(s; W, Φ)|W ∈ W(π ⊗π ′ , ψ), φ ∈ S(kn )}. Once again, in the case m < n we have that the space Ij (π, π ′ ) is independent of j and we denote it also by I(π, π ′ ). These are still independent of ψ. So we know from above that I(π, π ′ ) ⊂ M(π × π ′ ). The remainder of [38] is then devoted to showing the following. Theorem 3.7 I(π, π ′ ) = M(π × π ′ ). As a consequence of this, we draw the following useful Corollary. Corollary There is a Whittaker function W in W(π ⊗ π ′ , ψ) such that L(s, π × π ′ ) = Ψ(s; W ) if m < n or finite collectionof functions Wi ∈ W(π ⊗ P π ′ , ψ) and Φi ∈ S(kn ) such that L(s, π × π ′ ) = i Ψ(s; Wi , Φi ) if m = n.

In the cases of m = n − 1 or m = n, Jacquet and Shalika can indeed get the local L-function as a finite linear combination of integrals involving only K-finite functions in W(π, ψ) and W(π ′ , ψ −1 ), that is, without going to the completion of W(π, ψ) ⊗ W(π ′ , ψ −1 ), but this has not been published. As a final result, let us note that in [12] it is established that the linear functionals e(s; W ) = Ψ(s; W )L(s, π × π ′ )−1 and e(s; W, Φ) = Ψ(s; W, Φ) L(s, π × π ′ )−1 are continuous on W(π ⊗ π ′ , ψ), uniformly for s in compact sets. Since there is a choice of W ∈ W(π ⊗ π ′ , ψ) such that e(s; W ) ≡ 1 P or Wi ∈ W(π ⊗ π ′ , ψ) and Φi ∈ S(kn ) such that e(s; Wi , Φi ) ≡ 1, as a result of this continuity and the fact that the algebraic tensor product W(π, ψ) ⊗ W(π ′ , ψ −1 ) is dense in W(π ⊗ π ′ , ψ) we have the following result.

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Proposition 3.8 For any s0 ∈ C there are choices of W ∈ W(π, ψ), W ′ ∈ W(π ′ , ψ −1 ) and if necessary Φ such that e(s0 ; W, W ′ ) 6= 0 or e(s0 ; W, W ′ , Φ) 6= 0.

4

Global L-functions

Once again, we let k be a global field, A its ring of adeles, and fix a non-trivial continuous additive character ψ = ⊗ψv of A trivial on k. Let (π, Vπ ) be a cuspidal representation of GLn (A) (see Section 1 for all the implied assumptions in this terminology) and (π ′ , Vπ′ ) a cuspidal representation of GLm (A). Since they are irreducible we have restricted tensor product decompositions π ≃ ⊗′ πv and π ′ ≃ ⊗′ πv′ with (πv , Vπv ) and (πv′ , Vπv′ ) irreducible admissible smooth generic unitary representations of GLn (kv ) and GLm (kv ) [14,18]. Let ω = ⊗′ ωv and ω ′ = ⊗′ ωv′ be their central characters. These are both continuous characters of k× \A× . Let S be the finite set of places of k, containing the archimedean places S∞ , such that for all v ∈ / S we have that πv , πv′ , and ψv are unramified. For each place v of k we have defined the local factors L(s, πv × πv′ ) and ε(s, πv × πv′ , ψv ). Then we can at least formally define Y Y L(s, π × π ′ ) = L(s, πv × πv′ ) and ε(s, π × π ′ ) = ε(s, πv × πv′ , ψv ). v

v

We need to discuss convergence of these products. Let us first consider the convergence of L(s, π × π ′ ). For those v ∈ / S, so πv , πv′ , and ψv are ′ unramified, we know that L(s, πv × πv ) = det(I − qv−s Aπv ⊗ Aπv′ )−1 and that 1/2

the eigenvalues of Aπv and Aπv′ are all of absolute value less than qv . Thus the partial (or incomplete) L-function Y Y LS (s, π × π ′ ) = L(s, πv × πv′ ) = det(I − q −s Aπv ⊗ Aπv′ )−1 v∈S /

v∈S /

is absolutely convergent for Re(s) >> 0. Thus the same is true for L(s, π × π ′ ). For the ε–factor, we have seen that ε(s, πv × πv′ , ψv ) ≡ 1 for v ∈ / S so that the product is in fact a finite product and there is no problem with convergence. The fact that ε(s, π × π ′ ) is independent of ψ can either be checked by analyzing how the local ε–factors vary as you vary ψ, as is done in [7, Lemma 2.1], or it will follow from the global functional equation presented below.

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4.1

125

The Basic Analytic Properties

Our first goal is to show that these L-functions have nice analytic properties. Theorem 4.1 The global L–functions L(s, π × π ′ ) are nice in the sense that 1. L(s, π × π ′ ) has a meromorphic continuation to all of C, 2. the extended function is bounded in vertical strips (away from its poles), 3. they satisfy the functional equation L(s, π × π ′ ) = ε(s, π × π ′ )L(1 − s, π e×π e′ ).

To do so, we relate the L-functions to the global integrals. Let us begin with continuation. In the case m < n for every ϕ ∈ Vπ and ′ ϕ ∈ Vπ′ we know the integral I(s; ϕ, ϕ′ ) converges absolutely for all s. From the unfolding in Section 2 and the local calculation of Section 3 we know that for Re(s) >> 0 and for appropriate choices of ϕ and ϕ′ we have Y I(s; ϕ, ϕ′ ) = Ψv (s; Wϕv , Wϕ′v ) v

=

Y

!

Ψv (s; Wϕv , Wϕ′v ) LS (s, π × π ′ )

v∈S

= =

! Y Ψv (s; Wϕv , Wϕ′ ) v L(s, π × π ′ ) L(s, πv × πv′ ) v∈S ! Y ev (s; Wϕv , Wϕ′v ) L(s, π × π ′ ) v∈S

We know that each ev (s; Wv , Wv′ ) is entire. Hence L(s, π × π ′ ) has a meromorphic continuation. If m = n then for appropriate ϕ ∈ Vπ , ϕ′ ∈ Vπ′ , and Φ ∈ S(An ) we again have ! Y ′ ′ ev (s; Wϕv , Wϕ′v , Φv ) L(s, π × π ′ ). I(s; ϕ, ϕ , Φ) = v∈S

Once again, since each ev (s; Wv , Wv′ , Φv ) is entire, L(s, π × π ′ ) has a meromorphic continuation.

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Let us next turn to the functional equation. This will follow from the functional equation for the global integrals and the local functional equations. We will consider only the case where m < n since the other case is entirely analogous. The functional equation for the global integrals is simply ˜ − s; ϕ, I(s; ϕ, ϕ′ ) = I(1 e ϕ e′ ).

Once again we have for appropriate ϕ and ϕ′ ′

I(s; ϕ, ϕ ) =

Y

!

ev (s; Wϕv , Wϕ′ ′v )

L(s, π × π ′ )

v∈S

while on the other side ˜ − s; ϕ, I(1 e ϕ e′ ) =

Y

!

v∈S

fϕv , W f ′ ′ ) L(1 − s, π e˜v (1 − s; ρ(wn,m )W e×π e′ ). ϕv

However, by the local functional equations, for each v ∈ S we have e f f′ fv , W fv′ ) = Ψ(1 − s; ρ(wn,m )Wv , Wv ) e˜v (1 − s; ρ(wn,m )W L(1 − s, π e×π e′ )

Ψ(s; Wv , Wv′ ) L(s, π × π ′ ) ′ n−1 ′ = ωv (−1) ε(s, πv × πv , ψv )ev (s, Wv , Wv′ )

= ωv′ (−1)n−1 ε(s, πv × πv′ , ψv )

Combining these, we have ′

L(s, π × π ) =

Y

ωv′ (−1)n−1 ε(s, πv

×

!

πv′ , ψv )

v∈S

L(1 − s, π e×π e′ ).

Now, for v ∈ / S we know that πv′ is unramified, so ωv′ (−1) = 1, and also that ′ ε(s, πv × πv , ψv ) ≡ 1. Therefore Y Y ωv′ (−1)n−1 ε(s, πv × πv′ , ψv ) = ωv′ (−1)n−1 ε(s, πv × πv′ , ψv ) v ′

v∈S

= ω (−1)n−1 ε(s, π × π ′ ) = ε(s, π × π ′ ) and we indeed have L(s, π × π ′ ) = ε(s, π × π ′ )L(1 − s, π e×π e′ ).

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127

Note that this implies that ε(s, π × π ′ ) is independent of ψ as well. Let us now turn to the boundedness in vertical strips. For the global integrals I(s; ϕ, ϕ′ ) or I(s; ϕ, ϕ, Φ) this simply follows from the absolute convergence. For the L-function itself, the paradigm is the following. For every ′ , and Φ finite place v ∈ S we know that there is a choice of Wv,i , Wv,i v,i if necessary such that L(s, πv × πv′ ) = L(s, πv × πv′ ) =

X

X

Ψ(s; Wv,i , Wv′ ′ i )

or

Ψ(s; Wv,i , Wv′ ′ i , Φv,i ).

If m = n − 1 or m = n then by the unpublished work of Jacquet and Shalika mentioned toward the end of Section 3 we know that we have similar statements for v ∈ S∞ . Hence if m = n − 1 or m = n there are global choices ϕi , ϕ′i , and if necessary Φi such that L(s, π × π ′ ) =

X

I(s; ϕi , ϕ′i )

or

L(s, π × π ′ ) =

X

I(s; ϕi , ϕ′i , Φi ).

Then the boundedness in vertical strips for the L-functions follows from that of the global integrals. However, if m < n − 1 then all we know at those v ∈ S∞ is that there ′ −1 ˆ is a function Wv ∈ W(πv ⊗ πv′ , ψv ) = W(πv , ψv )⊗W(π v , ψv ) or a finite collection of such functions Wv,i and of Φv,i such that L(s, πv × πv′ ) = I(s; Wv )

or

L(s, πv × πv′ ) =

X

I(s; Wv,i , Φv,i ).

To make the above paradigm work for m < n − 1 we would need to rework ˆ π′ . This is the theory of global Eulerian integrals for cusp forms in Vπ ⊗V naturally the space of smooth vectors in an irreducible unitary cuspidal representation of GLn (A) × GLm (A). So we would need extend the global theory of integrals parallel to Jacquet and Shalika’s extension of the local integrals in the archimedean theory. There seems to be no obstruction to carrying this out, and then we obtain boundedness in vertical strips for L(s, π × π ′ ) in general. We should point out that if one approaches these L-functions by the method of constant terms and Fourier coefficients of Eisenstein series, then Gelbart and Shahidi have shown a wide class of automorphic L-functions, including ours, to be bounded in vertical strips [17].

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4.2

Poles of L-functions

Let us determine where the global L-functions can have poles. The poles of the L-functions will be related to the poles of the global integrals. Recall from Section 2 that in the case of m < n we have that the global integrals I(s; ϕ, ϕ′ ) are entire and that when m = n then I(s; ϕ, ϕ′ , Φ) can have at most simple poles and they occur at s = −iσ and s = 1 − iσ for σ real when π≃π e′ ⊗ | det |iσ . As we have noted above, the global integrals and global L-functions are related, for appropriate ϕ, ϕ′ , and Φ, by ! Y ev (s; Wϕv , Wϕ′ ′v ) L(s, π × π ′ ) I(s; ϕ, ϕ′ ) = v∈S

or I(s; ϕ, ϕ′ , Φ) =

Y

!

ev (s; Wϕv , Wϕ′ ′v , Φv ) L(s, π × π ′ ).

v∈S

On the other hand, we have seen that for any s0 ∈ C and any v there is a choice of local Wv , Wv′ , and Φv such that the local factors ev (s0 ; Wv , Wv′ ) 6= 0 or ev (s0 ; Wv , Wv′ , Φv ) 6= 0. So as we vary ϕ, ϕ′ and Φ at the places v ∈ S we see that division by these factors can introduce no extraneous poles in L(s, π × π ′ ), that is, in keeping with the local characterization of the Lfactor in terms of poles of local integrals, globally the poles of L(s, π × π ′ ) are precisely the poles of the family of global integrals {I(s; ϕ, ϕ′ )} or {I(s; ϕ, ϕ′ , Φ)}. Hence from Theorems 2.1 and 2.2 we have. Theorem 4.2 If m < n then L(s, π × π ′ ) is entire. If m = n, then L(s, π × π ′ ) has at most simple poles and they occur iff π ≃ π e′ ⊗ | det |iσ with σ real and are then at s = −iσ and s = 1 − iσ. If we apply this with π ′ = π e we obtain the following useful corollary.

Corollary L(s, π × π e) has simple poles at s = 0 and s = 1.

4.3

Strong Multiplicity One

Let us return to the Strong Multiplicity One Theorem for cuspidal representations. First, recall the statement: Theorem (Strong Multiplicity One) Let (π, Vπ ) and (π ′ , Vπ′ ) be two cuspidal representations of GLn (A). Suppose there is a finite set of places S such that for all v ∈ / S we have πv ≃ πv′ . Then π = π ′ .

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We will now present Jacquet and Shalika’s proof of this statement via Lfunctions [36]. First note the following observation, which follows from our analysis of the location of the poles of the L-functions. Observation For π and π ′ cuspidal representations of GLn (A), L(s, π×e π′ ) has a pole at s = 1 iff π ≃ π ′ . Thus the L-function gives us an analytic method of testing when two cuspidal representations are isomorphic, and so by the Multiplicity One Theorem, the same. Proof: If we take π and π ′ as in the statement of Strong Multiplicity One, we have that πv ≃ πv′ for v ∈ / S and hence Y Y e′ ) LS (s, π × π e) = L(s, πv × π ev ) = L(s, πv × π ev′ ) = LS (s, π × π v∈S /

v∈S /

Since the local L-factors never vanish and for unitary representations they have no poles in Re(s) ≥ 1 (since the local integrals have no poles in this region) we see that for s = 1 that L(s, π × π e′ ) has a pole at s = 1 iff S ′ L (s, π × π e ) does. Hence we have that since L(s, π × π e) has a pole at S S S s = 1, so does L (s, π × π e). But L (s, π × π e) = L (s, π × π e′ ), so that S ′ ′ both L (s, π × π e ) and then L(s, π × π e ) have poles at s = 1. But then the L-function criterion above gives that π ≃ π ′ . Now apply Multiplicity One. 2 In fact, Jacquet and Shalika push this method much further. If π is an irreducible automorphic representation of GLn (A), but not necessarily cuspidal, then it is a theorem of Langlands [44] that there are cuspidal representations, say τ1 , . . . , τr of GLn1 , . . . , GLnr with n = n1 + · · ·+ nr , such that π is a constituent of Ind(τ1 ⊗ · · · ⊗ τr ). Similarly, π ′ is a constituent of Ind(τ1′ ⊗ · · · ⊗ τr′ ′ ). Then the generalized version of the Strong Multiplicity One theorem that Jacquet and Shalika establish in [36] is the following. Theorem (Generalized Strong Multiplicity One) Given π and π ′ irreducible automorphic representations of GLn (A) as above, suppose that / S. Then there is a finite set of places S such that πv ≃ πv′ for all v ∈ r = r ′ and there is a permutation σ of the set {1, . . . , r} such that ni = n′σ(i) ′ . and τi = τσ(i) Note, the cuspidal representations τi and τi′ need not be unitary in this statement.

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Non-vanishing Results

Of interest for questions from analytic number theory, for example questions of equidistribution, are results on the non-vanishing of L-functions. The fundamental non-vanishing result for GLn is the following theorem of Jacquet and Shalika [35] and Shahidi [56, 57]. Theorem 4.3 Let π and π ′ be cuspidal representations of GLn (A) and GLm (A). Then the L-function L(s, π × π ′ ) is non-vanishing for Re(s) ≥ 1. The proof of non-vanishing for Re(s) > 1 is in keeping with the spirit of these notes [36, I, Theorem 5.3]. Since the local L-functions are never zero, to establish the non-vanishing of the Euler product for Re(s) > 1 it suffices to show that the Euler product is absolutely convergent for Re(s) > 1, and for this it is sufficient to work with the incomplete L-function LS (s, π × π ′ ) where S is as at the beginning of this Section. Then we can write Y Y LS (s, π × π ′ ) = L(s, πv × πv′ ) = det(I − qv−s Aπv ⊗ Aπv′ )−1 v∈S /

v∈S /

with absolute convergence for Re(s) >> 0. Q Recall that an infinite product (1 + an ) is absolutely convergent iff the P associated series log(1 + an ) is absolutely convergent. Let us write ′ µv,1 µv,1 .. .. and Aπv′ = Aπv = . . . µ′v,m

µv,n

1/2

1/2

and |µ′v,j | < qv . Then X log L(s, πv × πv′ ) = − log(1 − µv,i µ′v,j qv−s )

We have seen that |µv,i | < qv

i,j

=

∞ XX (µv,i µ′v,j )d i,j d=1

dqvds

=

∞ X tr(Adπv ) tr(Adπ′ ) v

d=1

dqvds

with the sum absolutely convergent for Re(s) >> 0. Then, still for Re(s) >> 0, ∞ tr(Ad ) tr(Ad ) XX πv πv′ . log(LS (s, π × π ′ )) = ds dqv v∈S / d=1

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e we find If we apply this to π ′ = π = π log(LS (s, π × π)) =

∞ XX | tr(Adπv )|2 dqvds v∈S / d=1

which is a Dirichlet series with non-negative coefficients. By Landau’s Lemma this will be absolutely convergent up to the its first pole, which we know is at s = 1. Hence this series, and the associated Euler product L(s, π × π e), is absolutely convergent for Re(s) > 1. An application of the Cauchy–Schwatrz inequality then implies that the series ∞ XX tr(Adπv ) tr(Adπv′ ) log(LS (s, π × π ′ )) = dqvds v∈S / d=1

is also absolutely convergent for Re(s) > 1. Thus L(s, π × π ′ ) is absolutely convergent and hence non-vanishing for Re(s) > 1. To obtain the non-vanishing on the line Re(s) = 1 requires the technique of analyzing L-functions via their occurrence in the constant terms and Fourier coefficients of Eisenstein series, which we have not discussed. They can be found in the references [35] and [56, 57] mentioned above.

4.5

The Generalized Ramanujan Conjecture (GRC)

The current version of the GRC is a conjecture about the structure of cuspidal representations. Conjecture (GRC) Let π be a (unitary) cuspidal representation of GLn (A) with decomposition π ≃ ⊗′ πv . Then the local components πv are tempered representations. However, it has an interesting interpretation in terms of L-functions which is more in keeping with the origins of the conjecture. If π is cuspidal, then at every finite place v where π v is unramified we have associated µv,1 .. a semisimple conjugacy class, say Aπv = so that . µv,n

L(s, πv ) = det(I −

qv−s Aπv )−1

n Y (1 − µv,i qv−s )−1 . = i=1

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If v is an archimedean place where πv is unramified, then we can similarly write n Y Γv (s + µv,i ) L(s, π) = i=1

where

( π −s/2 Γ( 2s ) Γv (s) = 2(2π)−s Γ(s)

if kv ≃ R . if kv ≃ C

Then the statement of the GRC in these terms is Conjecture (GRC for L-functions) If π is a cuspidal representation of GLn (A) and if v is a place where πv is unramified, then |µv,i | = 1 for v non-archimedean and Re(µv,i ) = 0 for v archimedean. Note that by including the archimedean places, this conjecture encompasses not only the classical Ramanujan conjectures but also the various versions of the Selberg eigenvalue conjecture [27]. −1/2 Recall that by the Corollary to Theorem 3.3 we have the bounds qv < 1/2 |µv,i | < qv for v non-archimedean, and a similar local analysis for v archimedean would give | Re(µv,i )| < 21 . The best bound for general GLn is due to Luo, Rudnick, and Sarnak [46]. They are the uniform bounds −( 21 −

qv

1 ) n2 +1

1 − 1 2 n2 +1

≤ |µv,i | ≤ qv

if v is non-archimedean

and

1 1 − for v archimedean. 2 n2 + 1 Their techniques are global and rely on the theory of Rankin–Selberg Lfunctions as presented here, a technique of persistence of zeros in families of L-functions, and a positivity argument. | Re(µv,i )| ≤

For GL2 there has been much recent progress. The best general estimates I am aware of at present are due to Kim and Shahidi [41], who use the holomorphy of the symmetric ninth power L-function for Re(s) > 1 to obtain − 91

qv

1

< |µv,i | < qv9

for i = 1, 2, and v non-archimedean,

and Kim and Sarnak, who obtain the analogous estimate for v archimedean (with possible equality) in the appendix to [39]. For some applications, the notion of weakly Ramanujan [8] can replace knowing the full GRC.

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Definition A cuspidal representation π of GLn (A) is called weakly Ramanujan if for every ǫ > 0 there is a constant cǫ > 0 and an infinite sequence of places {vm } with the property that each πvm is unramified and the Satake parameters µvm ,i satisfy −ǫ ǫ c−1 ǫ qvm < |µvm ,i | < cǫ qvm .

For example, if we knew that all cuspidal representations on GLn (A) were weakly Ramanujan, then we would know that under Langlands liftings between general linear groups, the property of occurrence in the spectral decomposition is preserved [8]. For n = 2, 3 our techniques let us show the following. Proposition 4.1 For n = 2 or n = 3 all cuspidal representations are weakly Ramanujan.

Proof: First, let π be a cuspidal representation or GLn (A). Recall that from the absolute convergence of the Euler product for L(s, π × π) we know X X | tr(Adπ )|2 v is absolutely convergent for Re(s) > 1, so that the series dqvds d v∈S / X | tr(Aπ )|2 v that in particular we have that is absolutely convergent for qvs v∈S /

Re(s) > 1. Thus, for a set of places of positive density, we have the estimate | tr(Aπv )|2 < qvǫ for each ǫ. Since Aπv = A−1 πv for components of cuspidal representations, we have the same estimate for | tr(A−1 πv )|. In the case of n = 2 and n = 3, these estimates and the fact that | det Aπv | = |ωv (̟v )| = 1 give us estimates on the coefficients of the characteristic polynomial for Aπv . For example, if n = 3 and the characteristic ǫ/2 polynomial of Aπv is X 3 + aX 2 + bX + c then we know |a| = | tr(Aπv )| < qv , ǫ/2 |b| = | tr(A−1 πv ) det(Aπv )| < qv , and |c| = | det(Aπv )| = 1. Then an application of Rouche’s theorem gives that the roots of this polynomial all lie in the circle of radius qvǫ as long as qv > 3. Applying this to both Aπv and A−1 πv we find that for our set primes of positive density above we have the estimate qv−ǫ < |µvm ,i | < qvǫ . Thus we find that for n = 2, 3 cuspidal representations of GLn are weakly Ramanujan. 2

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The Generalized Riemann Hypothesis (GRH)

This is one of the most important conjectures in the analytic theory of Lfunctions. Simply stated, it is Conjecture (GRH) For any cuspidal representation π, all the zeros of the L-function L(s, π) lie on the line Re(s) = 12 . Even in the simplest case of n = 1 and π = 1 the trivial representation this reduces to the Riemann hypothesis for the Riemann zeta function! For an interesting survey on these and other conjectures on L-functions and their relation to number theoretic problems, we refer the reader to the survey of Iwaniec and Sarnak [27].

5

Converse Theorems

Let us return first to Hecke. Recall that to a modular form f (τ ) =

∞ X

an e2πinτ

n−1

for, say, SL2 (Z) Hecke attached an L function L(s, f ) and they were related via the Mellin transform Z ∞ f (iy)y s d× y Λ(s, f ) = (2π)−s Γ(s)L(s, f ) = 0

and derived the functional equation for L(s, f ) from the modular transformation law for f (τ ) under the modular transformation law for the transformation τ 7→ −1/τ . In his fundamental paper [24] he inverted this process by taking a Dirichlet series D(s) =

∞ X an n=1

ns

and assuming that it converged in a half plane, had an entire continuation to a function of finite order, and satisfied the same functional equation as the L-function of a modular form of weight k, then he could actually reconstruct a modular form from D(s) by Mellin inversion Z 2+i∞ X 1 −2πny f (iy) = an e = (2π)−s Γ(s)D(s)y s ds 2πi 2−i∞ i

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135

and obtain the modular transformation law for f (τ ) under τ 7→ −1/τ from the functional equation for D(s) under s 7→ k − s. This is Hecke’s Converse Theorem. In this Section we will present some analogues of Hecke’s theorem in the context of L-functions for GLn . Surprisingly, the technique is exactly the same as Hecke’s, i.e., inverting the integral representation. This was first done in the context of automorphic representation for GL2 by Jacquet and Langlands [30] and then extended and significantly strengthened for GL3 by Jacquet, Piatetski-Shapiro, and Shalika [31]. For a more extensive bibliography and history, see [10]. This section is taken mainly from our survey [10]. Further details can be found in [7, 9].

5.1

The Results

Once again, let k be a global field, A its adele ring, and ψ a fixed non-trivial continuous additive character of A which is trivial on k. We will take n ≥ 3 to be an integer. To state these Converse Theorems, we begin with an irreducible admissible representation Π of GLn (A). In keeping with the conventions of these notes, we will assume that Π is unitary and generic, but this is not necessary. It has a decomposition Π = ⊗′ Πv , where Πv is an irreducible admissible generic representation of GLn (kv ). By the local theory of Section 3, to each Πv is associated a local L-function L(s, Πv ) and a local ε-factor ε(s, Πv , ψv ). Hence formally we can form Y Y L(s, Π) = L(s, Πv ) and ε(s, Π, ψ) = ε(s, Πv , ψv ). v

v

We will always assume the following two things about Π: 1. L(s, Π) converges in some half plane Re(s) >> 0, 2. the central character ωΠ of Π is automorphic, that is, invariant under k× . Under these assumptions, ε(s, Π, ψ) = ε(s, Π) is independent of our choice of ψ [7]. Our Converse Theorems will involve twists by cuspidal automorphic representations of GLm (A) for certain m. For convenience, let us set A(m)

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to be the set of automorphic representations of GLm (A), A0 (m) the set of m a cuspidal representations of GLm (A), and T (m) = A0 (d). d=1

Let π ′ = ⊗′ π ′ v be a cuspidal representation of GLm (A) with m < n. Then again we can formally define Y Y L(s, Π×π ′ ) = L(s, Πv ×π ′ v ) and ε(s, Π×π ′ ) = ε(s, Πv ×π ′ v , ψv ) v

v

since again the local factors make sense whether Π is automorphic or not. A consequence of (1) and (2) above and the cuspidality of π ′ is that both e × πe′ ) converge absolutely for Re(s) >> 0, where L(s, Π × π ′ ) and L(s, Π e and πe′ are the contragredient representations, and that ε(s, Π × π ′ ) is Π independent of the choice of ψ. We say that L(s, Π × π ′ ) is nice if it satisfies the same analytic properties it would if Π were cuspidal, i.e., e × πe′ ) have analytic continuations to entire 1. L(s, Π × π ′ ) and L(s, Π functions of s, 2. these entire continuations are bounded in vertical strips of finite width, 3. they satisfy the standard functional equation e × πe′ ). L(s, Π × π ′ ) = ε(s, Π × π ′ )L(1 − s, Π

The basic Converse Theorem for GLn is the following.

Theorem 5.1 Let Π be an irreducible admissible representation of GLn (A) as above. Suppose that L(s, Π × π ′ ) is nice for all π ′ ∈ T (n − 1). Then Π is a cuspidal automorphic representation. In this theorem we twist by the maximal amount and obtain the strongest possible conclusion about Π. The proof of this theorem essentially follows that of Hecke [24] and Weil [67] and Jacquet–Langlands [30]. It is of course valid for n = 2 as well. For applications, it is desirable to twist by as little as possible. There are essentially two ways to restrict the twisting. One is to restrict the rank of the groups that the twisting representations live on. The other is to restrict ramification. When we restrict the rank of our twists, we can obtain the following result.

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Theorem 5.2 Let Π be an irreducible admissible representation of GLn (A) as above. Suppose that L(s, Π × π ′ ) is nice for all π ′ ∈ T (n − 2). Then Π is a cuspidal automorphic representation. This result is stronger than Theorem 5.1, but its proof is a bit more delicate. The theorem along these lines that is most useful for applications is one in which we also restrict the ramification at a finite number of places. Let us fix a finite set of S of finite places and let T S (m) denote the subset of T (m) consisting of representations that are unramified at all places v ∈ S. Theorem 5.3 Let Π be an irreducible admissible representation of GLn (A) as above. Let S be a finite set of finite places. Suppose that L(s, Π × π ′ ) is nice for all π ′ ∈ T S (n − 2). Then Π is quasi-automorphic in the sense that there is an automorphic representation Π′ such that Πv ≃ Π′v for all v ∈ / S. Note that as soon as we restrict the ramification of our twisting representations we lose information about Π at those places. In applications we usually choose S to contain the set of finite places v where Πv is ramified. The second way to restrict our twists is to restrict the ramification at all but a finite number of places. Now fix a non-empty finite set of places S which in the case of a number field contains the set S∞ of all archimedean places. Let TS (m) denote the subset consisting of all representations π ′ in T (m) which are unramified for all v ∈ / S. Note that we are placing a grave restriction on the ramification of these representations. Theorem 5.4 Let Π be an irreducible admissible representation of GLn (A) as above. Let S be a non-empty finite set of places, containing S∞ , such that the class number of the ring oS of S-integers is one. Suppose that L(s, Π×π ′ ) is nice for all π ′ ∈ TS (n − 1). Then Π is quasi-automorphic in the sense that there is an automorphic representation Π′ such that Πv ≃ Π′v for all v ∈ S and all v ∈ / S such that both Πv and Π′v are unramified. There are several things to note here. First, there is a class number restriction. However, if k = Q then we may take S = S∞ and we have a Converse Theorem with “level 1” twists. As a practical consideration, if we let SΠ be the set of finite places v where Πv is ramified, then for applications we usually take S and SΠ to be disjoint. Once again, we are losing all information at those places v ∈ / S where we have restricted the ramification unless Πv was already unramified there.

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The proof of Theorem 5.1 essentially follows the lead of Hecke, Weil, and Jacquet–Langlands. It is based on the integral representations of Lfunctions, Fourier expansions, Mellin inversion, and finally a use of the weak form of Langlands spectral theory. For Theorems 5.2, 5.3, and 5.4, where we have restricted our twists, we must impose certain local conditions to compensate for our limited twists. For Theorem 5.2 and 5.3 there are a finite number of local conditions and for Theorem 5.4 an infinite number of local conditions. We must then work around these by using results on generation of congruence subgroups and either weak or strong approximation.

5.2

Inverting the Integral Representation

Let Π be as above and let ξ ∈ VΠ be a decomposable vector in the space VΠ of Π. Since Π is generic, then fixing local Whittaker models W(Πv , ψv ) at all places, compatibly normalized at the unramified places, we can associate Q to ξ a non-zero function Wξ (g) = Wξv (gv ) on GLn (A) which transforms by the global character ψ under left translation by Nn (A), i.e., Wξ (ng) = ψ(n)Wξ (g). Since ψ is trivial on rational points, we see that Wξ (g) is left invariant under Nn (k). We would like to use Wξ to construct an embedding of VΠ into the space of (smooth) automorphic forms on GLn (A). The simplest idea is to average Wξ over Nn (k)\ GLn (k), but this will not be convergent. However, if we average over the rational points of the mirabolic P = Pn then the sum X Uξ (g) = Wξ (pg) Nn (k)\ P(k)

is absolutely convergent. For the relevant growth properties of Uξ see [7]. Since Π is assumed to have automorphic central character, we see that Uξ (g) is left invariant under both P(k) and the center Zn (k). Suppose now that we know that L(s, Π × π ′ ) is nice for all π ′ ∈ T (m). Then we will hope to obtain the remaining invariance of Uξ from the GLn × GLm functional equation by inverting the integral representation for L(s, Π× π ′ ). With this in mind, let Q = Qm be the mirabolic subgroup of GLn which stabilizes the standard unit vector tem+1 , that is, the column vector all of whose entries are 0 except the (m + 1)th , which is 1. Note that if m = n − 1 then Q is nothing more than the opposite mirabolic P =t P−1 to P. If we let

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139

αm be the permutation matrix in GLn (k) given by 1 αm = Im In−m−1 −1 then Qm = α−1 m αn−1 Pαn−1 αm is a conjugate of P and for any m we have that P(k) and Q(k) generate all of GLn (k). So now set X Vξ (g) = Wξ (αm qg) N′ (k)\ Q(k)

where N′ = α−1 m Nn αm ⊂ Q. This sum is again absolutely convergent and is invariant on the left by Q(k) and Z(k). Thus, to embed Π into the space of automorphic forms it suffices to show Uξ = Vξ , for the we get invariance of Uξ under all of GLn (k). It is this that we will attempt to do using the integral representations. Now let (π ′ , Vπ′ ) be an irreducible subrepresentation of the space of automorphic forms on GLm (A) and assume ϕ′ ∈ Vπ′ is also factorizable. Let Z h n ′ Pm Uξ ϕ′ (h)| det(h)|s−1/2 dh. I(s; Uξ , ϕ ) = 1 GLm (k)\ GLm (A) This integral is always absolutely convergent for Re(s) >> 0, and for all s if π ′ is cuspidal. As with the usual integral representation we have that this unfolds into the Euler product Z h 0 ′ Wξ Wϕ′ ′ (h)| det(h)|s−(n−m)/2 dh I(s; Uξ , ϕ ) = 0 I n−m Nm (A)\ GLm (A) YZ hv 0 Wξv = Wϕ′ ′ v (hv )| det(hv )|vs−(n−m)/2 dhv 0 I n−m Nm (kv )\ GLm (kv ) v Y = Ψv (s; Wξv , Wϕ′ ′ v ). v

Note that unless π ′ is generic, this integral vanishes. Assume first that π ′ is cuspidal. Then from the local theory of Lfunctions from Section 3, for almost all finite places we have Ψv (s; Wξv , Wϕ′ ′ ) v = L(s, Πv ×π ′ v ) and for the other places Ψv (s; Wξv , Wϕ′ ′ v ) = ev (s; Wξv , Wϕ′ ′ v ) L(s, Πv ×π ′ v ) with the ev (s; Wξv , Wϕ′ ′ v ) entire and bounded in vertical strips. So in this case we have I(s; Uξ , ϕ′ ) = e(s)L(s, Π × π ′ ) with e(s) entire and

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bounded in vertical strips. Since L(s; Π × π ′ ) is assumed to be nice we may conclude that I(s; Uξ , ϕ′ ) has an analytic continuation to an entire function which is bounded in vertical strips. When π ′ is not cuspidal, it is a subrepresentation of a representation that is induced from (possibly non-unitary) P cuspidal representations σi of GLri (A) for ri < m with ri = m and is in fact, if our integral doesn’t vanish, the unique generic constituent of this induced representation. Then we can make a similar argument using this induced representation and the fact that the L(s, Π × σi ) are nice to again Q conclude that for all π ′ , I(s; Uξ , ϕ′ ) = e(s)L(s, Π × π ′ ) = e′ (s) L(s, Π × σi ) is entire and bounded in vertical strips. (See [7] for more details on this point.) Similarly, consider I(s; Vξ , ϕ′ ) for ϕ′ ∈ Vπ′ with π ′ an irreducible subrepresentation of the space of automorphic forms on GLm (A), still with Z h n ′ Pm Vξ ϕ′ (h)| det(h)|s−1/2 dh. I(s; Vξ , ϕ ) = 1 GLm (k)\ GLm (A) Now this integral converges for Re(s) 1, then either m1 ≤ n − 2 or mr ≤ n − 2 (or both). For simplicity assume mr ≤ n − 2. Let S be a finite set of places containing all archimedean places, v0 , v1 , SΠ , and Sσi for each i. Taking

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π′ = σ er ∈ T (n − 2), we have the equality of partial L-functions

LS (s, Π × π ′ ) = LS (s, Π0 × π ′ ) = LS (s, Π1 × π ′ ) Y Y = LS (s, σi × π ′ ) = LS (s + ti − tr , σi◦ × σ er◦ ). i

i

Now LS (s, σr × σ er ) has a pole at s = 1 and all other terms are non-vanishing at s = 1. Hence L(s, Π × π ′ ) has a pole at s = 1 contradicting the fact that L(s, Π × π ′ ) is nice. If m1 ≤ 2, then we can make a similar argument using e × σ1 ). So in fact we must have r = 1 and Π0 = Π1 = Ξ is cuspidal. L(s, Π Since Π0 agrees with Π at v1 and Π1 agrees with Π at v0 we see that in fact Π = Π0 = Π1 and Π is indeed cuspidal automorphic. 5.3.3

Theorem 5.3

Now consider Theorem 5.3. Since we have restricted our ramification, we no longer know that L(s, Π × π ′ ) is nice for all π ′ ∈ T (n − 2) and so Proposition 5.1 above is not immediately applicable. In this case, for each place v ∈ S we fix a vector ξv′ ∈ VΠv as in the above Lemma. (So we must assume we Q have chosen ψ so it is unramified at the places in S.) Let ξS′ = v∈S ξv′ ∈ ΠS . Consider now only vectors ξ of the form ξ S ⊗ ξS′ with ξ S arbitrary h ′ n and in VΠS and ξS fixed. For these vectors, the functions Pn−2 Uξ 1 h Pnn−2 Vξ are unramified at the places v ∈ S, so that the integrals 1 I(s; Uξ , ϕ′ ) and I(s; Vξ , ϕ′ ) vanish unless ϕ′ (h) is also unramified at those places in S. In particular, if π ′ ∈ T (n − 2) but π ′ ∈ / T S (n − 2) these integrals will vanish for all ϕ′ ∈ Vπ′ . So now, for this fixed class of ξ we actually have I(s; Uξ , ϕ′ ) = I(s; Vξ , ϕ′ ) for all ϕ′ ∈ Vπ′ for all π ′ ∈ T (n − 2). Hence, as before, Pnn−2 Uξ (In−1 ) = Pnn−2 Vξ (In−1 ) for all such ξ. Now we proceed as before. Our Fourier expansion argument is a bit more subtle since we have to work around our local conditions, which now have been imposed before this step, but we do obtain that Uξ (g) = Vξ (g) for all Q g ∈ G′ = ( v∈S K00,v (pnv v )) GS . The generation of congruence subgroups goes as before. We then use weak approximation as above, but then take for Π′ any constituent of the extension of ΠS to an automorphic representation of GLn (A). There is no use of strong multiplicity one nor any further use of the L-function in this case. More details can be found in [9].

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Let us now sketch the proof of Theorem 5.4. We fix a non-empty finite set of places S, containing all archimedean places, such that the ring oS of Sinteger has class number one. Recall that we are now twisting by all cuspidal representations π ′ ∈ TS (n − 1), that is, π ′ which are unramified at all places v∈ / S. Since we have not twisted by all of T (n − 1) we are not in a position to apply Proposition 5.1. To be able to apply that, we will have to place local conditions at all v ∈ / S. We begin by recalling the definition of the conductor of a representation Πv of GLn (kv ) and the conductor (or level) of Π itself. Let Kv = GLn (ov ) be the standard maximal compact subgroup of GLn (kv ). Let pv ⊂ ov be the unique prime ideal of ov and for each integer mv ≥ 0 set ∗ .. mv mv . ∗ K0,v (pv ) = g ∈ GLn (ov ) g ≡ (mod p ) ∗ 0 ··· 0 ∗

mv mv v and K1,v (pm v ) = {g ∈ K0,v (pv ) | gn,n ≡ 1 (mod pv ))}. Note that for mv = 0 we have K1,v (p0v ) = K0,v (p0v ) = Kv . Then for each local component v Πv of Π there is a unique integer mv ≥ 0 such that the space of K1,v (pm v )– fixed vectors in Πv is exactly one. For almost all v, mv = 0. We take the ideal Q mv v pm v = f(Πv ) as the conductor of Πv . Then the ideal n = f(Π) = v pv ⊂ o is called the conductor of Π. For each place v we fix a non-zero vector ξv◦ ∈ Πv v which is fixed by K1,v (pm v ), which at the unramified places is taken to be the vector with respect to which the restricted tensor product Π = ⊗′ Πv is ◦ ◦ v taken. Note that for g ∈ K0,v (pm v ) we have Πv (g)ξv = ωΠv (gn,n )ξv . Now fix a non-empty finite set of places S, containing the archimedean Q places if there are any. As is standard, we will let GS = v∈S GLn (kv ), Q (k ), Π = ⊗ Π , ΠS = ⊗′v∈S GS = v∈S / GL / Πv , etc. Then the compact Qn v mv S S v∈S v S ⊂ k or the ideal it determines nS = k ∩ kS nS ⊂ oS subring n = v∈S p v / Q mv is called the S–conductor of Π. Let KS1 (n) = v∈S / K1,v (pv ) and similarly S ◦ ◦ S for K0 (n). Let ξ = ⊗v∈S by KS1 (n) and / ξv ∈ Π . Then this vector is fixed Q transforms by a character under KS0 (n). In particular, since v∈S / GLn−1 (ov ) h embeds in KS1 (n) via h 7→ we see that when we restrict ΠS to GLn−1 1 the vector ξ ◦ is unramified. Now let us return to the proof of Theorem 5.4 and in particular the version of Proposition 5.1 we can salvage. For every vector ξS ∈ ΠS consider

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the functions UξS ⊗ξ ◦ and VξS ⊗ξ ◦ . When we restrict these functions to GLn−1 they become unramified for all places v ∈ / S. Hence we see that the integrals I(s; UξS ⊗ξ ◦ , ϕ′ ) and I(s; VξS ⊗ξ ◦ , ϕ′ ) vanish identically if the function ϕ′ ∈ Vπ′ is not unramified for v ∈ / S, and in particular if ϕ′ ∈ Vπ′ for π ′ ∈ T (n−1) but ′ π ∈ / TS (n − 1). Hence, for vectors of the form ξ = ξS ⊗ ξ ◦ we do indeed have that I(s; UξS ⊗ξ ◦ , ϕ′ ) = I(s; VξS ⊗ξ ◦ , ϕ′ ) for all ϕ′ ∈ Vπ′ and all π ′ ∈ T (n − 1). Hence, as in Proposition 5.1 we may conclude that UξS ⊗ξ ◦ (In ) = VξS ⊗ξ ◦ (In ) for all ξS ∈ VΠS . Moreover, since ξS was arbitrary in VΠS and the fixed vector ξ ◦ transforms by a character of KS0 (n) we may conclude that UξS ⊗ξ ◦ (g) = VξS ⊗ξ ◦ (g) for all ξS ∈ VΠS and all g ∈ GS KS0 (n). What invariance properties of the function UξS ⊗ξ ◦ have we gained from our equality with VξS ⊗ξ ◦ . Let us let Γi (nS ) = GLn (k) ∩ GS KSi (n) which we may view naturally as congruence subgroups of GLn (oS ). Now, as a function on GS KS0 (n), UξS ⊗ξ ◦ (g) is naturally left invariant under Γ0,P (nS ) = Z(k) P(k) ∩ GS KS0 (n) while VξS ⊗ξ ◦ (g) is naturally left invariant under Γ0,Q (nS ) = Z(k) Q(k) ∩ GS KS0 (n) where Q = Qn−1 . Similarly we set Γ1,P (nS ) = Z(k) P(k) ∩ GS KS1 (n) and Γ1,Q (nS ) = Z(k) Q(k) ∩ GS KS1 (n). The crucial observation for this Theorem is the following result. Proposition The congruence subgroup Γi (nS ) is generated by Γi,P (nS ) and Γi,Q (nS ) for i = 0, 1. This proposition is a consequence of results in the stable algebra of GLn due to Bass which were crucial to the solution of the congruence subgroup problem for SLn by Bass, Milnor, and Serre. This is the reason for the restriction to n ≥ 3 in the statement of Theorem 5.4. From this we get not an embedding of Π into a space of automorphic forms on GLn (A), but rather an embedding of ΠS into a space of classical automorphic forms on GS . To this end, for each ξS ∈ VΠS let us set ΦξS (gS ) = UξS ⊗ξ ◦ ((gS , 1S )) = VξS ⊗ξ ◦ ((gS , 1S )) for gS ∈ GS . Then ΦξS will be left invariant under Γ1 (nS ) and transform by a Nebentypus character χS under Γ0 (nS ) determined by the central character ωΠS of ΠS . Furthermore, it will transform by a character ωS = ωΠS under the center Z(kS ) of GS . The requisite growth properties are satisfied and hence the map ξS 7→ ΦξS defines an embedding of ΠS into the space A(Γ0 (nS )\ GS ; ωS , χS ) of classical automorphic forms on GS relative to the congruence subgroup Γ0 (nS ) with Nebentypus χS and central character ωS .

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We now need to lift our classical automorphic representation back to an adelic one and hopefully recover the rest of Π. By strong approximation for GLn and our class number assumption we have the isomorphism between the space of classical automorphic forms A(Γ0 (nS )\ GS ; ωS , χS ) and the KS1 (n) invariants in A(GLn (k)\ GLn (A); ω) where ω is the central character of Π. Hence ΠS will generate an automorphic subrepresentation of the space of automorphic forms A(GLn (k)\ GLn (A); ω). To compare this to our original Π, we must check that, in the space of classical forms, the ΦξS ⊗ξ ◦ are Hecke eigenforms for a classical Hecke algebra and that their Hecke eigenvalues agree with those from Π. We do this only for those v ∈ / S which are unramified, where it is a rather standard calculation. As we have not talked about Hecke algebras, we refer the reader to [7] for details. Now if we let Π′ be any irreducible subrepresentation of the representation generated by the image of ΠS in A(GLn (k)\ GLn (A); ω), then Π′ is automorphic and we have Π′v ≃ Πv for all v ∈ S by construction and Π′v ≃ Πv for all v ∈ / S ′ by the Hecke algebra calculation. Thus we have proven Theorem 5.4.

5.4

Converse Theorems and Liftings

In this section we would like to make some general remarks on how to apply these Converse Theorems to the problem of functorial liftings [3]. In order to apply these theorems, you must be able to control the global properties of the L-function. However, for the most part, the way we have of controlling global L-functions is to associate them to automorphic forms or representations. A minute’s thought will then lead one to the conclusion that the primary application of these results will be to the lifting of automorphic representations from some group H to GLn . Suppose that H is a split classical group, π an automorphic representation of H, and ρ a representation of the L-group of H. Then we should be able to associate an L-function L(s, π, ρ) to this situation [3]. Let us assume that ρ :L H → GLn (C) so that to π should be associated an automorphic representation Π of GLn (A). What should Π be and why should it be automorphic. We can see what Πv should be at almost all places. Since we have the (arithmetic) Langlands (or Langlands-Satake) parameterization of representations for all archimedean places and those finite places where the representations are unramified [3], we can use these to associate to πv and the map

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ρv :L Hv → GLn (C) a representation Πv of GLn (kv ). If H happens to be GLm then we in principle know how to associate the representation Πv at all places now that the local Langlands conjecture has been solved for GLm [23, 26], but in practice this is still not feasible. For other situations, we do not know what Πv should be at the ramified places. We will return to this difficulty momentarily. But for now, let’s assume we can finesse this local problem and arrive at a representation Π = ⊗′ Πv such that L(s, π, ρ) = L(s, Π). Π should then be the Langlands lifting of π to GLn associated to ρ. For simplicity of exposition, let us now assume that ρ is simply the standard embedding of L H into GLn (C) and write L(s, π, ρ) = L(s, π) = L(s, Π). We have our candidate Π for the lift of π to GLn , but how to tell whether Π is automorphic. This is what the Converse Theorem lets us do. But to apply them we must first be able to define and control the twisted L-functions L(s, π × π ′ ) for π ′ ∈ T with an appropriate twisting set T from one of our Converse Theorems. This is one reason why it is always crucial to define not only the standard L-functions for H, but also the twisted versions. If we know, from the theory of L-functions of H twisted by GLm for appropriate π ′ , that L(s, π × π ′ ) is nice and L(s, π × π ′ ) = L(s, Π × π ′ ) for twists, then we can use Theorem 5.1 or 5.2 to conclude that Π is cuspidal automorphic or Theorem 5.3 or 5.4 to conclude that Π is quasi-automorphic and at least obtain a weak automorphic lifting Π′ which is verifiably the correct representation at almost all places. At this point this relies on the state of our knowledge of the theory of twisted L-functions for H. Let us return now to the (local) problem of not knowing the appropriate local lifting πv 7→ Πv at the ramified places. We can circumvent this by a combination of global and local means. The global tool is simply the following observation. Observation Let Π be as in Theorem 5.3 or 5.4. Suppose that η is a fixed (highly ramified) character of k× \A× . Suppose that L(s, Π × π ′ ) is nice for all π ′ ∈ T ⊗ η, where T is either of the twisting sets of Theorem 5.3 or 5.4. Then Π is quasi-automorphic as in those theorems. The only thing to observe, say by looking at the local or global integrals, is that if π ′ ∈ T then L(s, Π×(π ′ ⊗η)) = L(s, (Π⊗η)×π ′ ) so that applying the Converse Theorem for Π with twisting set T ⊗η is equivalent to applying the Converse Theorem for Π ⊗ η with the twisting set T . So, by either Theorem 5.3 or 5.4, whichever is appropriate, Π ⊗ η is quasi-automorphic and hence Π is as well.

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Now, if we begin with π automorphic on H(A), we will take T to be the set of finite places where πv is ramified. For applying Theorem 5.3 we want S = T and for Theorem 5.4 we want S ∩ T = ∅. We will now take η to be highly ramified at all places v ∈ T . So at v ∈ T our twisting representations are all locally of the form (unramified principal series)⊗(highly ramified character). We now need to know the following two local facts about the local theory of L-functions for H. 1. Multiplicativity of γ-factors: If π ′ v = Ind(π ′ 1,v ⊗ π ′ 2,v ), with π ′ i,v and irreducible admissible representation of GLri (kv ), then γ(s, πv × π ′ v , ψv ) = γ(s, πv × π ′ 1,v , ψv )γ(s, πv × π ′ 2,v , ψv ) and L(s, πv × π ′ v )−1 should divide [L(s, πv × π ′ 1,v )L(s, π × π ′ 2,v )]−1 . If πv = Ind(σv ⊗ πv′ ) with σv an irreducible admissible representation of GLr (kv ) and πv′ an irreducible admissible representation of H′ (kv ) with H′ ⊂ H such that GLr × H′ is the Levi of a parabolic subgroup of H, then ev × π ′ v , ψv ). γ(s, πv × π ′ v , ψv ) = γ(s, σv × π ′ v , ψv )γ(s, πv′ × π ′ v , ψv )γ(s, σ

2. Stability of γ-factors: If π1,v and π2,v are two irreducible admissible representations of H(kv ), then for every sufficiently highly ramified character ηv of GL1 (kv ) we have γ(s, π1,v × ηv , ψv ) = γ(s, π2,v × ηv , ψv ) and L(s, π1,v × ηv ) = L(s, π2,v × ηv ) ≡ 1. Once again, for these applications it is crucial that the local theory of L-functions is sufficiently developed to establish these results on the local γ-factors. As we have seen in Section 3, both of these facts are known for GLn . To utilize these local results, what one now does is the following. At the places where πv is ramified, choose Πv to be arbitrary, except that it should have the same central character as πv . This is both to guarantee that the central character of Π is the same as that of π and hence automorphic and

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to guarantee that the stable forms of the γ–factors for πv and Πv agree. Now form Π = ⊗′ Πv . We choose our character η so that at the places v ∈ T we have that the L– and γ–factors for both πv ⊗ ηv and Πv ⊗ ηv are in their stable form and agree. We then twist by T ⊗ η for this fixed character η. If π ′ ∈ T ⊗ η, then for v ∈ T , π ′ v is of the form π ′ v = Ind(µv,1 ⊗ · · · ⊗ µv,m ) ⊗ ηv with each µv,i an unramified character of kv× . So at the places v ∈ T we have γ(s, πv × π ′ v ) = γ(s, πv × (Ind(µv,1 ⊗ · · · ⊗ µv,m ) ⊗ ηv )) Y = γ(s, πv ⊗ (µv,i ηv )) (by multiplicativity) Y = γ(s, Πv ⊗ (µv,i ηv )) (by stability)

= γ(s, Πv × (Ind(µv,1 ⊗ · · · ⊗ µv,m ) ⊗ ηv )) (by multiplicativity)

= γ(s, Πv × π ′ v ) and similarly for the L-factors. From this it follows that globally we will have L(s, π×π ′ ) = L(s, Π×π ′ ) for all π ′ ∈ T ⊗η and the global functional equation for L(s, π × π ′ ) will yield the global functional equation for L(s, Π × π ′ ). So L(s, Π × π ′ ) is nice and we may proceed as before. We have, in essence, twisted away all information about π and Π at those v ∈ T . The price we pay is that we also lose this information in our conclusion since we only know that Π is quasi-automorphic. In essence, the Converse Theorem fills in a correct set of data at those places in T to make the resulting global representation automorphic.

5.5

Some Liftings

To conclude, let us make a list of some of the liftings that have been accomplished using these Converse Theorems. Some have used the above trick of multiplicativity and stability of γ–factors to handle the ramified places. Others, principally those that involve GL2 , have adopted a technique of Ramakrishnan [51] involving a sequence of base changes and descents to get a more complete handle on the ramified places. 1. The symmetric square lifting from GL2 to GL3 by Gelbart and Jacquet [15]. 2. Non-normal cubic base change for GL2 by Jacquet, Piatetski-Shapiro, and Shalika [32].

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3. The tensor product lifting from GL2 × GL2 to GL4 by Ramakrishnan [51]. 4. The lifting of generic cusp forms from SO2n+1 to GL2n , with Kim, Piatetski-Shapiro, and Shahidi [6]. 5. The tensor product lifting from GL2 × GL3 to GL6 and the symmetric cube lifting from GL2 to GL4 by Kim and Shahidi [40]. 6. The exterior square lifting from GL4 to GL6 and the symmetric fourth power lift from GL2 to GL5 by Kim [39]. For the most part, it was Theorem 5.3 that was used in each case, with the exception of (4), where a simpler variant was used requiring twists by T S (n − 1). For the non-normal cubic base change both Theorem 5.3 with n = 3 and Theorem 5.1 with n = 2 were used.

Acknowledgments Most of what I know about L-functions for GLn I have learned through my years of work with Piatetski-Shapiro. I owe him a great debt of gratitude for all that he has taught me. For several years Piatetski-Shapiro and I have envisioned writing a book on L-functions for GLn [13]. The contents of these notes essentially follows our outline for that book. In particular, the exposition in Sections 1, 2, and parts of 3 and 4 is drawn from drafts for this project. The exposition in Section 5 is drawn from the survey of our work on Converse Theorems [10]. I would also like to thank Piatetski-Shapiro for graciously allowing me to present part of our joint efforts in these notes. I would also like to thank Jacquet for enlightening conversations over the years on his work on L-functions for GLn .

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[39] H. Kim, Functoriality for the exterior square of GL4 and the symmetric fourth of GL2 , submitted (2000). [40] H. Kim and F. Shahidi, Functorial products for GL2 × GL3 and functorial symmetric cube for GL2 , submitted (2000). [41] H. Kim and F. Shahidi, Cuspidality of symmetric powers of GL(2) with applications, submitted (2000). [42] S. Kudla, The local Langlands correspondence: the non-Archimedean case, Proc. Sympos. Pure Math., 55, part 2, (1994), 365–391. [43] R.P. Langlands, Euler Products, Yale Univ. Press, New Haven, 1971. [44] R.P. Langlands, On the notion of an automorphic representation, Proc. Sympos. Pure Math., 33, part 1, (1979), 203–207. [45] R.P. Langlands, On the classification of irreducible representations of real algebraic groups, in Representation Theory and Harmonic Analysis on Semisimple Lie Groups, AMS Mathematical Surveys and Monographs, No.31, 1989, 101–170. [46] W. Luo, Z. Rudnik, and P. Sarnak, On the generalized Ramanujan conjecture for GL(n), Proc. Symp. Pure Math., 66, part 2, (1999), 301– 310. [47] I.I. Piatetski-Shapiro, Euler Subgroups, in Lie Groups and Their Representations, edited by I.M. Gelfand. John Wiley & Sons, New York– Toronto, 1971, 597–620. [48] I.I. Piatetski-Shapiro, Multiplicity one theorems, Proc. Sympos. Pure Math., 33, Part 1 (1979), 209–212. [49] I.I. Piatetski-Shapiro, Complex Representations of GL(2, K) for Finite Fields K, Contemporary Math. Vol.16, AMS, Providence, 1983. [50] D. Ramakrishnan, Pure motives and automorphic forms, Proc. Sympos. Pure Math., 55, part 2, (1991), 411–446. [51] D. Ramakrishnan, Modularity of the Rankin-Selberg L-series, and multiplicity one for SL(2), Annals of Math., 152 (2000), 45–111.

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[52] R.A. Rankin, Contributions to the theory of Ramanujan’s function τ (n) and similar arithmetical functions, I and II, Proc. Cambridge Phil. Soc., 35 (1939), 351–372. ¨ [53] B. Riemann, Uber die Anzahl der Primzahlen unter einer gegebenen Gr¨osse, Monats. Berliner Akad. (1859), 671–680. [54] A. Selberg, Bemerkungen u ¨ber eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist, Arch. Math. Naturvid. 43 (1940), 47–50. [55] F. Shahidi, Functional equation satisfied by certain L-functions. Compositio Math., 37 (1978), 171–207. [56] F. Shahidi, On non-vanishing of L-functions, Bull. Amer. Math. Soc., N.S., 2 (1980), 462–464. [57] F. Shahidi, On certain L-functions. Amer. J. Math., 103 (1981), 297– 355. [58] F. Shahidi, Fourier transforms of intertwining operators and Plancherel measures for GL(n). Amer. J. Math., 106 (1984), 67–111. [59] F. Shahidi, Local coefficients as Artin factors for real groups. Duke Math. J., 52 (1985), 973–1007. [60] F. Shahidi, On the Ramanujan conjecture and finiteness of poles for certain L-functions. Ann. of Math., 127 (1988), 547–584. [61] F. Shahidi, A proof of Langlands’ conjecture on Plancherel measures; complementary series for p-adic groups. Ann. of Math., 132 (1990), 273–330. [62] J. Shalika, The multiplicity one theorem for GL(n), Ann. Math. 100 (1974), 171–193. [63] T. Shintani, On an explicit formula for class-1 “Whittaker functions” on GLn over P-adic fields. Proc. Japan Acad. 52 (1976), 180–182. [64] J. Tate, Fourier Analysis in Number Fields and Hecke’s Zeta-Functions (Thesis, Princeton, 1950), in Algebraic Number Theory, edited by J.W.S. Cassels and A. Frolich, Academic Press, London, 1967, 305–347.

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[65] J. Tate, Number theoretic background, Proc. Symp. Pure Math., 33, part 2, 3–26. [66] N. Wallach, Real Reductive Groups, I & II, Academic Press, Boston, 1988 & 1992. ¨ [67] A. Weil, Uber die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann., 168 (1967), 149–156. [68] A. Weil, Basic Number Theory, Springer–Verlag, Berlin, 1974. [69] A. Zelevinsky, Induced representations of reductive p–adic groups, II. ´ Norm. Sup., 4e Irreducible representations of GL(n), Ann. scient. Ec. s´erie, 13 (1980), 165–210.

Representation Theory of GL(n) over Non-Archimedean Local Fields Dipendra Prasad1∗ and A. Raghuram2† 1

2

Harish-Chandra Research Institute, Jhusi, Allahabad, India School of Mathematics, Tata Institute of Fundamental Research, Colaba, Mumbai, India

Lectures given at the School on Automorphic Forms on GL(n) Trieste, 31 July - 18 August 2000

LNS0821004

∗ †

[email protected] [email protected]

Contents 1 Introduction

163

2 Generalities on representations

163

3 Preliminaries on GLn (F )

168

4 Parabolic induction

172

5 Jacquet functors

176

6 Supercuspidal representations

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7 Discrete series representations

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8 Langlands classification

189

9 Certain classes of representations 9.1 Generic representations . . . . . . . . . 9.2 Tempered representations . . . . . . . . 9.3 Unramified or spherical representations . 9.4 Iwahori spherical representations . . . .

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10 Representations of local Galois Groups

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11 The local Langlands conjecture for GLn

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References

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Representation Theory of GL(n) over Non-Archimedean Local Fields 163

1

Introduction

The aim of these notes is to give an elementary introduction to representation theory of p-adic groups via the basic example of GL(n). We have emphasised those topics which are relevant to the theory of automorphic forms. To keep these notes to a reasonable size, we have omitted many proofs which would have required a lot more preparation. We have given references throughout the notes either to original or more authoritative sources. Most of the omitted proofs can be found in the fundamental papers of BernsteinZelevinsky [2], Bushnell-Kutzko [5], or Casselman’s unpublished notes [7] which any serious student of the subject will have to refer sooner or later. Representation theory of p-adic groups is a very active area of research. One of the main driving forces in the development of the subject is the Langlands program. One aspect of this program, called the local Langlands correspondence, implies a very intimate connection between representation theory of p-adic groups with the representation theory of the Galois group of the p-adic field. This conjecture of Langlands has recently been proved by Harris and Taylor and also by Henniart for the case of GL(n). We have given an introduction to Langlands’ conjecture for GL(n) as well as some representation theory of Galois groups in the last two sections of these notes.

2

Generalities on representations

Let G denote a locally compact totally disconnected topological group. In the terminology of [2] such a group is called an l-group. For these notes the fundamental example of an l-group is the group GLn (F ) of invertible n × n matrices with entries in a non-Archimedean local field F. By a smooth representation (π, V ) of G we mean a group homomorphism of G into the group of automorphisms of a complex vector space V such that for every vector v ∈ V, the stabilizer of v in G, given by stabG (v) = {g ∈ G : π(g)v = v}, is open. The space V is called the representation space of π. By an admissible representation (π, V ) of G we mean a smooth representation (π, V ) such that for any open compact subgroup K of G, the invariants in V under K, denoted V K , is finite dimensional. Given a smooth representation (π, V ) of G, a subspace W of V is said to be stable or invariant under G if for every w ∈ W and every g ∈ G we have π(g)w ∈ W. A smooth representation (π, V ) of G is said to be irreducible if the only G stable subspaces of V are (0) and V.

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If (π, V ) and (π ′ , V ′ ) are two (smooth) representations of G then by HomG (π, π ′ ) we denote the space of G intertwining operators from V into V ′ , i.e., HomG (π, π ′ ) is the space of all linear maps f : V → V ′ such that f (π(g)v) = π ′ (g)f (v) for all v ∈ V and all g ∈ G. Let (π, V ) be a smooth representation of G. Let V ∗ denote the space of linear functionals on V, i.e., V ∗ = HomC (V, C). There is an obvious action π ∗ of G on V ∗ given by (π ∗ (g)f )(v) = f (π(g−1 )v) for g ∈ G, v ∈ V and f ∈ V ∗ . However this representation is in general not smooth. Let V ∨ denote the subspace of V ∗ which consists of those linear functionals in V ∗ whose stabilizers are open in G under the above action. It is easily seen that V ∨ is stabilized by G and this representation denoted (π ∨ , V ∨ ) is called the contragredient representation of (π, V ). The following is a basic lemma in representation theory. Lemma 2.1 (Schur’s Lemma) Let (π, V ) be a smooth irreducible admissible representation of an l-group G. Then the dimension of HomG (π, π) is one. Proof : Let A ∈ HomG (π, π). Let K be any open compact subgroup of G. Then A takes the space of K-fixed vectors V K to itself. Choose a K such that V K 6= (0). Since V K is a finite dimensional complex vector space, there exists an eigenvector for the action of A on V K , say, 0 6= v ∈ V K with Av = λv. It follows that the kernel of (A − λ1V ) is a nonzero G invariant subspace of V. Irreducibility of the representation implies that A = λ1V . 2 Corollary 2.2 Any irreducible admissible representation of an abelian lgroup is one dimensional. Corollary 2.3 On an irreducible admissible representation (π, V ) of an lgroup G the centre Z of G , operates via a character ωπ , i.e., for all z ∈ Z we have π(z) = ωπ (z)1V . (This character ωπ is called the central character of π.) Exercise 2.4 (Dixmier’s lemma) Let V be a complex vector space of countable dimension. Let Λ be a collection of endomorphisms of V acting irreducibly on V. Let T be an endomorphism of V which commutes with every element of Λ. Then prove that T acts as a scalar on V. (Hint. Think of V as a module over the field of rational functions in oneovariable C(X) with n 1 X acting on V via T and use the fact that X−λ |λ ∈ C is an uncountable

Representation Theory of GL(n) over Non-Archimedean Local Fields 165 set consisting of linearly independent elements in the C-vector space C(X).) Deduce that if G is an l-group which is a countable union of compact sets and (π, V ) is an irreducible smooth representation of G then the dimension of HomG (π, π) is one. One of the most basic ways of constructing representations is by the process of induction. Before we describe this process we need a small digression on Haar measures. Let G be an l-group. Let dl x be a left Haar measure on G. So in particular dl (ax) = dl (x) for all a, x ∈ G. For any g ∈ G the measure dl (xg) (where x is the variable) is again a left Haar measure. By uniqueness of Haar measures there is a positive real number ∆G (g) such that dl (xg) = ∆G (g)dl (x). It is easily checked that g 7→ ∆G (g) is a continuous group homomorphism and is called the modular character of G. Further dr (x) := ∆G (x)−1 dl (x) is a right Haar measure. A left Haar measure is right invariant if and only if the modular character is trivial and in this case G is said to be unimodular. Example 2.5 Let F be a non-Archimedean local field. The group G = GLn (F ) is unimodular. (In general a reductive p-adic group is unimodular.) This may be seen as follows. Note that ∆G is trivial on [G, G] as the range of ∆G is abelian. Further by the defining relation ∆G is trivial on the centre Z. (Both these remarks are true for any group.) Hence ∆G is trivial on Z[G, G]. Observe that Z[G, G] is of finite index in G and since positive reals admits no non-trivial finite subgroups we get that ∆G is trivial. Exercise 2.6 Let B be the subgroup of all upper triangular matrices in G = GLn (F ). Then show that B is not unimodular as follows. Let d∗ y be a Haar measure on F ∗ and let dx be a Haar measure on F + . The normalized absolute value on F ∗ is defined by d(ax) = |a|F dx for all a ∈ F ∗ and x ∈ F. For b ∈ B given by y1 1 1 xi,j y2 , b= . . 1 yn show that a left Haar measure on B is given by the formula : Y Y dxi,j . db = d∗ yi i

1≤i<j≤n

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Deduce that the modular character ∆B of B is given by : n−1 3−n ∆B (diag(a1 , ..., an )) = |a1 |1−n F |a2 |F ...|an |F .

Let δB = ∆−1 B . Show that δB (b) = |det(Ad(b)|Lie(U ) )|F where Ad(b)|Lie(U ) is the adjoint representation of B on the Lie algebra Lie(U ) of U which is the unipotent radical of B. (In some places δB is used as the definition of the modular character, see for e.g., [1].) For any l-group G, let ∆G denote the modular character of G. Let B be a closed subgroup of G. Let ∆ = ∆G |B · ∆−1 B . Let (σ, W ) be a smooth representation of B. To this is associated the (normalized) induced representation IndG B (σ) whose representation space is : (1) f (bg) = ∆(b)1/2 σ(b)f (g), ∀b ∈ B, ∀g ∈ G f :G→W . (2) f (hu) = f (h) for all u ∈ Uf an open set in G

The group G acts on this space by right translation, i.e., given x ∈ G and f ∈ IndG B (σ) we have (x · f )(g) = f (gx) for all g ∈ G. We remark here that throughout these notes we deal with normalized induction. By normalizing we mean the ∆1/2 factor appearing in (1) above. This is done so that unitary representations are taken to unitary representations under induction. This simplifies some formulae like that which describes the contragredient of an induced representation and complicates some other formulae like Frobenius reciprocity. This induced representation admits a subrepresentation which is also another ‘induced’ representation called that obtained by compact induction. This is denoted as indG B (σ) whose representation space is given by G indG B (σ) = {f ∈ IndB (σ) : f is compactly supported modulo B}.

The following theorem summarises the basic properties of induced representations. Theorem 2.7 Let G be an l-group and let B be a closed subgroup of G. G 1. Both IndG B and indB are exact functors from the category of smooth representations of B to the category of smooth representations of G.

Representation Theory of GL(n) over Non-Archimedean Local Fields 167 2. Both the induction functors are transitive,i.e., if C is a closed subgroup B G of B then IndG B (IndC ) = IndC . A similar relation holds for compact induction. G ∨ ∨ 3. Let σ be a smooth representation of B. Then IndG B (σ) = indB (σ ).

4. (Frobenius Reciprocity) Let π be a smooth representation of G and σ be a smooth representation of B. 1/2 σ). (a) HomG (π, IndG B (σ)) = HomB (π|B , ∆ −1/2 σ, (π ∨ | )∨ ). (b) HomG (indG B B (σ), π) = HomB (∆

Both the above identifications are functorial in π and σ. The reader may refer to paragraphs 2.25, 2.28 and 2.29 of [2] for a proof of the above theorem. We recall some basic facts now about taking invariants under open compact subgroups. Let C be an open compact subgroup of an l-group G. Let H(G, C) be the space of compactly supported bi-C-invariant functions on G. This forms an algebra under convolution which is given by : Z f (xy −1 )g(y) dy (f ∗ g)(x) = G

with the identity element being vol(C)−1 eC where eC is the characteristic function of C as a subset of G. Let (π, V ) be a smooth admissible representation of G then the space of invariants V C is a finite dimensional module for H(G, C). The following proposition is not difficult to prove. The reader is urged to try and fix a proof of this. (See paragraph 2.10 of [2].) Proposition 2.8 Let the notations be as above. 1. The functor (π, V ) 7→ (π C , V C ) is an exact additive functor from the category of smooth admissible representations of G to the category of finite dimensional modules of the algebra H(G, C). 2. Let (π, V ) be an admissible representation of G such that V C 6= (0). Then (π, V ) is irreducible if and only if (π C , V C ) is an irreducible H(G, C) module. 3. Let (π1 , V1 ) and (π2 , V2 ) be two irreducible admissible representations of G such that both V1C and V2C are non-zero. We have π1 ≃ π2 as G representations if and only if π1C ≃ π2C as H(G, C) modules.

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Preliminaries on GLn (F )

In this section we begin the study of the group G = GLn (F ) and its representations. (Unless otherwise mentioned, from now on G will denote the group GLn (F ).) We begin by describing this group and some of its subgroups which are relevant for these notes. The emphasis is on various decompositions of G with respect to these subgroups. Let F be a non-Archimedean local field. Let OF be its ring of integers whose unique maximal ideal is PF . Let ̟F be a uniformizer for F, i.e., PF = ̟F OF . Let qF be the cardinality of the residue field kF = OF /PF . The group G = GLn (F ) is the group of all invertible n × n matrices with entries in F. A more invariant description is that it is the group of all invertible linear transformations of an n dimensional F vector space V and in this case it is denoted GL(V ). If we fix a basis of V then we can identify GL(V ) with GLn (F ). We let B denote the subgroup of G consisting of all upper triangular matrices. Let T denote the subgroup of all diagonal matrices and let U denote the subgroup of all upper triangular unipotent matrices. Note that T normalizes U and B is the semi-direct product T U. This B is called the standard Borel subgroup with U being its unipotent radical and T is called the diagonal torus. We let U − to be the subgroup of all lower triangular unipotent matrices. Let W , called the Weyl group of G, denote the group NG (T )/T where NG (T ) is the normalizer in G of the torus T. It is an easy exercise to check that NG (T ) is the subgroup of all monomial matrices and so W can be identified with Sn the symmetric group on n letters. We will usually denote a diagonal matrix in T as diag(a1 , ..., an ). We now introduce the Borel subgroup and more generally the parabolic subgroups in a more invariant manner. Let V be an n dimensional F vector space. Define a flag in V to be a strictly increasing sequence of subspaces W• = {W0 ⊂ W1 ⊂ · · · Wm = V }. The subgroup of GL(V ) which stabilizes the flag W• , i.e., with the property that gWi = Wi for all i is called a parabolic subgroup of G associated to the flag W• . If {v1 , ..., vn } is a basis of V then the stabilizer of the flags of the form W• = {(v1 ) ⊂ (v1 , v2 ) ⊂ · · · (v1 , v2 , ..., vn ) = V }

Representation Theory of GL(n) over Non-Archimedean Local Fields 169 is called a Borel subgroup. It can be seen that GL(V ) operates transitively on the set of such flags, and hence the stabilizer of any two such (maximal) flags are conjugate under GL(V ). If W• = {W0 ⊂ W1 ⊂ · · · Wm = V }, then inside the associated parabolic subgroup P , there exists the normal subgroup N consisting of those elements which operate trivially on Wi+1 /Wi for 0 ≤ i ≤ m − 1. The subgroup N is called the unipotent radical of P . It can be seen that there is a semi direct product decomposition P = M N with m−1 Y GL(Wi+1 /Wi ). M= i=0

The decomposition P = M N is called a Levi decomposition of P with N the unipotent radical, and M a Levi subgroup of P. We now introduce some of the open compact subgroups of G which will be relevant to us. We let K = GLn (OF ) denote the subgroup of elements G with entries in OF and whose determinant is a unit in OF . This is an open compact subgroup of G. The following exercise contains most of the basic properties of K. Exercise 3.1 Let V be an n dimensional F vector space. Let L be a lattice in V, i.e., an OF submodule of rank n. Show that stabG (L) is an open compact subgroup of G = GL(V ). If C is any open compact subgroup then show that there is a lattice L such that C ⊂ stabG (L). Deduce that up to conjugacy K is the unique maximal open compact subgroup of GLn (F ). For every integer m ≥ 1, the map OF → OF /Pm F induces a map K → m GLn (OF /PF ). The kernel of this map, denoted Km is called the principal congruence subgroup of level m. We also define K0 to be K. For all m ≥ 1, we have Km = {g ∈ GLn (OF ) : g − 1n ∈ ̟Fm Mn (OF )} . Note that Km is an open compact subgroup of G and gives a basis of neighbourhoods at the identity. Hence G is an l-group, i.e., a locally compact totally disconnected topological group. We are now in a position to state the main decomposition theorems which will be of use to us later in the study of representations of G.

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Theorem 3.2 (Bruhat decomposition) G=

a

BwB =

w∈W

a

BwU.

w∈W

Proof: This is an elementary exercise in basic linear algebra involving row reduction (on the left) and column reduction (on the right) by elementary operations. Since the rank of an element in G is n we end up with a monomial matrix with such operations and absorbing scalars into T ⊂ B we end up with an element in W. We leave the details to the reader. The disjointness of the union requires more work and is not obvious. We refer the reader to Theorem 8.3.8 of [14]. 2 Theorem 3.3 (Cartan decomposition) Let A = {diag(̟Fm1 , ..., ̟Fmn ) : mi ∈ Z, m1 ≤ m2 ≤ · · · ≤ mn }. a G= K ·a·K a∈A

Proof : Let g ∈ G. After fixing a basis for an n dimensional F vector space V we identify GL(V ) with GLn (F ). Let L be the standard lattice in V corresponding to this basis. Let L1 be the lattice g(L). Let r be the least integer such that ̟Fr L is contained in L1 and let L2 = ̟Fr L. The proof of the Cartan decomposition falls out from applying the structure theory of finitely generated torsion modules over principal ideal domains. In our context we would apply this to the torsion OF module L1 /L2 . We urge the reader to fill in the details. 2 Theorem 3.4 (Iwasawa decomposition) G=K ·B Proof : We sketch a proof for this decomposition for GL2 (F ). The proof for GLn (F ) uses induction on n and the same matrix manipulations as in the GL2 (F ) case is used in reducing from n to n− 1. Assume that G = GL2 (F ). Let g = ac db ∈ G. If a = 0 then write g as w(w−1 g) ∈ K · B. If a 6= 0 and if a−1 c ∈ OF then 1 0 ∗ ∗ g= ∈ K · B. ca−1 1 0 ∗

Representation Theory of GL(n) over Non-Archimedean Local Fields 171 If a−1 c ∈ / OF then replace g by kg where 1 1 + ̟Fr k= ∈ K. 1 ̟Fr We may choose r large enough such that the (modified) a and c satisfy the property a−1 c ∈ OF which gets us to the previous case and we may proceed as before. 2 Theorem 3.5 (Iwahori factorization) For m ≥ 1, Km = (Km ∩ U − ) · (Km ∩ T ) · (Km ∩ U ). Proof : This is an easy exercise in row and column reduction. See paragraph 3.11 of [2]. 2 We are now in a position to begin the study of representations of G = GLn (F ). The purpose of these notes is to give a detailed account of irreducible representations of G with emphasis on those topics which are relevant to the theory of automorphic forms. The first point to notice is that most representations we deal with are infinite dimensional. We leave this as the following exercise. Exercise 3.6 1. Show that the derived group of GLn (F ) is SLn (F ), the subgroup of determinant one elements. Any character χ of F ∗ gives a character g 7→ χ(det(g)) of G = GLn (F ) and every character of G looks like this. 2. Show that a finite dimensional smooth irreducible representation of G is one dimensional and hence is of the form g 7→ χ(det(g)). Note that any representation π of G can be twisted by a character χ denoted as π ⊗ χ whose representation space is the same as π and is given by (π ⊗ χ)(g) = χ(det(g))π(g). It is trivial to see that π is irreducible if and only if π ⊗ χ is irreducible and that ωπ⊗χ = ωπ χn as characters of the centre Z ≃ F ∗ . For any complex number s we will denote π(s) to be the representation π ⊗ | · |sF . Even if one is primarily interested in irreducible representations, some natural constructions such as parabolic induction defined in the next section, force us to also consider representations which are not irreducible. However

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they still would have certain finiteness properties. We end this section with a quick view into these finiteness statements. Let (π, V ) be a smooth representation of G. We call π to be a finitely generated representation if there exist finitely many vectors v1 , ..., vm in V such that the smallest G invariant subspace of V containing these vectors is V itself. We say that π has finite length if there is a sequence of G invariant subspaces (0) = V0 ⊂ V1 ⊂ · · · ⊂ Vm = V such that each successive quotient Vi+1 /Vi for 0 ≤ i ≤ m − 1 is an irreducible representation of G. The following theorem is a non-trivial theorem with a fairly long history and various mathematician’s work has gone into proving various parts of it (especially Harish-Chandra, R. Howe, H. Jacquet and I.N. Bernstein). (See Theorem 4.1 of [2]. The introduction of this same paper has some relevant history.) Theorem 3.7 Let (π, V ) be a smooth representation of G = GLn (F ). Then the following are equivalent : 1. π has finite length. 2. π is admissible and is finitely generated. Exercise 3.8

1. Prove the above theorem for G = GL1 (F ) = F ∗ .

2. Show that the sum of all characters of F ∗ on which the uniformizer ̟F acts trivially is an admissible representation which is not of finite length. 3. Let f be the characteristic function of OF× thought of as a subset of F ∗ . Let V be the representation of F ∗ generated by f inside the regular representation of F ∗ on Cc∞ (F ∗ ). Show that V gives an example of a smooth finitely generated representation which is not of finite length.

4

Parabolic induction

One important way to construct representations of G is by the process of parabolic induction. Let P = M N be a parabolic subgroup of G = GLn (F ). Recall that P is the stabilizer of some flag and its unipotent radical is the subgroup which

Representation Theory of GL(n) over Non-Archimedean Local Fields 173 acts trivially on all successive quotients. More concretely, for a partition n = n1 + n2 + · · · + nk of n, let P = P (n1 , n2 , ..., nk ) be the standard parabolic subgroup given by block upper triangular matrices : g1 ∗ ∗ ∗ g ∗ ∗ 2 : gi ∈ GLn (F ) . P = i . . gk

The unipotent radical of P is the block upper triangular unipotent matrices given by : 1n1 ∗ ∗ ∗ 1 ∗ ∗ n 2 NP = N = . . 1nk and the Levi subgroup of P is the block diagonal subgroup : g 1 k Y g 2 GLni (F ). MP = M = : gi ∈ GLni (F ) ≃ . i=1 gk

Let (ρ, W ) be a smooth representation of M. Since M is the quotient of P by N we can inflate ρ to a representation of P (sometimes referred to as ‘extending it trivially across N’) also denoted ρ. Now we can consider IndG P (ρ). (See § 2.) We say that this representation is parabolically induced from M to G. To recall, IndG P (ρ) consists of all locally constant functions f : G → Vρ such that f (pg) = δP (p)1/2 ρ(p)f (g) where δP (p) = |det(Ad(p)|Lie(N ) )|F . (See Exercise 2.6.) The following theorem contains the basic properties of parabolic induction. Theorem 4.1 Let P = M N be a parabolic subgroup of G = GLn (F ). Let (ρ, W ) be a smooth representation of M. 1. The functor ρ 7→ IndG P (ρ) is an exact additive functor from the category of smooth representations of M to the category of smooth representations of G.

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G 2. IndG P (ρ) = indP (ρ). G ∨ ∨ 3. IndG P (ρ ) ≃ IndP (ρ) .

4. If ρ is unitary then so is IndG P (ρ). 5. If ρ is admissible then so is IndG P (ρ). 6. If ρ is finitely generated then so is IndG P (ρ). Proof : General properties of induction stated in Theorem 2.7 gives (1). Iwasawa decomposition (Theorem 3.4) implies (2). Theorem 2.7 and (2) imply (3). For a proof of (4) observe that given an M invariant unitary structure (see Definition 7.1) on W we can cook up a G invariant unitary structure on IndG P (ρ) by integrating functions against each other which is justified by (2). We urge the reader to fill in the details. We now prove (5). It suffices to prove the space of vectors fixed by Km (for any m) is finite dimensional where Km is the principal congruence subgroup of level m. Let f ∈ IndG P (ρ) which is fixed by Km . Note that the Iwasawa decomposition (Theorem 3.4) gives that P \G/Km is a finite set. Let g1 , ..., gr be a set of representatives for this double coset decomposition. We may and shall choose these elements to be in K. The function f is completely determined by its 1/2 values on the elements gi . Since we have f (mgi k) = δP (m)ρ(m)f (gi ) for all m ∈ M and k ∈ Km we get that f (gi ) is fixed by M ∩ (gi Km gi−1 ) = M ∩ Km which is simply the principal congruence subgroup of level m for M. Hence each of the f (gi ) takes values in a finite dimensional space by admissibility Km is finite dimensional. of ρ which implies that IndG P (ρ) Now for the proof of (6) we give the argument to show that the space of locally constant functions on P \G is a finitely generated representation of G. The argument for general parabolically induced representations is similar. To prove that the space of locally constant functions on P \G is finitely generated, it is sufficient to treat the case when P is the Borel subgroup B of the group of upper triangular matrices. (We do this because the essence of the argument is already seen in the case of B\G. It is however true that a subrepresentation of a finitely generated representation of G is itself finitely generated. See Theorem 4.19 of [2].) The proof of finite generation of the space of locally constant functions on B\G depends on the Iwahori factorization (Theorem 3.5). Recall that if

Representation Theory of GL(n) over Non-Archimedean Local Fields 175 Km is a principal congruence subgroup of level m then we have Km = (Km ∩ U − ) · (Km ∩ T ) · (Km ∩ U ). − = K ∩ U − to We note that there are elements in T which shrink Km m the identity. For example if we take the matrix

µ = diag(1, ̟F , ̟F2 , ..., ̟Fn−1 ) − to the identity, then the powers of µ have the property that they shrink Km −i − i i.e., limi→∞ µ Km µ = {1}. Let χX denote the characteristic function of a subset X of a certain −i . This ambient space. Look at the translates of χB·Km − by the powers µ will give us, µ−i · χBKm − = χ − i = χ − i BKm µ Bµ−i Km µ −i , we get the characteristic function of Therefore translating χBKm − by µ BC for arbitrarily small open compact subgroups C. These characteristic functions together with their G translates clearly span all the locally constant functions on B\G, completing the proof of (6). 2

We would like to emphasize that even if ρ is irreducible the representation IndG P (ρ) is in general not irreducible. However the above theorem assures us, using Theorem 3.7, that it is of finite length. In general, understanding when the induced representation is irreducible is an extremely important one and this is well understood for G. We will return to this point in the section on Langlands classification (§ 8). One instance of this is given in the next example. Example 4.2 (Principal series) Let χ1 , χ2 , ..., χn be n characters of F ∗ . Let χ be the character of B associated to these characters, i.e., a1 ∗ ∗ ∗ a2 ∗ ∗ = χ1 (a1 )χ2 (a2 )...χn (an ). χ . . an The representation π(χ) = π(χ1 , ..., χn ) of G obtained by parabolically inducing χ to G is called a principal series representation of G. It turns out that π(χ) is reducible if and only if there exist i 6= j such that χi = χj | · |F where |·|F is the normalized multiplicative absolute value on F. In particular if χ1 , ..., χn are all unitary then π(χ) is a unitary irreducible representation.

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Jacquet functors

Parabolic induction constructs representations of GLn (F ) from representations of its Levi subgroups. There is a dual procedure, more precisely, an adjoint functor, which constructs representations of Levi subgroups from representations of GLn (F ). The importance and basic properties of this construction was noted by Jacquet for the GLn (F ) case, which was generalized to all reductive groups by Harish-Chandra. Definition 5.1 Let P = M N be the Levi decomposition of a parabolic subgroup P of G = GLn (F ). For a smooth representation (ρ, V ) of P , define ρN to be the largest quotient of ρ on which N operates trivially. Let ρ(N ) = V (N ) = {n · v − v|n ∈ N, v ∈ V }. Then ρN = V /V (N ). This ρN is called the Jacquet functor of ρ. If ρ is a smooth representation of G then the Jacquet functor ρN of ρ is just that of ρ restricted to P. It is easily seen that the Jacquet functor ρN is a representation for M. This is seen by noting that M normalizes N. There is an easy and important characterization of this subspace V (N ) and this is given in the following lemma. Lemma 5.2 For a smooth representation V of N , V (N ) is exactly the space of vectors v ∈ V such that Z n · v dn = 0, KN

where KN is an open compact subgroup of N and dn is a Haar measure on N . (The integral is actually a finite sum.) Proof : The main property of N used in this lemma is that it is a union of open compact subgroups. Clearly the integral of the vectors of the form n · v − v on an open compact subgroup of N containing n is zero. On the other hand if the integral equals zero, then v ∈ V (N ) as it reduces to a similar conclusion about finite groups, which is easy to see. 2 The following theorem contains most of the basic properties of Jacquet functors. The reader is urged to compare this with Theorem 4.1.

Representation Theory of GL(n) over Non-Archimedean Local Fields 177 Theorem 5.3 Let (π, V ) be a smooth representation of G. Let P = M N be a parabolic subgroup of G. Then 1. The Jacquet functor V → VN is an exact additive functor from the category of smooth representations of G to the category of smooth representations of M. 2. (Transitivity) Let Q ⊂ P be standard parabolic subgroups of G with Levi decompositions P = MP NP and Q = MQ NQ . Hence MQ ⊂ MP , NP ⊂ NQ , and MQ (NQ ∩ MP ) is a parabolic subgroup of MP with MQ as a Levi subgroup and NQ ∩ MP as the unipotent radical, and we have (πNP )NQ ∩MP ≃ πNQ . 3. (Frobenius Reciprocity) For a smooth representation σ of M, 1/2

HomG (π, IndG P (σ)) ≃ HomM (πN , σδP ). 4. If π is finitely generated then so is πN . 5. If π is admissible then so is πN . Proof : For the proof of (1) it suffices to prove that V1 ∩ V (N ) = V1 (N ) which is clear from the previous lemma. We urge the reader to fix proofs of (2) and (3). For (2) note that NQ = NP (NQ ∩ MP ). For (3) it is an exercise in modifying the proof of the usual Frobenius reciprocity while using the definitions of parabolic induction and Jacquet functors. An easy application of Iwasawa decomposition (Theorem 3.4) gives (4). Statement (5) is originally due to Jacquet. We sketch an argument below as it is important application of Iwahori factorization. Let P = P (n1 , ..., nk ) be a standard parabolic subgroup and let P = M N be its Levi decomposition. The principal congruence subgroup Km of level m admits an Iwahori factorization which looks like + − 0 Km Km Km = Km + = K ∩ N and K 0 = K ∩ M and K − = K ∩ N − where where Km m m m m m P − = M N − is the opposite parabolic subgroup obtained by simply taking transposes of all elements in P. Let A : V → VN be the canonical projection. Under this projection we show that 0 A(V Km ) = (VN )Km .

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Clearly, using admissibility of π, this would prove admissibility of πN because 0 is the principal congruence subgroup of level m of M and these form a Km basis of neighbourhoods at the identity for M. 0 Since A is a P equivariant map, we get that A(V Km ) ⊂ (VN )Km . Let 0 v¯ ∈ (VN )Km . Choose v ∈ V such that A(v) = v¯. For any z ∈ ZM , the centre of M, given by m1 ̟F 1n1 ̟Fm2 1n2 z= . mk ̟F 1nk let t(z) = maxi,j |mi − mj |. Let v1 = z −1 v. Then A(v1 ) = A(z −1 v) = 0. z −1 A(v) = z −1 v¯. Since z is central in M , A(v1 ) is also fixed by Km − z −1 ⊂ stab (v). Hence K − will fix Choose t(z) ≫ 0 such that zKm G m −1 − and A(v ) is fixed by K 0 and v1 = z v. To summarize, v1 is fixed by Km 1 m + . (The latter because N acts trivially on V .) Km N m ∩ P ) = 1. Let v2 = R Choose a Haar measure on G such that vol(K−1 ¯. The upshot is that Km π(k)v1 dk. It is easy to check that A(v2 ) = z v 0

given v¯ ∈ (VN )Km there exists z ∈ ZM such that z −1 v¯ ∈ A(V Km ). 0 Now if v¯1 , ..., v¯r are any linearly independent vectors in (VN )Km then there exists z ∈ ZM such that z −1 v¯1 , ..., z −1 v¯ are in A(V Km ) and are also 0 linearly independent. This implies that the dimension of (VN )Km is bounded 2 above by that of A(V Km ) and hence they must be equal. We now compute the Jacquet functor of the principal series representation introduced in Example 4.2. As a notational convenience for a smooth representation π of finite length of any l-group H, we denote by πss the semisimplification of π, i.e., if π = π0 ⊃ π1 ⊃ · · · ⊃ πn = {0} with each πi /πi+1 n−1 irreducible, then πss = ⊕i=0 (πi /πi+1 ). By the Jordan-Holder theorem, πss is independent of the filtration π = π0 ⊃ π1 ⊃ · · · ⊃ πn = {0}.

Theorem 5.4 (Jacquet functor for principal series) Let π be the prinGL (F ) cipal series representation π(χ) = IndB n (χ). Then the Jacquet functor of π with respect to the Borel subgroup B = T U is given as a module for T as X 1/2 (πU )ss ≃ χw δB w∈W

χw

where denotes the character of the torus obtained by twisting χ by the element w in the Weyl group W , i.e χw (t) = χ(w(t)).

Representation Theory of GL(n) over Non-Archimedean Local Fields 179 Proof : The representation space of π can be thought of as a certain space of “functions on G/B twisted by the character χ”; more precisely, π can be thought of as the space of locally constant functions on G/B with values in a sheaf Eχ obtained from the character χ of the Borel subgroup B. If Y is a closed subspace of a topological space X “of the kind that we are considering here”, e.g. locally closed subspaces of the flag variety, then there is an exact sequence, 0 → Cc∞ (X − Y, Eχ |X−Y ) → Cc∞ (X, Eχ ) → Cc∞ (Y, Eχ |Y ) → 0. It follows that Mackey’s theory (originally for finite groups) about restriction of an induced representation to a subgroup holds good for p-adic ` groups too. Hence using the Bruhat decomposition GL(n) = w∈W BwB, and denoting B ∩ wBw−1 to be T · Uw , we have X 1/2 w (ResB IndG indB B (χ))ss = B∩wBw −1 ((χδB ) ) w∈W

=

X

1/2

w indB T ·Uw ((χδB ) )

w∈W

=

X

1/2

Cc∞ (U/Uw , (χδB )w ).

w∈W

We now note that the largest quotient of Cc∞ (U/Uw ) on which U operates trivially is one dimensional (obtained by integrating a function with respect to a Haar measure on U/Uw ) on which the action of the torus T is the sum of positive roots which are not in Uw which can be seen to be [δB · −1 −w .) This implies that stands for the w translate of δB (δB )−w ]1/2 . (Here δB 1/2 w ∞ the largest quotient of Cc (U/Uw , (χδB ) ) on which U operates trivially is 1/2 the 1 dimensional T -module on which T operates by the character χw δB . Hence, i h X GL (F ) 1/2 IndB n (χ) ≃ χw δB . U ss

w∈W

2 GL (F )

GLn (F )

Corollary 5.5 If HomGLn (F ) (IndB n (χ), IndB χ′ = χw for some w in W , the Weyl group.

(χ′ )) is nonzero then

Proof : This is a simple consequence of the Frobenius reciprocity combined with the calculation of the Jacquet functor done above. 2

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Example 5.6 Let π(χ) denote a principal series representation of GL2 (F ). Then 1/2

1/2

1. If χ 6= χw , then π(χ)U ≃ χδB ⊕ χw δB . 2. If χ = χw , then π(χ)U is a non-trivial extension of T -modules: 1/2

1/2

0 → χδB → π(χ)U → χδB → 0. Exercise 5.7 With notation as in the previous exercise, prove that a principal series representation of GL2 (F ) induced from a unitary character is irreducible. Exercise 5.8 Let G be an abelian group with characters χ1 and χ2 . Prove that if χ1 6= χ2 , then any exact sequence of G-modules, 0 → χ1 → V → χ2 → 0, splits. We end this section with a theorem about Jacquet modules for parabolically induced representations. This theorem is at the basis for considering supercuspidal representations (which are those representations for which all Jacquet modules are trivial). Theorem 5.9 Let P = M N be a parabolic subgroup of G = GLn (F ). Let σ be a smooth irreducible representation of M. Let (π, V ) be any ‘subquotient’ of IndG P (ρ). Then the Jacquet module πN of π is non-zero. Proof : Note that if π is subrepresentation of the induced representation then it easily follows from Frobenius reciprocity. Indeed, we have 1/2

(0) 6= HomG (π, IndG P (ρ)) = HomM (πN , δP ρ) which implies the πN 6= (0). In general if π is a subquotient then there is a ‘trick’ due to Harish-Chandra (see Corollary 7.2.2 in [7]) using which π can be realized as a subrepresentation of a Weyl group ‘twist’ of the induced representation, which will bring us to the above case. Since this trick is very important and makes its presence in quite a few arguments we delineate it as the following theorem. 2

Representation Theory of GL(n) over Non-Archimedean Local Fields 181 We will need a little bit of notation before we can state this theorem. Let P = M N be a parabolic subgroup of G. Let x ∈ NG (M ) - the normalizer in G of M. Then we can consider the representation xρ of M whose representation space is same as that of ρ and the action is given by xρ(m) = ρ(x−1 mx). x Now we can consider the induced representation IndG P ( ρ). Note that this representation is in general not equivalent to the original induced representation. Theorem 5.10 Let P = M N be a parabolic subgroup of G = GLn (F ). Let σ be an smooth irreducible representation of M. Let (π, V ) be any irreducible subquotient of IndG P (σ). Then there exists an element w ∈ W ∩NG (M ) (where w W is the Weyl group) such that π is a subrepresentation of IndG P ( σ).

6

Supercuspidal representations

A very important and novel feature of p-adic groups (compared to real reductive groups) is the existence of supercuspidal representations. We will see that these representations are the building blocks of all irreducible admissible representations of p-adic groups. A complete set of supercuspidals for GLn (F ) was constructed by Bushnell and Kutzko in their book [5]. The local Langlands correspondence, proved by Harris and Taylor and also by Henniart [9], interprets supercuspidal representations of GLn (F ) in terms of irreducible n dimensional representations of the Galois group of F. Before we come to the definition of a supercuspidal representation, we need to define the notion of a matrix coefficient of a representation. For a smooth representation (π, V ) of GLn (F ) recall that (π ∨ , V ∨ ) denotes the contragredient representation of (π, V ). For vectors v in π and v ∨ in π ∨ , define the matrix coefficient fv,v∨ to be the function on GLn (F ) given by fv,v∨ (g) = hv ∨ , π(g)vi. Theorem 6.1 Let (π, V ) be an irreducible admissible representation of G = GLn (F ). Then the following are equivalent : 1. One matrix coefficient of π is compactly supported modulo the centre. 2. Every matrix coefficient of π is compactly supported modulo the centre. 3. The Jacquet functors of π (for all proper parabolic subgroups) are zero. 4. The representation π does not occur as a subquotient of any representation parabolically induced from any proper parabolic subgroup.

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A representation satisfying any one of the above conditions of the theorem is called a supercuspidal representation. We refer the reader to paragraph 3.21 of [2] for a proof of this theorem. We note that every irreducible admissible representation of GL1 (F ) = F ∗ is supercuspidal. However for GLn (F ) with n ≥ 2 since parabolic induction is of no use in constructing supercuspidal representations, a totally new approach is needed. One way to construct supercuspidal representations is via induction from certain finite dimensional representations of compact open subgroups. This has been a big program in recent times which has been completed for the case of GLn by Bushnell and Kutzko in their book [5]. They prove that any supercuspidal representation of GLn (F ) is obtained by induction from a finite dimensional representation of certain open compact modulo centre subgroup of GLn (F ). Here is a sample of such a construction in the simplest possible situation. These are what are called depth zero supercuspidal representations of GLn (F ). By a cuspidal representation of the finite group GLn (Fq ) we mean an irreducible representation for which all the Jacquet functors are zero (equivalently there are no non-zero vectors fixed by N (Fq ) for unipotent radicals N of any proper parabolic subgroup). Cuspidal representations of GLn (Fq ) are completely known by the work of J.Green [8] in the 50’s. Theorem 6.2 Consider a representation of GLn (Fq ) to be a representation of GLn (OF ) via the natural surjection from GLn (OF ) to GLn (Fq ). If σ is an irreducible cuspidal representation of GLn (Fq ) thought of as a representation of GLn (OF ) and χ is a character of F ∗ whose restriction to OF× is the same as the central character of σ then χ · σ is a representation of F ∗ GLn (OF ). GLn (F ) (χ · σ). Then π Let π be the compactly induced representation indF ∗ GL n (OF ) is an irreducible admissible supercuspidal representation of GLn (F ). The proof we give is based on Proposition 1.5 in [6]. To begin with we need a general lemma which describes the restriction of an induced representation in one special context that we are interested in. Lemma 6.3 Let H be an open compact-mod-centre subgroup of a unimodular l-group G. Let (σ, W ) be a smooth finite dimensional representation of H. Let π = indG H (σ) be the compact induction of σ to a representation π of G. Then the restriction of π to H is given by : M g π|H = indG (σ) ≃ IndH H H∩g −1 Hg ( σ|H∩g −1 Hg ) H g∈H\G/H

Representation Theory of GL(n) over Non-Archimedean Local Fields 183 where gσ is the representation of g−1 Hg given by gσ(g−1 hg) = σ(h) for all h ∈ H. Proof of Lemma 6.3 : For g ∈ G, let Vg = C ∞ (HgH, σ) denote the space of smooth functions f on HgH such that f (h1 gh2 ) = σ(h1 )f (gh2 ). Since HgH is open in G and is also compact mod H we get a canonical injection Vg → indG H (σ) given by extending functions by zero outside HgH. All the inclusions Vg → π gives us a canonical map M

Vg → π.

g∈H\G/H

This map is an isomorphism of H modules. This can be seen as follows : This map is clearly injective. (Consider supports of the functions.) Surjectivity follows from the definition of π. Note also that Vg is H stable if H acts by right shifts and the map Vg → π is H equivariant. g The map sending f → f¯ from Vg to IndH H∩g −1 Hg ( σ|H∩g −1 Hg ) where f¯(h) = f (gh) is easily checked to be a well-defined H equivariant bijection. This proves the lemma. 2 Proof of Theorem 6.2 : Let H denote the subgroup F ∗ GLn (OF ) which is an open and compact-mod-centre subgroup of G = GLn (F ). For brevity let σ also denote the representation χ · σ of H. We will prove irreducibility, admissibility and supercuspidality of π separately, which will prove the theorem. Irreducibility: We begin by proving irreducibility of π. We make the following claim : Claim : For all g ∈ G − H, we have HomH∩g−1 Hg (σ, gσ) = (0). Assuming the claim for the time being we prove the theorem as follows. The claim and the previous lemma implies that σ appears in π with multiplicity one, i.e., dim(HomH (σ, π|H)) = 1. Now suppose π is not irreducible then there exists an exact sequence of non-trivial G modules : 0 −→ π1 −→ π −→ π2 −→ 0. We note that since H is compact-mod-centre, any representation (like σ, π1 , π2 and π) on which F ∗ acts by a character is necessarily semi-simple as an HG module. Since π = indG H σ it embeds in IndH (σ) as a G module and hence so

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does π1 . Using Frobenius reciprocity 2.7.4(a) we get that HomH (π1 , σ) 6= (0), i.e., σ occurs in π1 . By Frobenius reciprocity 2.7.4(b) ∨ ∨ HomG (indG H σ, π2 ) = HomH (σ, (π2 |H ) ).

Since H is an open subgroup ((π2∨ )H )∨ = π2 as H-modules. Thus we get that σ occurs in π2 also but this contradicts the fact that σ occurs with multiplicity one in π. It suffices now to prove the claim. Since g ∈ G − H, we can assume without loss of generality, using the Cartan decomposition (see Theorem 3.3), m that g = diag(̟Fm1 , ..., ̟F n−1 , 1) where m1 ≥ ... ≥ mn−1 ≥ 0. Choose k such that mk ≥ 1 and mk+1 = 0. Suppose that HomH∩g−1 Hg (σ, gσ) 6= (0). Then we also have HomNk (O)∩g−1 Nk (O)g (σ, gσ) 6= (0) where Nk is the unipotent radical of the standard parabolic subgroup corresponding to the partition n = k + (n − k). We have Nk (O) ∩ gNk (O)g−1 ⊂ Nk (P) - the notations being obvious. Since σ is inflated from GLn (Fq ), gσ is trivial on Nk (O) ∩ g−1 Nk (O)g, i.e., HomNk (O)∩g−1 Nk (O)g (σ, 1) 6= (0). This contradicts the fact that (σ, W ) is a cuspidal representation of GLn (Fq ). (We urge the reader to justify this last statement.) Admissibility: We now prove that π is an admissible representation. It suffices to show that for m ≥ 1, π Km is finite dimensional, where Km is the principal congruence subgroup of level m. Note that π Km consists of locally constant functions f : G → W such that f (hgk) = σ(h)f (g), ∀h ∈ H, ∀g ∈ G, ∀k ∈ Km . Using the Cartan decomposition (see Theorem 3.3), we may choose representatives for the double cosets H\G/Km which look like g = a · k, where a = diag(̟Fm1 , · · · , ̟Fmn ) with 0 = m1 ≤ m2 ≤ · · · ≤ mn , and k ∈ K/Km . For such an a, let t(a) = max{mi+1 − mi : 1 ≤ i ≤ n − 1}. Note that if t(a) ≥ m, and if t(a) = mj+1 − mj , then Uj (O) ⊂ gKm g−1 ∩ H where Uj is the unipotent radical of the maximal parabolic corresponding to the partition n = j + (n − j). Therefore, σ(u) · f (g) = f (ug) = f (g · g−1 ug) = f (g), i.e., f (g) is a vector in W which is fixed by Uj (O) and hence Uj (Fq ). Cuspidality of (σ, W ) implies that f (g) = 0. This shows that if 0 6= f ∈ π Km , then f can be supported only on double cosets HakKm with t(a) < m, which is a finite set, proving the finite dimensionality of π Km .

Representation Theory of GL(n) over Non-Archimedean Local Fields 185 Supercuspidality: Let P = M · N be a parabolic subgroup of G. To prove that π is supercuspidal, we need to show that the Jacquet module πN = (0). It suffices to show that (πN )∗ = (0). Note that HomN (π, C) ∼ = (π ∗ )N is the space of N -fixed vectors in the vector space dual of π. We will show that (π ∗ )N = 0. ∗ Since π = indG H (σ), the dual vector space π may be identified with ∞ ∗ C (H\G, σ ) which is the space of locally constant functions φ on G with values in W ∗ such that φ(hg) = σ ∗ (h)φ(g). (We urge the reader to justify this identification; it boils down to saying that the dual of a direct sum of vector spaces is the direct product of vector spaces.) Hence (π ∗ )N consists of locally constant functions φ : G → W ∗ such that φ(hgn) = σ ∗ (h)φ(g),

∀h ∈ H, ∀g ∈ G, ∀n ∈ N.

Using Iwasawa decomposition, we may take representatives for double cosets H\G/N to lie in M . For m ∈ M , note that N (O) = N ∩ H = H ∩ mN m−1 . Hence for all h ∈ N (O), we have σ ∗ (h)φ(m) = φ(hm) = φ(m · m−1 hm) = φ(m), i.e., φ(m) ∈ W ∗ is fixed by N (Fq ). Cuspidality of (σ, W ) implies cuspidality of (σ ∗ , W ∗ ) which gives that φ(m) = 0. Hence (π ∗ )N = (0). 2 Exercise 6.4 With notations as in Theorem 6.2 show that GL (F )

GL (F )

n n (χ · σ). (χ · σ) = IndF ∗ GL indF ∗ GL n (OF ) n (OF )

We next have the following basic theorem which justifies the assertion made in the beginning of this section that supercuspidal representations are the building blocks of all irreducible representations. This theorem will be refined quite a bit in the section on Langlands classification. Theorem 6.5 Let π be an irreducible admissible representation of G = GLn (F ). Then there exists a Levi subgroup M and a supercuspidal representation ρ of M such that π is a subrepresentation of the representation of G obtained from ρ by the process of parabolic induction.

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Proof : Let P be a parabolic subgroup of GLn (F ) which is smallest for the property that the Jacquet functor with respect to P is non-zero. (So P = G is a possibility which occurs if and only if the representation is supercuspidal.) If P = M N , it is clear from theorem 5.3 that any irreducible subquotient of πN is a supercuspidal representation of M . Let ρ be an irreducible quotient of πN as an M module. Since HomM (πN , ρ) 6= (0), it follows from the Frobenius reciprocity that GLn (F )

HomGLn (F ) (π, IndP

−1/2

(δP

ρ)) 6= (0),

which gives a realization of π inside a representation parabolically induced from a supercuspidal representation. 2 Given the previous theorem one might ask if given an irreducible π ‘how many’ induced representations can it occur in? The next theorem says that π occurs in essentially only one such induced representation. Such a uniqueness assertion then allows us to talk of the supercuspidal support of the given representation. This is the following theorem due to Bernstein and Zelevinsky. (see Theorem 2.9 of [3].) We need some notation to state the theorem. If π is a representation of finite length then let JH(π) denote the set of equivalence classes of irreducible subquotients of π. Further, let JH0 (π) denote the set of all irreducible subquotients of π counted with multiplicity. So each ω in JH(π) is contained in JH0 (π) with some multiplicity. Theorem 6.6 Let P = M N and P ′ = M ′ N ′ be standard parabolic subgroups of G = GLn (F ). Let σ (resp. σ ′ ) be an irreducible supercuspidal G ′ ′ representation of M (resp. M ′ ). Let π = IndG P (σ) and π = IndP ′ (σ ). Then the following are equivalent. 1. There exists w ∈ W such that wM w−1 = M ′ and w σ = σ ′ . 2. HomG (π, π ′ ) 6= (0). 3. JH(π) ∩ JH(π ′ ) is not empty. 4. JH0 (π) = JH0 (π ′ ).

7

Discrete series representations

Note that one part of Theorem 6.1 says that every matrix coefficient of an irreducible supercuspidal representation is compactly supported modulo the

Representation Theory of GL(n) over Non-Archimedean Local Fields 187 centre. In general analytic behaviour of matrix coefficients dictate properties of the representation. Now we consider a larger class of representations which are said to be in the discrete series for G. These have a characterization in terms of their matrix coefficients being square integrable modulo the centre. We need a few definitions now. Albeit we have mentioned the notion of a unitary representation before (see Theorem 4.1) we now give a definition. Definition 7.1 Let (π, V ) be a smooth representation of G. We say π is a unitary representation of G if V has an inner product h , i (also called an unitary structure) which is G invariant, i.e., hπ(g)v, π(g)wi = hv, wi for all g ∈ G and all v, w ∈ V. In general, V equipped with h , i need only be a pre-Hilbert space. Definition 7.2 Let (π, V ) be a smooth irreducible representation of G. We say π is essentially square integrable if there is a character χ : F ∗ → R>0 such that |fv,v∨ (g)|2 χ(det g) is a function on Z\G for every matrix coefficient fv,v∨ of π, and Z |fv,v∨ (g)|2 χ(det g) dg < ∞.

Z\G

If χ can be taken to be trivial then π is said to be a square integrable representation and in this case it is said to be in the discrete series for G. Exercise 7.3 1. Let ν be a unitary character of F ∗ . Let L2 (Z\G, ν) denote the space of measurable functions f : G → C such that f (zg) = ν(z)f (g) for all z ∈ Z and g ∈ G and Z |f (g)|2 dg < ∞. Z\G

Show that L2 (Z\G, ν) is a unitary representation of G where G acts on these functions by right shifts. 2. Let (π, V ) be a discrete series representation. Check that its central character ωπ is a unitary character. Let 0 6= l ∈ V ∨ . Show that the map v 7→ fv,l is a non-zero G equivariant map from π into L2 (Z\G, ωπ ). Hence deduce that π is a unitary representation which embeds as a subrepresentation of L2 (Z\G, ωπ ). The classification of discrete series representations is due to Bernstein and Zelevinsky. We summarize the results below. To begin with we need a

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result on when a representation parabolically induced from a supercuspidal representation is reducible. (See Theorem 4.2 of [3].) Theorem 7.4 Let P = P (n1 , ..., nk ) = M N be a standard parabolic subgroup of G. Let σ = σ1 ⊗ · · · ⊗ σk be an irreducible representation of M with every σi an irreducible supercuspidal representation of GLni (F ). The parabolically induced representation IndG P (σ) is reducible if and only if there exist 1 ≤ i, j ≤ k with i 6= j, ni = nj and σi ≃ σj (1) = σj | · |F . We now consider some very specific reducible parabolically induced representations. Let n = ab and let σ be an irreducible supercuspidal representation of GLa (F ). Let P = M N be the standard parabolic subgroup corresponding to n = a + a + ... + a the sum taken b times, so M = GLa (F ) × ... × GLa (F ) the product taken b times. Let ∆ denote the segment ∆ = (σ, σ(1), ..., σ(b − 1)) thought of as the representation σ ⊗ σ(1) ⊗ ... ⊗ σ(b − 1) of M. Let IndG P (∆) denote the corresponding parabolically induced representation of G. By Theorem 7.4 this representation is reducible. We are now in a position to state the following theorem. (This theorem, as recorded in Theorem 9.3 of Zelevinsky’s paper [15], is normally attributed to Bernstein.) Theorem 7.5 With notations as above, we have : 1. For any segment ∆ the induced representation IndG P (∆) has a unique irreducible quotient. This irreducible quotient will be denoted Q(∆). 2. For any segment ∆ the representation Q(∆) is an essentially square integrable representation. Every essentially square integrable representation of G is equivalent to some Q(∆) for a uniquely determined ∆, i.e., for a uniquely determined a, b and σ. 3. The representation Q(∆) is square integrable if and only if it is unitary and also if and only if σ((b − 1)/2) is unitary. Example 7.6 (Steinberg Representation) One important example of a discrete series representation is obtained as follows. Take a = 1 and b = n with the notations as above. Hence the parabolic subgroup P is just (1−n)/2 as a (supercuspidal) the standard Borel subgroup B. Let σ = | · |F

Representation Theory of GL(n) over Non-Archimedean Local Fields 189 representation of GLa (F ) = F ∗ . Then the representation IndG P (∆) is just the regular representation of G on smooth functions on B\G. The corresponding Q(∆) is called the Steinberg representation of G. We will denote the Steinberg representation for GLn (F ) by Stn . It is a square integrable representation with trivial central character. It can also be defined as the alternating sum X Stn = (−1)rank(P ) Cc∞ (P \G), B⊂P

where B is a fixed Borel subgroup of G, P denotes a parabolic subgroup of G containing B, and rank(P ) denotes the rank of [M, M ] where M is a Levi subgroup of P . The essentially square integrable representations Q(∆) are also called generalized Steinberg representations. Exercise 7.7 Show that the Jacquet module with respect to B of the Steinberg representation of GLn (F ) is one dimensional. What is the character of T on this one dimensional space? Exercise 7.8 Show that the trivial representation of GLn (F ) is not essentially square integrable.

8

Langlands classification

In this section we state the Langlands classification for all irreducible admissible representations of GLn (F ). The supercuspidal representations were introduced in § 6. Then we used supercuspidal representations to construct all the essentially square integrable representations (the Q(∆)’s) in § 7. Not every representation is accounted for by this construction as for instance the trivial representation is not in this set (unless n = 1!). The Langlands classification builds every irreducible representation starting from the essentially square integrable ones. The theorem stated below is due to Zelevinsky. (See Theorem 6.1 of [15].) Before we can state the theorem we need the notion of when two segments are linked. Let ∆ = (σ, ..., σ(b − 1)) and ∆′ = (σ ′ , ..., σ ′ (b′ − 1)) be two segments where σ (resp. σ ′ ) is an irreducible supercuspidal representation of GLa (F ) (resp. GLa′ (F )). We say ∆ and ∆′ are linked if neither of them is included in the other and their union is a segment (so in particular a = a′ ). We say that ∆ precedes ∆′ if they are linked and there is a positive integer r such that σ ′ = σ(r).

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Theorem 8.1 (Langlands classification) For 1 ≤ i ≤ k, let ∆i be a segment for GLni (F ). Assume that for i < j, ∆i does not precede ∆j . Let n = n1 + ... + nk . Let G = GLn (F ) and let P be the parabolic subgroup of G corresponding to this partition. Then : 1. The parabolically induced representation IndG P (Q(∆1 )⊗···⊗Q(∆k )) admits a unique irreducible quotient which will be denoted Q(∆1 , ..., ∆k ). 2. Any irreducible representation of G is equivalent to some Q(∆1 , ..., ∆k ) as above. 3. If ∆′1 , ..., ∆′k′ is another set of segments satisfying the hypothesis then we have Q(∆1 , ..., ∆k ) ≃ Q(∆′1 , ..., ∆′k′ ) if and only if k = k′ and ∆i = ∆′s(i) for some permutation s of {1, ..., k}. Exercise 8.2 Give a realization of the trivial representation of G as a representation Q(∆1 , ..., ∆k ) in the Langlands classification. Exercise 8.3 (Representations of GL2 (F )) In this exercise all the irreducible admissible representations of GL2 (F ) are classified. Let G = GL2 (F ). 1. Let χ be a character of F ∗ . Then g 7→ χ(det(g)) gives a character of G. Show that these are all the finite dimensional irreducible admissible representations of G. 2. Let χ be a character of G as above. Let St2 be the Steinberg representation of G and let St2 (χ) = St2 ⊗ χ be the twist by χ. Show that any principal series representation is either irreducible or is reducible in which case, up to semi-simplification it looks like χ ⊕ St2 (χ) for some character χ. 3. Use the Langlands classification and make a list of all irreducible admissible representations of G. Show that any irreducible admissible representation is either one dimensional or is infinite dimensional and in which case it is either supercuspidal, or of the form St2 (χ) or is an irreducible principal series representation π(χ1 , χ2 ).

Representation Theory of GL(n) over Non-Archimedean Local Fields 191

9

Certain classes of representations

In this section we examine certain important classes of representations of G = GLn (F ). These notions are especially important in the theory of automorphic forms. (For readers familiar with such terms, local components of global representations tend to have such properties.)

9.1

Generic representations

Recall our notation that U is the unipotent radical of the standard Borel subgroup B = T U where T is the diagonal torus. (So U is the subgroup of all upper triangular unipotent matrices.) Since T normalizes U we get an ˆ consisting of all characters of U. A maximal orbit action of T on the space U ˆ is called a generic orbit and any character of U in a generic orbit of T on U is called a generic character of U. It is easily seen that a character of U is generic if and only if its stabilizer in T is the centre Z. We can construct (generic) characters of U as follows. Let u = u(i, j) be any element in U. Let ψF be a non-trivial additive character of U. Let a = (a1 , ..., an−1 ) be any (n − 1) tuple of elements of F. We get a character of U by u 7→ ψa (u) = ψF (a1 u(1, 2) + a2 u(2, 3) + ... + an−1 u(n − 1, n)). The reader is urged to check that every character of U is one such ψa . Further it is easily seen that ψa is generic if and only if each ai is non-zero. We will now fix one generic character of U , denoted Ψ. Fix for once and for all a non-trivial additive character ψF of F such that the maximal fractional ideal of F on which ψF is trivial is OF . We fix the following generic character of U given by: u 7→ Ψ(u) = ψF (u(1, 2) + ... + u(n − 1, n)). Now consider the representation IndG U (Ψ) which is the induction from U to G of the character Ψ. Note that the representation space of IndG U (Ψ) consists of all smooth functions f on G such that f (ng) = Ψ(n)f (g) for all n ∈ U and g ∈ G. Definition 9.1 An irreducible admissible representation (π, V ) is said to be generic if HomG (π, IndG U (Ψ)) is non-zero. If π is generic then using Frobenius reciprocity we get that there is a non-zero linear functional ℓ : V → C such

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that ℓ(π(n)v) = Ψ(n)ℓ(v) for all v ∈ V and n ∈ U. Such a linear functional is called a Whittaker functional for π. If π is generic then the representation space V may be realized on a certain space of functions f with the property that f (ng) = Ψ(n)f (g) for all n ∈ U and g ∈ G and the action of G on V is by right shifts on this realization. Such a realization is called a Whittaker model for π. One of the main results on generic representations is the following theorem (due to Shalika [13]) which says that Whittaker models are unique when they exist. This theorem is important for the global theory, for instance, to prove multiplicity one for automorphic forms on GL(n). Also, one way to attach local Euler factors for generic representations of G is via Whittaker models. Theorem 9.2 (Multiplicity one for Whittaker models) Let (π, V ) be an irreducible admissible representation of GLn (F ). Then the dimension of the space of Whittaker functionals is at most one, i.e., dimC (HomG (π, IndG U (Ψ))) ≤ 1. Put differently, if π admits a Whittaker model then it admits a unique one. The question now arises as to which representations are actually generic. The first theorem in this direction was due to Gelfand and Kazhdan which says that every irreducible admissible supercuspidal representation is generic. The classification of all generic representations of GLn (F ) is a theorem due to Zelevinsky (see Theorem 9.7 of [15]) which states that π is generic if and only if it is irreducibly induced from essentially square integrable representations. In particular one has that discrete series representations are always generic. We would like to point out that this phenomenon is peculiar to GLn (F ) and is false for general reductive p-adic groups. Theorem 9.3 (Generic representations) Let π = Q(∆1 , ..., ∆k ) be an irreducible admissible representation of GLn (F ). Then π is generic if and only if no two of the ∆i are linked in which case IndG P (Q(∆1 ) ⊗ · · · ⊗ Q(∆k )) is an irreducible representation, and π == Q(∆1 , ..., ∆k ) = IndG P (Q(∆1 ) ⊗ · · · ⊗ Q(∆k )). Exercise 9.4 1. Use the definition of genericity to show that one dimensional representations are never generic. Now verify the above theorem for one dimensional representations.

Representation Theory of GL(n) over Non-Archimedean Local Fields 193 2. Let G = GL2 (F ). Show that an irreducible admissible representation is generic if and only if it is infinite dimensional.

9.2

Tempered representations

The notion of a tempered representation is important in the global theory of automorphic forms. The importance is evidenced by what is called the generalized Ramanujan conjecture which says that every local component of a cuspidal automorphic representation of GL(n) is tempered. In this section we give the definition of temperedness and state a theorem due to Jacquet [11] which says when a representation is tempered. (It is closely related to Zelevinsky’s theorem in the previous section.) It says that a representation with unitary central character is tempered if and only if it is irreducibly induced from a discrete series representation (as against essentially square integrable representations to get the generic ones). Definition 9.5 An irreducible admissible representation (π, V ) is tempered if ωπ the central character of π is unitary and if one (and equivalently every) matrix coefficient fv,v∨ is in L2+ǫ (Z\G) for every ǫ > 0. Every unitary supercuspidal representation of G is tempered, since matrix coefficients of supercuspidals are compactly supported modulo centre and hence in L2+ǫ (Z\G). One may drop the unitarity and rephrase this as every supercuspidal being essentially tempered with an obvious meaning given to the latter. Theorem 9.6 (Tempered representations) Let π = Q(∆1 , ..., ∆k ) be an irreducible admissible representation of GLn (F ). Then π is tempered if and only if one actually has π = IndG P (Q(∆1 ) ⊗ · · · ⊗ Q(∆k )) with every Q(∆i ) being square-integrable. From Theorem 9.3 and Theorem 9.6 we get that every tempered representation of GLn (F ) is generic. We emphasize this point in light of the global context. A theorem of Shalika [13] says that every local component of a cuspidal automorphic representation of GLn is generic. The generalized Ramanujan conjecture for GLn says that every such local component is tempered. The reader is urged to construct examples of generic representations of GLn (F ) which are not tempered.

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Unramified or spherical representations

The notion of unramified representations is again of central importance in the global theory as almost all local components of global representations are unramified. An unramified representation is a GLn analogue of an unramified character of F ∗ which is just a character trivial on the units OF× . Such a character is called unramified because every character corresponding to an unramified extension via local class field theory is unramified (the norm map being surjective on the units). Recall that K = GLn (OF ) is the (up to conjugacy) unique maximal compact subgroup of G = GLn (F ). We let HK denote the spherical Hecke algebra of G which is the space of compactly supported bi-K-invariant functions on G. This is an algebra under convolution Z f (xy −1 )g(y) dy (f ∗ g)(x) = G

where dy is a Haar measure on G normalized such that vol(K) = 1. The identity element in HK is just the characteristic function of K. Exercise 9.7 (Gelfand) Show that HK is a commutative algebra. Consider the transpose map on G to show that it induces a map on HK which is both an involution and an anti-involution. (Hint: Use Theorem 3.3.) Actually more is known about this Hecke algebra and we state the following theorem of Satake regarding the structure of HK . Theorem 9.8 (Satake Isomorphism) The spherical Hecke algebra HK is canonically isomorphic to the Weyl group invariants of the space of Laurent polynomials in n variables, i.e., −1 W HK ≃ C[t1 , t−1 1 , ..., tn , tn ] .

Definition 9.9 Let (π, V ) be an irreducible admissible representation of G. It is said to be unramified if it has a non-zero vector fixed by K. The space of K fixed vectors denoted V K is a module for the spherical Hecke algebra HK . Let (π, V ) be an irreducible unramified representation of G. We know that V K is a module for HK . We can use Proposition 2.8 and the fact that HK is commutative to get that the space of fixed vectors is actually one

Representation Theory of GL(n) over Non-Archimedean Local Fields 195 dimensional and hence this gives a character of the spherical Hecke algebra which is called the spherical character associated to π. Using this proposition again gives that the spherical character uniquely determines the unramified representation. We now construct all the unramified representations of G. Let χ1 , ..., χn be n unramified characters of F ∗ (i.e., they are all trivial on the units of F ). Let π = π(χ1 , ..., χn ) be the corresponding principal series representation of GLn (F ). It is an easy exercise using the Iwasawa decomposition to see that π admits a non-zero vector fixed by K which is unique up to scalars. Hence π admits a unique subquotient which is unramified. It turns out that if the characters χi are ordered to satisfy the ‘does not precede’ condition of Theorem 8.1 then the unique irreducible quotient Q(χ1 , ..., χn ) of π is actually unramified and so is the unique unramified subquotient of π. We now state the main theorem classifying all the unramified representations of GLn (F ). Theorem 9.10 (Spherical representations) Let χ1 , ..., χn denote n unramified characters of F ∗ which are ordered to satisfy the ‘does not precede’ condition of Theorem 8.1. Then the representation Q(χ1 , ..., χn ) is an unramified representation of GLn (F ). Further, every irreducible admissible unramified representation is equivalent to such a Q(χ1 , ..., χn ). Proof : We briefly sketch the proof of the second assertion in the theorem. We show that the Q(χ1 , ..., χn ) exhaust all the unramified representations of G. This is done by appealing to Proposition 2.8 and the Satake isomorphism. The proof boils down to showing that every character of the spherical Hecke algebra is accounted for by one of the characters coming from a Q(χ1 , ..., χn ) and so by appealing to Statement (3) of Proposition 2.8 we would be done. Now via the Satake isomorphism, a character of HK determines and is determined by a set of n non-zero complex numbers which are the values taken by the character on the variables t1 , ..., tn . Now it is easy to check that the spherical character of Q(χ1 , ..., χn ) (or π(χ1 , ..., χn ) which is more easier to work with) takes the value χi (̟F ) on the variable ti . The remark that an unramified character is completely determined by its value on a uniformizer which can be any non-zero complex number finishes the proof. 2 Example 9.11 Let π = Q(χ1 , χ2 ) be an irreducible admissible unitary unramified generic representation of GL2 (F ). Then for i = 1, 2 it can be shown

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that q −1/2 ≤ |χi (̟F )| ≤ q 1/2 . In particular, there are such representations which are not tempered. The Ramanujan conjecture for GL2 (F ) would assert that if π is a local component of a cuspidal automorphic representation of GL2 then π is tempered. Note that almost all such local components are unitary unramified and generic and the Ramanujan conjecture would boil down to showing that |χi (̟F )| = 1. See the discussion on pp.332-334 of [1] for related matters and in particular as to how the Langlands program implies the Ramanujan conjecture.

9.4

Iwahori spherical representations

This is a class of representations which may not have an immediate motivation from the global theory but is important all the same for various local reasons. (Although justifying this claim may take us outside the scope of these notes as it requires more knowledge of the structure theory of p-adic groups.) The Iwahori subgroup I of G = GLn (F ) that we will be looking at is the inverse image of the standard Borel subgroup under the canonical map from GLn (OF ) → GLn (Fq ). By definition, I is a subgroup of K. We will be looking at those representations of G which have vectors fixed by I. For this reason, as before, we need to consider the Iwahori Hecke algebra HI which is the space of compactly supported bi-I-invariant functions on G with the algebra structure being given by convolution. Definition 9.12 An irreducible admissible representation (π, V ) of G is called Iwahori spherical if it has a non-zero vector fixed by the Iwahori subgroup I. In this case the space of fixed vectors V I is a simple module denoted (π I , V I ) for the Iwahori Hecke algebra HI . To begin with, unlike the spherical case, the Iwahori Hecke algebra is not commutative and admits simple modules which are not one dimensional. The structure of HI is also a little complicated to describe and would take us outside the scope of these notes. (See § 3 of Iwahori-Matsumoto [10] for the original description of this algebra. See also § 3 of Borel [4] for a more pleasanter-to-read version of the structure of HI .) We record the following rather special lemma on representations of the Iwahori Hecke algebra. See Proposition 3.6 of [4], although Borel deals only with semi-simple groups the proof goes through mutatis mutandis to our case. We need some notation.

Representation Theory of GL(n) over Non-Archimedean Local Fields 197 For any g ∈ G, let eIgI be the characteristic function of IgI as a subset of G. Clearly eIgI ∈ HI . If we normalize the Haar measure on G such that vol(I) = 1, then eI is the identity of element of HI . Lemma 9.13 Let (σ, W ) be a finite dimensional representation of HI . Then for any g ∈ G the endomorphism σ(eIgI ) is invertible. The main fact about Iwahori spherical representations is the following theorem due to Borel and Casselman. (See Theorem 3.3.3. of [7] and a strengthening of it in Lemma 4.7 of [4].) Recall our notation that B = T U is the standard Borel subgroup of G. The Iwahori subgroup I admits an ‘Iwahori-factorization’ with respect to B as I = I −I 0I + where I − = I ∩ U − , I 0 = I ∩ T which is the maximal compact subgroup of T and I + = I ∩ U. Theorem 9.14 Let (π, V ) be an admissible representation of G. Then the canonical projection from V to the Jacquet module VU induces an isomor0 phism from V I onto (VU )I . Proof : Let A : V → VU be the canonical projection map. By the proof of 0 statement (5) of Theorem 5.3 we have that A(V I ) = (VU )I . We now have to show that A is injective on V I . Suppose v ∈ V I is such that A(v) = 0. Then v ∈ V I ∩ V (U ). Choose a compact open subgroup U1 of U such that v ∈ V (U1 ) which gives that π(e R U1 )v = 0. (For any compact subset C of G we let π(eC )v stand for C π(g)v dg.) Choose r ≫ 0 such that µ−r U1 µr ⊂ I + where µ = diag(1, ̟F , ..., ̟Fn−1 ). We then get that π(eI + )(π(µ−r )v) = 0. The Iwahori factorization gives that π(eI )π(µ−r )v = 0. What we have shown is that there exists a g ∈ G such that π(eI )π(g)v = 0. Since v ∈ V I this implies that π(eIgI )v = 0. Now appealing to Lemma 9.13 we get that v = 0. 2 Theorem 9.15 (Iwahori spherical representations) Let χ1 , ..., χn be unramified characters of F ∗ . Let π(χ) = π(χ1 , ..., χn ) be the corresponding principal series representation. Then any irreducible subquotient of π(χ) is Iwahori spherical and every Iwahori spherical representation arises in this manner.

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Proof : Let π be any irreducible subquotient of π(χ). We know from Theorem 5.9 that the Jacquet module πU is non-zero. We also know from the computation of the Jacquet module of π(χ) in Theorem 5.4 that I 0 acts trivially on π(χ)U since every χi is unramified and hence I 0 acts trivially on 0 πU , i.e., (πU )I 6= (0). Using Theorem 9.14 we get therefore that π I 6= (0), i.e., π is Iwahori spherical. Now let π be any Iwahori spherical representation of G. We know that 0 I πU 6= (0) by the above theorem and hence there is an unramified character χ of T such that HomT (πU , χ) 6= (0). Appealing to Frobenius reciprocity finishes the proof. 2 Corollary 9.16 Let (π, V ) be any irreducible admissible representation of G = GLn (F ). Then the dimension of space of fixed vectors under I is bounded above by the order of the Weyl group W, i.e., dimC (V I ) ≤ n! Equivalently, the dimension of any simple module for the Iwahori Hecke algebra HI is bounded above by n!. Proof : The corollary would follow if we show that the space of I fixed vectors of an uramified principal series π(χ) is exactly n! which is the order of the Weyl group. This follows from Theorem 9.14 and Theorem 5.4. 2

10

Representations of local Galois Groups

The Galois group GF = Gal(F¯ /F ) has distinguished normal subgroups IF , the Inertia subgroup, and PF the wild Inertia subgroup which is contained in IF . The Inertia subgroup sits in the following exact sequence, ¯ q /Fq ) → 1. 1 → IF → Gal(F¯ /F ) → Gal(F ¯ q /Fq ) is given by the natural Here the mapping from Gal(F¯ /F ) to Gal(F action of the Galois group of a local field on its residue field. The Inertia group can be thought of as the Galois group of F¯ over the 1/d maximal unramified extension F un of F . Let F t = ∪(d,q)=1 F un (̟F ). The field F t is known to be the maximal tamely ramified extension of F un (An extension is called tamely ramified if the index of ramification is coprime to the characteristic of the residue field.) We have,

Representation Theory of GL(n) over Non-Archimedean Local Fields 199

Gal(F¯ /F )/IF IF /PF

ˆ ≃ Z Y ≃ Z× ℓ . ℓ6=p

One defines the Weil group WF of F to be the subgroup of Gal(F¯ /F ) ¯ q /Fq ) is an integral power of the Frobenius autowhose image inside Gal(F q morphism x → x . Representations of the Weil group which are continuous on the inertia subgroup are exactly those representations of the Weil group for which the image of the Inertia subgroup is finite. The Weil group is a dense subgroup of the Galois group hence an irreducible representation of the Galois group defines an irreducible representation of the Weil group. It is easy to see that a representation of the Weil group can, after twisting by a character, be extended to a representation of the Galois group. Local class field theory implies that the maximal abelian quotient of the Weil group of F is naturally isomorphic to F ∗ , and hence 1 dimensional representations of WF are in bijective correspondence with characters of GL1 (F ) = F ∗ . It is this statement of abelian class field theory which is generalized by the local Langlands correspondence. However, there is a slight amount of change one needs to make, and instead of taking the Weil group, one needs to take what is called the Weil-Deligne group whose representations are the same as representations of WF on a vector space V together with a nilpotent endomorphism N such that wN w−1 = |w|N ¯ q /Fq ) is the i-th power of the where |w| = q −i if the image of w in Gal(F Frobenius. One can identify representations of the Weil-Deligne group to representations of WF ×SL2 (C) via the Jacobson-Morozov theorem. It is usually much easier to work with WF × SL2 (C) but the formulation with the nilpotent operators appears more naturally in considerations of ℓ-adic cohomology of Shimura varieties where the nilpotent operator appears as the ‘monodromy’ operator. We end with the following important proposition for which we first note that any field extension K of degree n of a local field F gives rise to an inclusion of the Weil group WK inside WF as a subgroup of index n. Since

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characters of WK are, by local class field theory, identified to characters of K ∗ , a character of K ∗ gives by induction a representation of WF of dimension n. Proposition 10.1 If (n, q) = 1, then any irreducible representation of WF of dimension n is induced from a character χ of K ∗ for a field extension K of degree n. Proof : Since irreducible representations of the Weil group and Galois groups are the same, perhaps after a twist, we will instead work with the Galois group. Let ρ be an irreducible representation of the Galois group Gal(F¯ /F ) of dimension n > 1. We will prove that there exists an extension L of F of degree greater than 1, and an irreducible representation of Gal(F¯ /L) which induces to ρ. This will complete the proof of the proposition by induction on n. The proof will be by contradiction. We will assume that ρ is not induced from any proper subgroup. We recall that there is a filtration on the Galois group PF ⊂ IF ⊂ GF = Gal(F¯ /F ). Since PF is a pro-p group, all its irreducible representations of dimension greater than 1 are powers of p. Since n is prime to p, this implies that there is a 1 dimensional representation of PF which appears in ρ restricted to PF . Since PF is a normal subgroup of the Galois group, it implies that the restriction of ρ to PF is a sum of characters. All the characters must be the same by Clifford theory. (Otherwise, the representation ρ is induced from the stabilizer of any χ-isotypical component.) So, under the hypothesis that ρ is not induced from any proper subgroup, PF acts via scalars on ρ. Since the exact sequence, 1 → PF → IF → IF /PF → 1 is a split exact sequence, let M be a subgroup in IF which goes isomorphically to IF /PF . Take a character of the abelian group M appearing in ρ. As PF operates via scalars, the corresponding 1 dimensional space is invariant under IF . It follows that ρ restricted to IF contains a character. Again IF being normal, ρ restricted to I is a sum of characters which must be all the same under the assumption that ρ is not induced from any proper subgroup. Since G/IF is pro-cyclic, this is not possible. 2

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11

The local Langlands conjecture for GLn

It is part of abelian class field theory that for a local field F , the characters of F ∗ can be identified to the characters of the Weil group WF of F . Langlands visualised a vast generalisation of this in the late 60’s to non-abelian representations of WF which is now a theorem due to M.Harris and R.Taylor, and another proof was supplied shortly thereafter by G.Henniart. The general conjecture of Langlands uses a slight variation of the Weil group, called the Deligne-Weil group, denoted WF′ and defined to be WF′ = WF × SL2 (C). Theorem 11.1 (Local Langlands Conjecture) There exists a natural bijective correspondence between irreducible admissible representations of GLn (F ) and n-dimensional representations of the Weil-Deligne group WF′ of F which are semi-simple when restricted to WF and algebraic when restricted to SL2 (C). The correspondence reduces to class field theory for n = 1, and is equivariant under twisting and taking duals. The correspondence in the conjecture is called the local Langlands correspondence, and the n-dimensional representation of WF′ associated to a representation π of GLn (F ) is called the Langlands parameter of π. The Langlands correspondence is supposed to be natural in the sense that there are L-functions and ǫ-factors attached to pairs of representations of WF′ , and also to pairs of representations of GLn (F ), characterising these representations, and the correspondence is supposed to be the unique correspondence preserving these. We will not define L-functions and ǫ-factors here, but refer the reader to the excellent survey article of Kudla [12] on the subject. The results of Bernstein-Zelevinsky reviewed in section 7 and 8 reduce the Langlands correspondence between irreducible representations of GLn (F ) and representations of WF′ to one between irreducible supercuspidal representations of GLn (F ) and irreducible representations of WF , as can be seen as follows. An n-dimensional representation σ of the Weil-Deligne group WF′ of F which is semi-simple when restricted to WF and algebraic when restricted to SL2 (C) is of the form σ=

i=r X i=1

σi ⊗ Sp(mi )

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where σi are irreducible representations of WF of dimension say ni , and Sp(mi ) is the unique mi dimensional irreducible representation of SL2 (C). Assuming the Langlands correspondence between irreducible supercuspidal representations of GLn (F ) and irreducible representations of WF , we have representations πi of GLni (F ) naturally associated to the representations σi of dimension ni of WF . To the product mi ni , and a supercuspidal representation πi of GLni (F ), Theorem 7.5 associates an essentially square integrable representation to the segment (πi (− mi2−1 ), · · · , πi ( mi2+1 )), which we denote by Stmi (πi ). Now order the representations so that for i < j, the segment m −1 m +1 (πi (− mi2−1 ), · · · , πi ( mi2+1 )), does not precede (πj (− j2 ), · · · , πj ( j2 )). By Theorem 8.1, the parobolically induced representation IndG P (Stm1 (π1 ) ⊗ · · · ⊗ Stmr (πr )), admits a unique irreducible quotient which is the representation of GLn (F ) P associated by the Langlands correspondence to σ = i=r i=1 σi ⊗ Sp(mi ).

Example 11.2 The irreducible admissible representations π of GL2 together with the associated representation σπ of the Weil-Deligne group is as follows. 1. Principal series representation, induced from a pair of characters (χ1 , χ2 ) of F ∗ . These representations are irreducible if and only if χ1 χ−1 2 6= ±1 | · | . For irreducible principal series representation, the associated Langlands parameter is χ1 ⊕ χ2 , where χ1 , χ2 are now being treated as characters of WF . 2. Twists of Steinberg. The Langlands parameter of the Steinberg is the standard 2 dimensional representation of SL2 (C). 3. Twists of the trivial representation. The Langlands parameter of the trivial representation is | · |1/2 ⊕ | · |−1/2 . 4. The rest, which is exactly the set of supercuspidal representations of GL2 (F ). In odd residue characteristic, these representations can be constructed by what is called the Weil representation, and the representation of GL2 (F ) so constructed are parametrized by characters of the invertible elements of quadratic field extensions of F . By proposition 10.1, every 2 dimensional representation of WF , when the residue characteristic is odd, is also given by induction of such a character on a quadratic extension which is the Langlands parameter.

Representation Theory of GL(n) over Non-Archimedean Local Fields 203 Acknowledgments These notes are based on the lectures given by the first author in the workshop ‘Automorphic Forms on GL(n)’ organized by Professors G. Harder and M.S. Raghunathan at ICTP, Trieste in August 2000. The authors would like to thank the organizers for inviting them for this nice conference; the European commission for financial support; Professor L. G¨ottsche for having taken care of all the details of our stay as well as the conference so well.

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References [1] D. Bump, Automorphic Forms and Representations, Cambridge studies in Advanced Mathematics 55, (1997). [2] I.N. Bernstein and A.V. Zelevinsky, Representation Theory of GL(n, F ) where F is a non-Archimedea local field, Russian Math. Survey, 31:3, 1-68, (1976). [3] I.N. Bernstein and A.V. Zelevinsky, Induced representations of reduc´ tive p-adic groups. I, Ann. Sci. Ecole Norm. Sup., (4) 10, no. 4, 441–472 (1977). [4] A. Borel, Admissible representations of a semi-simple group over a local field with vectors fixed under am Iwahori subgroup, Inv. Math., 35, 233-259 (1976). [5] C.J. Bushnell and P.C. Kutzko, The admissible dual of GL(N ) via compact open subgroups, Princeton university press, Princeton, (1993). [6] H. Carayol, Representations cuspidales du groupe lineaire, Ann. Sci. ´ Ecole. Norm. Sup., (4), Vol 17, 191-225 (1984). [7] W. Casselman, An introduction to the theory of admissible representations of reductive p-adic groups, Unpublished notes. [8] J.A. Green,The characters of the finite general linear groups, Trans. Amer. Math. Soc. 80, 402–447 (1955). [9] G. Henniart, Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique. (French) [A simple proof of the Langlands conjectures for GL(n) over a p-adic field], Invent. Math., 139, no. 2, 439–455 (2000). [10] N. Iwahori and H. Matsumoto, On some Bruhat decompositions and the structure of the Hecke algebra rings of p-adic Chevalley groups, Publ. Math. I.H.E.S., 25, 5-48 (1965). [11] H. Jacquet, Generic representations, in Non-Commutative harmonic analysis, LNM 587, Springer Verlag, 91-101 (1977).

Representation Theory of GL(n) over Non-Archimedean Local Fields 205 [12] Stephen S. Kudla, The local Langlands correspondence: the nonArchimedean case, Motives (Seattle, WA, 1991), 365–391, Proc. Sympos. Pure Math., 55, Part 2, Amer. Math. Soc., Providence, RI (1994). [13] J.A. Shalika, The multiplicity one theorem for GLn , Ann. of Math., Vol 100, 171-193 (1974). [14] T.A. Springer, Linear Algebraic Groups, 2nd Edition, Progress in Mathematics, Vol 9, Birkh¨auser, (1998). [15] A.V. Zelevinsky, Induced representations of reductive p-adic groups II, ´ Ann. Sci. Ecole Norm. Sup., (4) Vol 13, 165-210 (1980).

The Langlands Program (An overview) G. Harder∗

Mathematisches Institut der Universit¨ at Bonn, Bonn, Germany

Lectures given at the School on Automorphic Forms on GL(n) Trieste, 31 July - 18 August 2000

LNS0821005

∗

[email protected]

Contents Introduction

211

I. A Simple Example

211

II. The General Picture

223

References

234

The Langlands Program (An overview)

211

Introduction The Langlands program predicts a correspondence between two types of objects. On the one side we have automorphic representations π and on the other side we have some arithmetic objects M which may be called motives or even objects of a more general nature. Both these objects produce Lfunctions and the correspondence should be defined by the equality of these L-functions. A special case is the Weil-Taniyama conjecture which has been proved by Wiles-Taylor and others. I. A Simple Example On his home-page under www.math.ias.edu Langlands considers a couple of very explicit and simple examples of this correspondence and here I reproduce one of these examples together with some further explanation. This example is so simple that the statement of the theorem can be explained to everybody who has some basic education in mathematics. The first object is a pair of integral, positive definite, quaternary quadratic forms P (x, y, u, v) = x2 + xy + 3y 2 + u2 + uv + 3v 2 Q(x, y, u, v) = 2(x2 + y 2 + u2 + v 2 ) + 2xu + xv + yu − 2yv These forms have discriminant 112 and I mention that these two quadratic forms Q, P are the only integral, positive definite quaternary forms with discriminant 112 . This may not be so easy to verify but it is true. (Rainer Schulze-Pillot pointed out that this is actually not true; there is a third form S(x, y, u, v) = x2 + 4(y 2 + u2 + v 2 ) + xu + 4yu + 3yv + 7uv but the two forms above are sufficient for the following considerations (see [He1])). This pair will give us automorphic forms, we come to this point later. The second object is an elliptic curve E, for us this is simply a polynomial G(x, y) = y 2 + y − x3 + x2 + 10x + 20. This object is a diophantine equation, for any commutative ring R with identity we can consider the set of solutions {(a, b) ∈ R2 |G(a, b) = 0}

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In the case where R is a field k we add a point at infinity (we should consider ˜ y, z) = y 2 z +yz 2 −x3 +x2 z +10xz 2 +20z 3 ) the homogenized polynomial G(x, and define E(k) = {(a, b) ∈ k2 |G(a, b) = 0} ∪ {∞} ˜ b, c) = 0}/k∗ . = {(a, b, c) ∈ k3 \ {(0, 0, 0)} | G(a, We come back to the first object. For any integer n we can define the numbers r(P, n) = #{γ ∈ Z4 |P (γ) = n} , r(Q, n) = #{γ ∈ Z4 |Q(γ) = n} in classical terms: We consider the number of representations of n by the two forms. We can encode these numbers in generating series X X Θ(P, t) = r(P, n)tn = tP (γ) n

Θ(Q, t) =

γ∈Z4

X X r(Q, n)tn = tQ(γ) n

γ∈Z4

Of course it is not so difficult to write a few terms of these series Θ(P, t) = 1 + 4t + 4t2 + 8t3 + 20t4 + 16t5 + 32t6 + 16t7 + 36t8 + 28t9 + 40t10 + 4t11 + 64t12 + 40t13 + 64t14 + 56t15 + 68t16 + 40t17 + 100t18 + 48t19 + 104t20 + . . . Θ(Q, t) = 1 + 12t2 + 12t3 + 12t4 + 12t5 + 24t6 + 24t7 + 36t8 + 36t9 + 48t10 + 72t12 + 24t13 + 48t14 + 60t15 + 84t16 + 48t17 + 84t18 + 48t19 + 96t20 + . . . . Now we return to our second object. For any prime p we can reduce our polynomial G(x, y) mod p and we can look at the solutions of our equation G(x, y) = 0 in the field Fp with p elements. Actually this equation defines what is called a curve over Fp and if we add the point at infinity we get a projective curve. We say that this curve is smooth over Fp (or we say that we have good reduction) if for any point in the algebraic closure ¯ p ) the vector of partial derivatives (a, b) ∈ E(F (

∂G ∂G (a, b), (a, b)) 6= 0. ∂x ∂y

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A simple calculation shows that we get a smooth curve over Fp except for p = 11. For any p we may ask: What is the number of solutions of our equation over Fp and this means we want to know what #E(Fp ) is. To get a rough idea of what will happen we do the following: We choose an a ∈ Fp and to find a point (a, b) ∈ E(Fp ) we have to solve the quadratic equation y 2 + y = a3 − a2 − 10a − 20 in Fp . If p 6= 2 then this equation has a solution in Fp if and only if the element a3 − a2 − 10a − 20 + 1/4 is a square in Fp . Now we know that exactly half the elements in F∗p are squares and hence our chance to hit a square is roughly 1/2. But if we hit a square then we get two solutions for our equation -unless the number above should be zero- therefore we can expect that the number of solutions is roughly p. For p 6= 11 we define the number ap by #E(Fp ) = p + 1 − ap , so this number ap measures the deviation from our expectation. We have the celebrated theorem by Hasse √ For p 6= 11 we have the estimate | ap | ≤ 2 p . Again we can produce a list of values of ap for small primes 2 3 5 7 13 17 19 −2 −1 1 −2 4 −2 0 Now we can formulate a theorem which is a special case of the Langlands correspondence but which was certainly known to Eichler: Theorem For all p 6= 11 we have 1 ap = (r(P, p) − r(Q, p)) 4 This is a surprising statement which is formulated in completely elementary terms. We have two diophantine problems of rather different nature, why are they related by the theorem above? I would like to say that the theorem in the form as it stands looks like a miracle. One possible interpretation is that it provides an elementary formula for the numbers ap . But from the computational side it seems to me that the

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ap are easier to compute than the representation numbers. I come back to this further down. The theorem becomes comprehensible if we establish the connection to modular forms. The following considerations go back into the 19-th century. We consider the two generating functions for our two quadratic forms. We make a substitution t → e2πiz and we observe that the functions z 7→ Θ(P, z), z 7→ Θ(Q, z) are holomorphic functions on the upper half plane H = {z | ℑ(z) > 0}. It is a classical result that these two functions are in fact modular forms of weight 2 for the congruence subgroup a b Γ0 (11) = { | a, b, c, d ∈ Z, c ≡ 0 mod 11}. c d This means that they satisfy (? = P, Q) Θ(?,

az + b ) = (cz + d)2 Θ(?, z) cz + d

for γ=

a b ∈ Γ0 (11) c d

and in addition a certain growth condition for ℑ(z) → ∞ is satisfied. [This can be verified by a classical calculation. First of all it is easy to see that in both cases the forms are invariant under z 7→ z + 1 and then the Poisson summation formula implies the rule Θ(?, z) =

1 1 Θ(?, ). 2 11z −11z

(I skip the computation, it is based on the observation that for x ∈ R4 the function X x 7→ e2πizQ(x+ω) ω∈Z4

is periodic with period Z4 . Hence it has a Fourier expansion, writing down this expansion, putting x = 0 and another small manipulation yields the assertion). Now the modularity follows. (See also [He1])] Hence we get two modular forms of weight 2 for the group Γ0 (11) and by classical dimension formulae we know that they span the vector space

The Langlands Program (An overview)

215

of these modular forms. We know that this space of modular forms is also spanned by two other forms: One of them is the Eisenstein series E(z) =

X

γ,c≡0

mod 11

1 1 − 2 (cz + d) 11

X

γ,c6≡0

mod 11

1 (cz + d)2

(this is a difference of two divergent series and this difference makes sense (this is Hecke so we are in the 20-th century)) and the other one is a cusp form, which in this case is 2πiz

f (z) = e

∞ Y

(1 − e2πniz )2 (1 − e2π11niz )2

n=1

(also classical we have the Dedekind η-function η(z) = eπiz/12 e2πniz )) It is now also in [He1] that

Q∞

n=1 (1

−

1 f (z) = (Θ(P, z) − Θ(Q, z)). 4 A small digression: Of course we would like have information on the individual Theta series. In this context we still have another theorem by Siegel. Our two quadratic forms are in fact in the same genus, that means over any p-adic ring Zp they become equivalent (but of course they are not equivalent over Z). Then we have a very general theorem by C.L. Siegel which asserts that the sum over the Theta series over a genus where the summands are multiplied by suitable weight factors (densities) gives us an Eisenstein series. In our special situation we find ∞

X 1 1 σn e2πinz Θ(P, z) + Θ(Q, z) = E(z) = 4 6 n=0

where the coefficients σn are given rather explicitly, for instance for a prime p 6= 11 we have σp = p + 1. I will say more about the other coefficients in a minute (See A below). At this point I want to meditate a second. Here are two important points to observe.

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A) If we look at the problem to understand the representation numbers we want to know the r(P, n) for all integers n. If we go back to our elliptic curve we get only numbers for each prime p, (p 6= 11). Here the theory of automorphic forms provides another remarkable and fundamental fact. The coefficients in the two series ∞

X 1 τn e2πinz f (z) = (Θ(P, e2πz ) − Θ(Q, e2πiz )) = 4 n=1

and

∞

X 1 1 E(z) = Θ(P, e2πiz ) + Θ(Q, e2πiz ) = σn e2πinz 4 6 n=0

behave multiplicatively and I explain what this means: If we have any series P F (z) = λn e2πinz then we build formally the Mellin transform. This is a Dirichlet series, it is defined by LF (s) =

∞ X

λn n−s .

n=1

(Let us ignore convergence problems, this construction has also been discussed in J. Cogdell’s lecture). Now multiplicativity means in our case that the two Mellin transforms have an Euler product expansion Lf (s) = (

Y

p6=11

LE (s) = (

Y

p6=11

1 1 − τp

p−s

+

p1−2s

)

1 1 − 11−s

1 1 ) −s 1−2s 1 − (p + 1)p + p 1 − 11−s

and this is equivalent to some recursion formulae namely τnm = τn τm , σnm = σn σm if n, m are coprime and for p 6= 11 τpr+1 = τpr τp + pτpr−1 , σpr+1 = σpr σp + pσpr−1 if r ≥ 1 and hence we know the σ, τ if we know them for prime indices. This follows from the theory of the Hecke operators which was actually designed for proving such multiplicativity formulae (See [He2]). The two

The Langlands Program (An overview)

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functions are eigenfunctions for this Hecke algebra and this is equivalent to the multiplicativity of the coefficients. This makes it also clear that our two functions are the only ones which have multiplicative coefficients. B) Now we have the formula 1 1 r(P, p) + r(Q, p) = p + 1 4 6 and together with our theorem we can say that we can express the representation numbers in terms of p and ap . Combined with the theorem by Hasse we get a consequence for the asymptotic behavior of the representation numbers and this was an application Eichler had in mind. From √ |τp | = | 41 (r(P, p) − r(Q, p))| ≤ 2 p we get the asymptotic formulae r(P, p) =

12 p + O(p1/2 ) 5

12 p + O(p1/2 ). 5 Now we return to our elliptic curve and I want to give a very sketchy outline of the proof of the theorem. We consider the Riemann surface Γ0 (11)\H. It was known to Fricke that this is a curve of genus 1 over C from which two points are removed. These two points are the cusps of the action of Γ0 (11) on H, they can be represented by 0, i∞. The curve of genus 1 can also be interpreted as Γ0 (11)\H∗ where H∗ = H ∪ Q ∪ {∞} = H ∪ P1 (Q) where this space is endowed with a suitable topology. Fricke found an equation for this curve which after some manipulation can be transformed into r(Q, p) =

y 2 + y = x3 − x2 − 10x − 20 and in modern language this means that that we have a model X0 (11)/Spec(Z) of our complex curve which has good reduction at all primes p 6= 11. The Hecke operators Tp are so-called correspondences, they can be interpreted as curves Tp ⊂ Γ0 (11)\H∗ × Γ0 (11)\H∗ which consist of the following points: If a first coordinate is represented by z ∈ H then the second coordinate is represented by one of the points {pz, z/p, (z +1)/p, . . . , (z +p−1)/p}; so in general there are p + 1 second coordinates corresponding to a first coordinate and vice versa. (Of course one has to check that replacing z by another representative gives the same set of corresponding points). These Hecke operators extend to correspondences also called Tp on the model X0 (11). To

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see that this is so one has to go to the modular interpretation of X0 (11), this means roughly that X0 (11) is the parameter space for the elliptic curves with a cyclic subgroup of order 11. Then Eichler showed that these correspondences Tp have a reduction mod p and this reduction is given by the congruence formula ([Ei]) Tp

mod p = Fp +t Fp .

¯ p ) × E(F ¯ p )} and t Fp is given Here Fp is given by the graph {(x, xp ) ∈ E(F p ¯ ¯ by {(x , x) ∈ E(Fp ) × E(Fp )}. Using the trace formula the coefficient τp can be expressed in terms of the fixed points of Tp . But mod p the fixed points of Fp and t Fp are the points in E(Fp ), and this gives a very rough indication how the theorem can be proved. The Taniyama-Shimura - Weil conjecture Wiles, Taylor and others proved the general Taniyama-Weil conjecture. I want to give some indication of the content of this general theorem. The precise statement needs some finer concepts and results from the theory of automorphic forms and the arithmetic of elliptic curves. A congruence subgroup Γ ⊂ SL2 (Z) is a subgroup of finite index which can be defined by congruence conditions on the entries. To any integer N we define the subgroup n a b o Γ(N ) = ∈ SL2 (Z) | a ≡ d ≡ 1 mod N , c d and a given subgroup Γ is called a congruence subgroup, if we can find an integer N such that Γ(N ) ⊂ Γ ⊂ SL2 (Z). Such a group operates on the upper half plane H and the quotient Γ\H carries the structure of a Riemann surface, more precisely we can compactify it to a compact Riemann surface by adding a finite number of points. These finitely many points are called the cusps. A holomorphic modular form for a given congruence subgroup Γ of weight k > 0 is a holomorphic function on the upper half plane f : H −→ C which satisfies f

az + b cz + d

= (cz + d)k f (z)

The Langlands Program (An overview)

219

for all γ ∈ Γ, and which satisfies a growth condition in the cusps (see 4.11 (2) in [V]). Now I need some results and concepts which I cannot explain in detail. On the space of modular forms of weight k for Γ0 (N ) we have an action of a commutative algebra T which is generated by operators Tp for the primes p 6 |N . In this space of modular forms we have the subspace of cusp forms. These are forms which vanish at infinity (see [V], .....), and this subspace is invariant under the Hecke operators. It is a classical result of Hecke that this space of cusp forms is a direct sum of spaces of common eigenforms for the Hecke operators. A modular form f for Γ0 (N ) is called a new form if i) The form f and certain transforms of it is not a modular form for a congruence subgroup Γ0 (N ′ ) where N ′ | N and N ′ < N . ii) The form f is an eigenform for all the Hecke operators Tp where p 6 |N . It requires a little bit of work to show that this is a reasonable concept. To such a new form f we can attach an L-function Y Lp (f, s), L(f, s) = p

where we have attached a local Euler factor Lp (s) to any prime p: i) For the primes p 6 |N our form is an eigenform for Tp , i.e. Tp f = ap f and we put Lp (f, s) =

ap ∈ C,

1 . 1 − ap p−s + pk−1−2s

ii) For the primes p | N and p 6= 2, 3 we have ( 1 2 1−εp p−s εp = ±1 if p 6 |N Lp (s) = 1 if p2 | N. The determination of the εp requires some knowledge of local representation theory. iii) For the primes p | N and p = 2 or 3 we also have ( 1 1−εp p−s εp = ±1 Lp (s) = 1

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but here the formulation of the conditions for the cases is even more subtle. Now it is a general theorem that the completed L function Λ(f, s) = · L(f, s) is a holomorphic function in the entire plane and satisfies the functional equation Γ(s) πs

Λ(2 − s) = N 1−s W (f )N s−1 Λ(s), where W (f ) = ±1. This is a so-called automorphic L-function. Now I explain how we can attach an L function to an elliptic curve E over Q. Let us consider an elliptic curve over Q. This is simply an equation G(x, y, z) = y 2 z + a1 xyz + a3 yz 2 − x3 − a2 x2 z − a4 xz 2 − a6 z 3 = 0 with rational coefficients a1 , a3 , a2 , a4 , a6 . (This is a traditional notation, these ai are not the ap which occur in the local L-factors.) We assume that this equation defines a non singular curve and this means that for any solution (x0 , y0 , z0 ) ∈ C3 , (x0 , y0 , z0 ) 6= 0 we have ∂G ∂G ∂G (x0 , y0 , z0 ), (x0 , y0 , z0 ), (x0 , y0 , z0 ) 6= (0, 0, 0). ∂x ∂y ∂z This is equivalent to the non vanishing of the discriminant ∆ = ∆(a1 , a2 , a3 , a4 , a6 ), this is a complicated expression in the coefficients. ([Mod], the articles of Tate and Deligne (Formulaire)). A special point is the point at infinity O = (0, 1, 0). It is the only point with z0 = 0. Now we can perform substitutions in the variables, and we get new Weierstraß equations. There is a so-called minimal Weierstraß equation y2z + a ˜1 xyz + a ˜ 3 y = x3 + a ˜2 x2 + a ˜4 x + a ˜6 , where all the a ˜i ∈ Z, and where the discriminant ∆ is minimal. (See Silverman [Si], Chap. III, § 1, VIII, § 8, [Hu], Chap. 5, § 2 and [Mod], articles by Tate and Deligne.) We have an algorithm – which is implemented in Pari – which produces this minimal equation.

The Langlands Program (An overview)

221

If now p is a prime, we can consider the reduction mod p. This gives us an equation over the finite field Fp , and we say that our equation has good reduction mod p, if the reduced equation defines an elliptic curve, i.e. it is smooth. If the mod p reduced curve is not smooth then we have exactly one singular point P0 = (x0 , y0 , z0 ) ∈ E(Fp ) which is different from the point O, we may assume z0 = 1. Then we can introduce variables u = x−x0 , v = y−y0 and our equation mod p becomes αu2 + βuv + γv 2 + higher order terms = 0. It is not hard to see that the quadratic leading term is not identically zero. Then we have two possibilities: i) Over Fp2 we have αu2 + βuv + γv 2 = (u − ξ1 v)(u − ξ2 v) where ξ1 6= ξ2 ii) The quadratic form is itself a square, i.e. αu2 + βuv + γv 2 = (u − ξv)2 In the first case we say that E has multiplicative reduction mod p, in the second case we say that E has potentially good reduction at p. (Potentially good is much more unpleasant than multiplicative reduction.) If we are in the case i) we define εp =

1 ifξ1 , ξ2 ∈ Fp , −1 else.

From this type of bad reduction we can produce a number np (E) > 0. If p 6= 2, 3 then np (E) =

1 multiplicative reduction, 2 potentially good reduction.

If we have p = 2 or p = 3 then the rule is more complicated, in this case we need something finer than the minimal Weierstrass-equation, we need the Neron model (see [O]) to produce np (E). We define Y N= pnp (E) , p

where p runs over the primes with bad reduction. This number N is the conductor of our curve.

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Now in [We] Weil defines an Euler factor Lp (E, s) for any prime p. If we have good reduction at p, we define as before Lp (E, s) =

1 1 − ap p−s + p1−2s

where E(Fp ) = p + 1 − ap . If we have bad reduction, then the Euler factor depends on the type of the bad reduction. The precise rule is (see [We] 2). ( 1 if we have potentially good reduction, Lp (E, s) = 1 1−εp p−s if we have multiplicative reduction. Then we can define L(E, s) =

Γ(s) Y Lp (E, s). πs p

(Here Γ(s) is the Γ-function.) Then the theorem of Wiles-Taylor asserts that to any elliptic curve E/Q with conductor N we can find a new form f on Γ0 (N ) such that Lf (s) = L(E, s). Our first example is a special case of this theorem. Converse theorems I come back to the L-function to a newform f . I introduced the Mellin transform very formally but as explained in Cogdell’s lecture we can also define it by the integral Z i∞ Γ(s) dy Λ(f, s) = f (iy)y s L(f, s) = s (2π) y 0 where f has to be suitably normalized. Now we can conclude from the theory of automorphic forms that our newform satisfies f (−

1 ) = W (f )N z 2 f (z), Nz

where W (f ) = ±1. We apply this to the integral representation: We choose a positive real number A > 0 and split the integral into an integral from A

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to ∞ and the integral from 0 to A. Into the second term we plug in the above transformation formula and get Z ∞ Z ∞ dy 1−s s dy + W (f )N f (iy)y 2−s f (iy)y Λ(f, s) = y y 1/(N A) A Γ(s) From this integral representation we can derive that (2π) s L(f, s) has an analytic continuation into the entire plane and that we have the functional equation Λ(2 − s) = W (f )N 1−s Λ(s).

Already Hecke observed that under certain circumstances we can go the other way round. If we have a Dirichlet series D(s) =

∞ X an n=1

ns

which defines a holomorphic function, which satisfies some boundedness conditions and satisfies a suitable functional equation then it comes from a modular form. Hecke considered the case of Sl2 (Z) and Weil generalized it in ([We]) but he had to assume that these properties remained true if the series is twisted by Dirichlet characters. (See Venkataramas’s and Cogdell’s lectures.) Such a theorem is called a converse theorem. Of course one would like to prove that the L-function of an elliptic curve has these nice analytic properties and then we could get a proof of Wiles theorem. But this is not the way it works. II. The General Picture Now I want to give some vague idea of the general Langlands program. I must confess that my own understanding is very limited. But on the other hand the entire picture is so vast and a precise formulation requires an explanation of so many subtle notions that I believe that a very rough approximation may be even more helpful than a precise presentation. The first datum is a reductive group G/Q, we may very well think that G = Gln . During this summer school we have seen that automorphic cusp forms should be understood as irreducible subrepresentations of the adele group G(A) occurring in the space of cusp forms. So this is an irreducible submodule Hπ ⊂ L20 (G(Q)\G(A))

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where the subscript π stands for the isomorphism class of our module. Several lecturers told us that such a π is in fact a restricted tensor product of local representations πv of G(Qv ) and we write π=

O′

πv .

These local representations have to satisfy some constraints. For instance for almost all finite primes πp has to be in the unramified principal series (see Prasad’s notes and below) and they must be unitary. In Raghunathan’s lecture it was explained that M L20 (G(Q)\G(A)) = m(π)Hπ and we state a fundamental problem: N′ πv given to Let us assume that there is a restricted product π = us, which fulfills the above constraints. When does π occur in the space of automorphic forms and what is m(π)? Of course this question is rather vague because we should know how π is given to us, i.e. what is the rule which produces the local data {πv }. The speculative answer to this question is, that the rule should come from some kind of arithmetic object. The classical case again We come back briefly to the special case Gl2 . In our example a modular form was a holomorphic function f on the upper half plane which satisfied f (γ(z)) = (cz + d)2 f (z) for γ in some congruence subgroup Γ ⊂ Sl2 (Z). In addition we required that it should be an eigenform for the so-called Hecke operators and I explained briefly that this was equivalent to the requirement that the Mellin transform of the Fourier expansion has an Euler product expansion. Actually the Hecke operators Tp are only defined for primes p not dividing the so-called level N of our form. In our example we had N = 11. Hence we see that f provides a collection of local data {τp }p,p6|N the eigenvalues of Tp . In our example we had in fact a rather simple rule which provided the local data, we took the difference of the representation numbers. If we want to translate from the classical language to the modern language then we have to assign a representation π(f ) of Gl2 (A) to our classical

The Langlands Program (An overview)

225

modular form: This representation should occur in the space of cusp forms L20 (Gl2 (Q)\Gl2 (A)). I do not construct it but I make a list of its properties which define it uniquely. If we write O′ π(f ) = πv then i) At the finite primes p not dividing the level the representation π(f )p is in the unramified principal series and hence a unitarily induced representation G(Q )

IndB(Qpp ) λp t1 u where λp is a quasicharacter λp ( ) = |t1 |s1 |t2 |s2 . It gives two numbers 0 t2 p 0 1 0 αp = λp ( ) and βp = λp ( ). 0 1 0 p Then these numbers are related to the p-th Fourier coefficient of f by the formula √ τp = p(αp + βp ) and αp βp = ω(p). Here ω is the so-called central character, it is the restriction of π(f ) to the center. ii) In our special situation where f is holomorphic of weight two the representation π(f )∞ of Gl2 (R) will be the first discrete series representation. If we have holomorphic modular form of weight k we get the (k − 1)-th representation of the discrete series a infinity and in the formula for the ap k−1 √ the p gets changed into p 2 . The second player in the game is our elliptic curve E/Q. This elliptic curve yields an object h1 (E), this is a motive. It is not entirely clear what this means but it creates some other objects A) A compatible system of ℓ-adic representations of the Galois group ¯ Gal(Q/Q).

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B) The Betti cohomology H 1 (E(C), Z) together with a so-called Hodge filtration on H 1 (E(C), C). I want to say a word about A). For any prime ℓ we can look on the ℓn division points ¯ | ℓn x = 0} E[ℓn ] = {x ∈ E(Q) ¯ is an abelian group and and at this point I assume that we know that E(Q) that E[ℓn ] is isomorphic to Z/ℓn Z × Z/ℓn Z. Of course these division points will have coordinates in larger and larger extensions of Q if n goes to infinity. ¯ This means that we have a natural action of Gal(Q/Q) on all these groups and if we form the projective limit Tℓ = lim E[ℓn ] ← n

the result is a free Zℓ -module of rank 2 together with a continuous action of the Galois group. I explained what it means that E has good reduction at a prime p. It is not so difficult to see that for a prime ℓ which is different from p the action of the Galois group is unramified at this prime p, in other words the inertia group acts trivially. Hence we can define a conjugacy class [Fp ] defined by the action of the Frobenius at p and the characteristic polynomial det(Id − Fp p−s |Tℓ (E)) ∈ Zℓ [p−s ] is a well-defined quantity. Now it follows from the Lefschetz fixed point formula that in fact det(Id − Fp p−s |Tℓ (E)) = 1 − ap p−s + p1−2s . This has important consequences 1) det(Id − Fp p−s |Tℓ (E)) ∈ Z[p−s ] 2) det(Id − Fp p−s |Tℓ (E)) does not depend on ℓ. Finally we have that 3) det(Id − Fp p−s |Tℓ (E)) is defined outside a finite set of primes S ∪ {ℓ}.

The Langlands Program (An overview)

227

These three properties of our different Galois modules (ℓ varies) are the defining properties for compatible systems of Galois modules. Hence we can reformulate the specific result in the first section: In our example the modular form of weight two and the elliptic curve provide a collection of local data ∞) A representation π∞ and a real Hodge structure on H 1 (E(C), Z) ⊗ C For almost all primes an unramified local representation πp of Gl2 (Qp ) and an unramified two dimensional representation ρ(πp ) of the Galois group ¯ p /Qp ) such that (in the notation used in the example) Gal(Q p) the automorphic Euler factor L(πp , s) = (1 − τp p−s + p1−2s ) is equal to the arithmetic L- factor det(Id − Fp p−s |Tℓ (E)). This means that in our example we have a second rule which produces the local components of a cusp form. This rule is provided by the elliptic curve. In this particular case it is also possible to establish the local correspondence also for the ramified primes, this has been shown by Langlands, Deligne and Carayol. It is now Langlands’ idea that such a correspondence between automorN phic representations π = ′ πv and some kind of arithmetic objects M(π) should always exist. The ideas of what nature these objects are, are also conjectural. Satake’s theorem Let us assume that we picked a prime p such that G × Qp is split. If G = Gln this can be any prime. Let Kp = G(Zp ) be the maximal compact subgroup defined by some Chevalley scheme structure G/Zp , if G = Gln this could be Gln (Zp ). To these data we attach the Hecke algebra Hp = C(Kp \G(Qp )/Kp ) : It consists of the C valued functions on G(Qp ) which are compactly supported and biinvariant under Kp and the algebra structure is given by convolution. We choose a Borel subgroup B ⊂ G and a maximal torus T ⊂ B such that T (Qp ) ∩ K = T (Zp ) is the maximal compact subgroup our torus T (Qp ). Let X∗ (T ) = Hom(Gm , T ) be the module of cocharacters, let W be the Weyl group. We introduce the module of unramified characters on the torus, this is Homunram (T (Qp ), C∗ ) = Hom(T (Qp )/T (Zp ), C∗ ) = Hom(X∗ (T ), C∗ ) = Λ(T ).

228

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We also view λ ∈ Λ(T ) as a character λ : B(Qp ) → C∗ , λ 7→ λ(b) = bλ . We will consider the group of characters Hom(T ×Q Qp , Gm ) = X ∗ (T )Qp as a subgroup of Λ(T ). An element γ ∈ X ∗ (T ) defines a homomorphism T (Qp ) → Q∗p and this gives us the following element {x 7→ |γ(x)|p } ∈ Λ(T ) which we denote by |γ|. Since we have the Iwasawa decomposition G(Qp ) = B(Qp )Kp we can attach to any λ ∈ Λ(T ) a spherical function φλ (g) = φλ (bp kp ) = (λ + |ρ|)(bp ) where ρ ∈ Λ(T ) is the half sum of positive roots. This spherical function is of course an eigenfunction for Hp under convolution, i.e. for hp ∈ Hp Z ˆ p (λ)φλ (g) φλ (gx−1 )hp (x)dx = h ˆ p (λ) is a homomorphism from Hp to C. and hp 7→ h The theorem of Satake asserts that this provides an identification ∼

Hom(Hp , C) → Λ(T )/W. To such a character we can attach an induced representation G(Q )

IndB(Qpp ) (λ) = {f : G(Qp ) → C | f (bg) = (λ + |ρ|)(b)f (g)} where in addition f |K is locally constant. These representations are called the principal series representations. We denote these irreducible modules by πp = πp (λp ) and λp is the so-called Satake parameter of πp . Let us now assume for simplicity that our group G/Q is split, for instance G = Gln /Q. In this case we may choose a split torus T /Q. We have the canonical isomorphism ∼

Hom(X∗ (T ), C∗ ) → X ∗ (T ) ⊗ C∗ and the character module X ∗ (T ) can be interpreted as the cocharacter module of the dual torus Tˆ. If we interchange the roots and the coroots then ˆ which is now a Tˆ becomes the maximal split torus of the dual group G, reductive group over C. If our group is G = Gln /Q then the dual group is Gln (C).

The Langlands Program (An overview)

229

A general philosophy Now we come back to our automorphic form π. If we write it as a restricted tensor product, then almost all the components are in the unramified principal series and now we can view the collection of unramified components {πp (λp )} as a collection of semi simple conjugacy classes in the dual group. Now Langlands philosophy assumes the existence of a very big group L and I cannot say exactly what properties this group should have. It certainly ¯ should somehow have the Weil group W (Q/Q) in it. This Weil group is some kind of complicated modification of the Galois group. We have also ¯ p /Qp ) and these are easier to explain: The group the local Weil groups W (Q ¯ ¯ W (Qp /Qp ) ⊂ Gal(Qp /Qp ) and consists of those elements whose image in ¯ p /Fp ) is an integral power of the Frobenius. Gal(F The arithmetic object M(π) attached to π should be a representation ˆ ρ(π) : L → G which at least fulfills the following requirement: At any prime p at which π is unramified the representation ρ(π) is also “unramified”. The structure of L should be such that for an unramified πp it provides an unramified representation ¯ p /Qp ) → G(C) ˆ ρ(πp ) : W (Q such that the image of the Frobenius Fp under ρ(πp ) is in the conjugacy class of the Satake parameter of πp . An unramified representation of the Weil group is of course a represen¯ p /Qp ) in Gal(F ¯ p /Fp ), therefore it is tation of the image Z =< Fp > of W (Q enough to know the image of the Frobenius Fp . Local Langlands correspondence Of course we can also consider ramified representations ¯ p /Qp ) → G(C) ˆ ρ : W (Q and the general Langlands programme predicts also a correspondence between these representations ρ and the admissible irreducible representations of G(Qp ). Actually the situation is more complicated than that, one has to replace the Weil group be the Weil-Deligne group. This is a difficult subject, we have seen a little bit of the difficulties when we discussed the Euler

230

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factors of the automorphic and the arithmetic L-function in our discussion of the Taniyama-Weil conjecture. The local Langlands conjecture is proved for the group Gln by work of Laumon-Rapoport-Stuhler ([L-R-S]) in characteristic p > 0 and by HarrisTaylor ([H-T]) in characteristic 0. This is discussed in Wedhorns lectures at this summer school ([Wed]). Representations with cohomology and motives I want to discuss a special case in which I feel a bit happier. Among the representations of G(R) there is a certain class consisting of representations π∞ which have non trivial cohomology. This means that there is a finite dimensional, irreducible rational G-module E such that H • (g, K∞ , π∞ ⊗ E) 6= {0}. Then E is determined by π∞ and for any choice of E the number of such π∞ is finite. We say that an automorphic representation π is cohomological if the component π∞ has cohomology in some module E. In this case one might speculate whether we can attach a motive or better a family of motives to it. A motive is still a conjectural object but certainly simpler in nature than L. I want to give a rough idea what a motive should be. First of all I refer to Delignes theorem that for a smooth projective scheme X/Q the ℓ adic co¯ Qℓ ) provide a compatible system of Galois modules. homology groups H i (X, A motive M is a piece in the cohomology which is defined by a projector obtained from correspondences. (In the classical case these correspondences are provided by Hecke operators). Then it is clear that M also provides a compatible system of Galois representations ¯ ¯ , Qℓ )) ρ(M ) : Gal(Q/Q) → Gl(H(M and the Euler factor at an unramified prime is defined as before by ¯ , Qℓ )) ∈ Z[p−s ]. det(Id − Fp p−s | H(M If we have an unramified principal series representation πp (λp ) and we ˆ choose in addition a finite dimensional irreducible representation r : G(C) → Gln (C), then we define the Euler factor L(πp (λp ), r, s) = det(Id − r(πp (λp ))p−s ).

The Langlands Program (An overview)

231

If we know all these Euler factors for all choices of r then we know the conˆ jugacy class of πp (λp ) viewed as an element in G(C). Now we can speculate To any cohomological π and which occurs in the space of (cuspidal) auˆ tomorphic forms on G and to any representation r : G(C) → Gln (C) we can find a motive M (π, r) such that for all unramified primes p we have an equality of local Euler factors L(r(πp (λp ), r, s) = det(Id − ρ(M )(Fp )p−s ) There should also be a matching between π∞ and the Hodge structure on the Betti cohomology of the motive. This system of ℓ representations (now r varies) should have the property that is compatible with the operations in linear algebra: If we decompose a tensor product r1 ⊗r2 into irreducibles then the Galois representations should decompose accordingly, at least if we pass to a subgroup of finite index in the Galois group. Already in the formulation we need the properties of compatible system. The right-hand side has a property which we a priori can not expect from the left hand side: Why should the automorphic Euler factors be in Z[p−s ]?? Can such a statement ever be true? Here the assumption that π∞ has cohomology helps. Using the rational (or even the integral structure) on the cohomology we can show that in fact that L(πp , r, s) viewed as polynomial in p−s has coefficients which are algebraic integers and which all lie in a finite extension of Q which depends on πf . We say that a cohomological form π is rational if these coefficients are in Z (this was so in our example). Otherwise we say that π is defined over F if F ⊂ C is generated by the coefficients of all our Euler factors. Then we can add to our assumption in our statement above that π should be rational. Otherwise we have to invent the notion of a motive with coefficients in F . This notion has been introduced by Deligne and then we can formulate the above assertion using this concept. Of course one can ask the question in the opposite direction: Given a motive is there somewhere an automorphic cohomological representation π such that M = M (π, r) for some r? Can we find such a representation even in the space of automorphic forms on Gln ? The theorem of Wiles is a special case where the answer to this question turns out to be yes.

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Functoriality If we believe in this kind of correspondence between automorphic forms and some sort of arithmetic objects, then we get remarkable consequences for automorphic forms. Let us just stick to the cohomological case. If we have two such motives we can form their product, which for the Galois modules amounts to take their tensor product. Going backwards we should be able to construct an automorphic form π1 × π2 on some bigger group. This is the principle of functoriality, which is suggested by the philosophy. Let me give an example. We consider holomorphic modular forms of weight 2, we can even go back to our example. We have seen that our modular form provides a compatible system of two dimensional ℓ-adic representations ¯ ¯ Qℓ )) ρ(H 1 (E)) : Gal(Q/Q) :→ Gl(H 1 (E, Now we take symmetric powers of these representations, this means that we take the k-fold tensor product of these representations first and this amounts to taking the k-fold product of h1 (E) by itself. Then we have an action of the symmetric group and we can take the symmetric part. In terms of the Galois representations this means that we get an representation on the symmetric tensors ¯ ¯ Qℓ ))) ρ(Symk (H 1 (E))) : Gal(Q/Q) :→ Gl(Symk (H 1 (E, and it is certainly a legitimate question whether this comes from an automorphic form. In this particular case we can look at our problem from a different point of view. We look at the L function (let us stick to our example) Y 1 1 LE (s) = ( ) −s 1−2s 1 − τp p + p 1 − 11−s p6=11

and we rewrite the Euler factors L(πp , s) =

1 1 − τp

p−s

+

p1−2s

)=

1 (1 − αp

p−s )(1

−α ¯ p p−s )

and we mention that it follows from Hasse’s theorem that α ¯ p is in fact the complex conjugate of αp . Now we form a new L-function, we pick a k > 1 and write a local L-factor at p 1 L(πp , r, s) = k−1 k −s (1 − αp p )(1 − αp α ¯ p p−s ) . . . (1 − α ¯ kp p−s )

The Langlands Program (An overview)

233

We can form a global L-function attached to the k-th symmetric power L(π, r, s) =

Y

p;p6|N

1 (1 −

αkp p−s )(1

−

αk−1 ¯ p p−s ) . . . (1 p α

−α ¯ kp p−s )

LN (π, r, s),

where I do not say anything about the factors at the ramified primes. In our case where N = 11 the Euler factor at 11 should not depend on k. Of course we can ask whether this is again an L-function attached to an automorphic cusp form on Glk+1 . This has been shown by Gelbart and Jacquet for k = 2. Here we are again in the situation where we could try to apply converse theorems, but we do not have methods to verify the necessary analytic properties of the L-Functions (see Cogdell’s Notes). But the cases k = 3, 4 have been treated successfully by Shahidi and Kim. We come to the concept of base change. Let us assume we have a (cuspidal) automorphic form π on some reductive group over Q. Let us assume we attached to it a representation ˆ ρ(π) : L → G(C) of our group L. Let us assume that we have a field extension K/Q, then it should be possible to restrict the group L to K and we would get a restricted representation ˆ ρ(π)K : LK → G(C). (This is another of the requirements one should put on L, if we work with motives then we would just extend the motive or restrict the Galois repre¯ sentations to Gal(Q/K)). Hence we should expect that this restriction of the representation ρ(π)K would provide an automorphic form on the group G × K which then would be the lift of π to G × K . The existence of such a lifting has indeed been proved for solvable extensions by Langlands in the case G = Gl2 /F and by Arthur and Clozel for G = Gln /F . This result plays a fundamental role in the proof of the Taniyama-Weil conjecture for elliptic curves and the local Langlands correspondence for Gln by Harris and Taylor.

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References [A-C]

Arthur, J.; Clozel, L.; Simple algebras, base change, and the advanced theory of the trace formula. Annals of Mathematics Studies, 120. Princeton University Press, Princeton, NJ, 1989. xiv+230 pp.

[Ca]

Carayol, H.; Formes automorphes et repr´esentations galoisiennes. (French) [Automorphic forms and Galois representations] Seminar on Number Theory, 1981/1982, Exp. No. 31, 20 pp., Univ. Bordeaux I, Talence, 1982.

[Co]

Cogdell, J.; Notes on L-functions for Gln , this volume.

[De1]

Deligne, P.; Formes modulaires et repr´esentations ℓ-adiques. S´eminaire Bourbaki, 1968/69, Exp. 335.

[Ei]

Eichler, M.; Quatern¨are quadratische Formen und die Riemannsche Vermutung f¨ ur die Kongruenzetafunktion, Arch. 5 (1954), 355-366.

[H-T]

Harris, M.; Taylor, R.; The geometry and cohomology of some simple Shimura varieties. With an appendix by Vladimir G. Berkovich. Annals of Mathematics Studies, 151. Princeton University Press, Princeton, NJ, 2001. viii+276 pp.

[He1]

Hecke, E.; Analytische Arithmetik der positiven quadratischen Formen, Kgl. Danske Selskab, Mathematisk-fysiske Meddelser. XIII, 12, 1940, 134 S.

[He2]

¨ Hecke, E.; Uber Modulformen und Dirichletsche Reihen mit Eulerscher Produktentwicklung I,II (Mathematische Annalen Bd. 114, 1937, S.1-28, S. 316-351).

[He3]

E. Hecke, E.; Mathematische Werke, Vandenhoeck & Ruprecht, G¨ottingen, 1959.

[Hu]

Husemoller, D.; Elliptic curves. With an appendix by Ruth Lawrence. Graduate Texts in Mathematics, 111. Springer-Verlag, New York, 1987. xvi+350 pp.

[La1]

Langlands, R.; Euler products. A James K. Whittemore Lecture in Mathematics given at Yale University, 1967. Yale Mathematical Monographs, 1. Yale University Press, New Haven, Conn.-London, 1971. v+53 pp.

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[La2]

Langlands, R.; Modular forms and ℓ-adic representations. Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), pp. 361-500. Lecture Notes in Math., Vol. 349, Springer, Berlin, 1973.

[La3]

Langlands, R.; Base change for Gl(2), Ann. of Math. Studies 96, Princeton University Press, 1980.

[La4]

Langlands, R.; Automorphic representations, Shimura varieties, and motives. Ein M¨archen. Automorphic forms, representations and Lfunctions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, pp. 205–246, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I.

[L-R-S] Laumon, G.; Rapoport, M.; Stuhler, U. D-elliptic sheaves and the Langlands correspondence. Invent. Math. 113 (1993), no. 2, 217-338. [Mod] Modular Functions of One Variable IV, ed. Proc. Int. summer school, Antwerp, ed. B.J. Birch and W. Kyuk, Springer Lecture Notes 476. [O]

Ogg, A. Elliptic curves and wild ramification, Am. Journal of Math. 89, p. 1-21.

[P-R]

Prasad, D. - A. Raghuram; Representation theory of GL(n) over non-Archimedean local fields, this volume.

[Si]

Silverman, J. H.; The arithmetic of elliptic curves. Corrected reprint of the 1986 original. Graduate Texts in Mathematics, 106. SpringerVerlag, New York, 1992. xii+400 pp.

[V]

Venkataramana, T.; Classical Modular Forms, this volume.

[Wed]

Wedhorn, T.; The local Langlands correspondence for GL(n) over p-adic fields, this volume.

[We]

¨ Weil, A.; Uber die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann., 168, 1967, S. 149-156.

[Wi]

Wiles, A.; Modular elliptic curves and Fermats Last Theorem. Ann. of Math., 142 (1995), 443 -551.

The Local Langlands Correspondence for GL(n) over p-adic Fields Torsten Wedhorn* Mathematisches Institut der Universit¨ at zu K¨ oln, K¨ oln, Germany

Lectures given at the School on Automorphic Forms on GL(n) Trieste, 31 July - 18 August, 2000 LNS0821006

* [email protected]

Abstract This work is intended as an introduction to the statement and the construction of the local Langlands correspondence for GL(n) over p-adic fields. The emphasis lies on the statement and the explanation of the correspondence.

Contents Introduction

241

Notations

246

1. The local Langlands correspondence

247

1.1 The local Langlands correspondence for GL(1) . . . . . . . . . . . . . 247 1.2 Formulation of the local Langlands correspondence. . . . . . . . . .251 2. Explanation of the GL(n)-side 254 2.1 Generalities on admissible representations. . . . . . . . . . . . . . . . . . .254 2.2 Induction and the Bernstein-Zelevinsky classification for GL(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 2.3 Square integrable and tempered representations . . . . . . . . . . . . . 266 2.4 Generic representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 2.5 Definition of L- and epsilon-factors . . . . . . . . . . . . . . . . . . . . . . . . . 271 3. Explanation of the Galois side 275 3.1 Weil-Deligne representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 3.2 Definition of L- and epsilon-factors . . . . . . . . . . . . . . . . . . . . . . . . . 278 4. Construction of the correspondence 282 4.1 The correspondence in the unramified case . . . . . . . . . . . . . . . . . . 282 4.2 Some reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 4.3 A rudimentary dictionary of the correspondence . . . . . . . . . . . . 286 4.4 The construction of the correspondence after Harris and Taylor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 5. Explanation of the correspondence 292 5.1 Jacquet-Langlands theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 5.2 Special p-divisible O-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 5.3 Deformation of p-divisible O-modules . . . . . . . . . . . . . . . . . . . . . . . 306 5.4 Vanishing cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 5.5 Vanishing cycles on the universal deformation of special p-divisible O-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 Bibliography

315

Local Langlands Correspondence for GL(n) over p-adic Fields

241

Introduction

Let K be a local field, i.e. K is either the field of real or complex numbers (in which case we call K archimedean) or it is a finite extension of Q p (in which case we call K p-adic) or it is isomorphic to IFq ((t)) for a finite field IFq (in which case we call K a local function field). The local Langlands conjecture for GLn gives a bijection of the set of equivalence classes of admissible representations of GLn (K) with the set of equivalence classes of n-dimensional Frobenius semisimple representations of the Weil-Deligne group of K. This bijection should be compatible with L- and ε-factors. For the precise definitions see chap. 2 and chap. 3. If K is archimedean, the local Langlands conjecture is known for a long time and follows from the classification of (infinitesimal) equivalence classes ˇ of admissible representations of GLn (K) (for K = C this is due to Zelobenko and Na˘ımark and for K = IR this was done by Langlands). The archimedean case is particularly simple because all representations of GLn (K) can be built up from representations of GL1 (IR), GL2 (IR) and GL1 (C). See the survey article of Knapp [Kn] for more details about the local Langlands conjecture in the archimedean case. If K is non-archimedean and n = 1, the local Langlands conjecture is equivalent to local abelian class field theory and hence is known for a long time (due originally to Hasse [Has]). Of course, class field theory predates the general Langlands conjecture. For n = 2 the local (and even the global Langlands conjecture) are also known for a couple of years (in the function field case this is due to Drinfeld [Dr1][Dr2], and in the p-adic case due to Kutzko [Kut] and Tunnel [Tu]). Later on Henniart [He1] gave also a proof for the p-adic case for n = 3. If K is a local function field, the local Langlands conjecture for arbitrary n has been proved by Laumon, Rapoport and Stuhler [LRS] generalizing Drinfeld‘s methods. They use certain moduli spaces of “D-elliptic sheaves” or “shtukas” associated to a global function field. Finally, if K is a p-adic field, the local Langlands conjecture for all n has been proved by Harris and Taylor [HT] using Shimura varieties, i.e. certain moduli spaces of abelian varieties. A few months later Henniart gave a much

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simpler and more elegant proof [He4]. On the other hand, the advantage of the methods of Harris and Taylor is the geometric construction of the local Langlands correspondence and that it establishes many instances of compatibility between the global and the local correspondence. Hence in all cases the local Langlands conjecture for GL(n) is now a theorem! We remark that in all cases the proof of the local Langlands conjecture for n > 1 uses global methods although it is a purely local statement. In fact, even for n = 1 (i.e. the case of local class field theory) the first proof was global in nature. This work is meant as an introduction to the local Langlands correspondence in the p-adic case. In fact, approximately half of it explains the precise statement of the local Langlands conjecture as formulated by Henniart. The other half gives the construction of the correspondence by Harris and Taylor. I did not make any attempt to explain the connections between the local theory and the global theory of automorphic forms. In particular, nothing is said about the proof that the constructed map satisfies all the conditions postulated by the local Langlands correspondence, and this is surely a severe shortcoming. Hence let me at least here briefly sketch the idea roughly: Let F be a number field which is a totally imaginary extension of a totally real field such that there exists a place w in F with Fw = K. The main idea is to look at the cohomology of a certain projective system X = (Xm )m of projective (n − 1)-dimensional F -schemes (the Xm are “Shimura varieties of PEL-type”, i.e. certain moduli spaces of abelian varieties with polarizations and a level structure depending on m). This system is associated to a reductive group G over Q such that G ⊗Q Q p is equal to Q× p × GLn (K) × anisotropic mod center factors. More precisely, these anisotropic factors are algebraic groups associated to skew fields. They affect the local structure of the Xm only in a minor way, so let us ignore them for the rest of this overview. By the general theory of Shimura varieties, to every absolutely irreducible representation ξ of G ¯ ℓ -sheaf Lξ on X where ℓ 6= p is some over Q there is associated a smooth Q ¯ ℓ -vector fixed prime. The cohomology H i (X, Lξ ) is an infinite-dimensional Q space with an action of G(Af ) × Gal(F¯ /F ) where Af denotes the ring of

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finite adeles of Q. We can choose ξ in such a way that H i (X, Lξ ) = 0 for i 6= n − 1. We have a map from the set of equivalence classes of irreducible admissible representations Π of G(Af ) to the set of finite-dimensional representations of WK ⊂ Gal(F¯ /F ) where WK denotes the Weil group of K by sending Π to Rξ (Π) = HomG(Af ) (Π, H n−1 (X, Lξ )). For every such Π the decomposition G(Af ) = Q × p × GLn (K) × remaining components gives a decomposition Π = Π0 ⊗ Πw ⊗ Πw . If π is a supercuspidal representation of GLn (K) then we can find a Π as above such that Πw ∼ = πχ where χ is an unramified character of K × , such that Π0 is unramified and such that Rξ (Π) 6= 0. ˜ m of Xm ⊗F K over OK and consider the Now we can choose a model X completions Rn,m of the local rings of a certain stratum of the special fibre ˜ m . These completions carry canonical sheaves ψi (namely the sheaf of of X m vanishing cycles) and their limits ψi are endowed with a canonical action × GLn (K) × D × 1/n × WK where D 1/n is the skew field with invariant 1/n and i i center K. If ρ is any irreducible representation of D × 1/n , ψ (ρ) = Hom(ρ, ψ ) is a representation of GLn (K) × WK . Via Jacquet-Langlands theory we can associate to every supercuspidal representation π of GLn (K) an irreducible representation ρ = jl(π ∨ ) of D × 1/n . Now there exists an n-dimensional representation r(π) of WK which satisfies

[π ⊗ r(π)] =

n−1 X

(−1)n−1−i [ψi (jl(π ∨ ))]

i=0

and n · [Rξ (Π) ⊗ χ(Π0 ◦ NmK/Q p )] ∈ ZZ[r(π)] where [ ] denotes the associated class in the Grothendieck group. To show × this one gives a description of H n−1 (X, Lξ )ZZp in which the [ψi (ρ)] occur.

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This way one gets sufficient information to see that the map π 7→ r(π ∨ ⊗ | |

1−n 2

)

defines the local Langlands correspondence. I now briefly describe the contents of the various sections. The first chapter starts with an introductory section on local abelian class field theory which is reformulated to give the local Langlands correspondence for GL1 . The next section contains the formulation of the general correspondence. The following two chapters intend to explain all terms and notations used in the formulation of the local Langlands correspondence. We start with some basic definitions in the theory of representations of reductive p-adic groups and give the Langlands classification of irreducible smooth representations of GLn (K). In some cases I did not find references for the statements (although everything is certainly well known) and I included a short proof. I apologize if some of those proofs are maybe somewhat laborious. After a short interlude about generic and square-integrable representations we come to the definition of L− and ε-factors of pairs of representations. In the following chapter we explain the Galois theoretic side of the correspondence. The fourth chapter starts with the proof of the correspondence in the unramified case. Although this is not needed in the sequel, it might be an illustrating example. After that we return to the general case and give a number of sketchy arguments to reduce the statement of the existence of a unique bijection satisfying certain properties to the statement of the existence of a map satisfying these properties. The third section contains a small “dictionary” which translates certain properties of irreducible admissible representations of GLn (K) into properties of the associated Weil-Deligne representation. In the fourth section the construction of the correspondence is given. It uses Jacquet-Langlands theory, and the cohomology of the sheaf of vanishing cycles on a certain inductive system of formal schemes. These notions are explained in the last chapter. Nothing of this treatise is new. For each of the topics there is a number of excellent references and survey articles. In many instances I just copied them (up to reordering). In addition to original articles my main sources, which can (and should) be consulted for more details, were [CF], [AT], [Neu],

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[Ta2] (for the number theoretic background), [Ca], [BZ1], [Cas1], [Ro] (for the background on representation theory of p-adic groups), and [Kud] (for a survey on “non-archimedean local Langlands”). Note that this is of course a personal choice. I also benefited from the opportunity to listen to the series of lectures of M. Harris and G. Henniart on the local Langlands correspondence during the automorphic semester at the IHP in Paris in spring 2000. I am grateful to the European network in Arithmetic Geometry and to M. Harris for enabling me to participate in this semester. This work is intended as a basis for five lectures at the summer school on “Automorphic Forms on GL(n)” at the ICTP in Triest. I am grateful to M.S. Raghunathan and G. Harder for inviting me to give these lectures. Further thanks go to U. G¨ ortz, R. Hill, N. Kr¨ amer and C. M¨ uller who made many helpful remarks on preliminary versions. Finally I would like to thank the ICTP to provide a pleasant atmosphere during the summer school.

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Notations Throughout we fix the following notations and conventions • p denotes a fixed prime number. • K denotes a p-adic field, i.e. a finite extension of Q p . • vK denotes the discrete valuation of K normalized such that it sends uniformizing elements to 1. • OK denotes the ring of integers of K. Further pK is the maximal ideal of OK and πK a chosen generator of pK . • κ denotes the residue field of OK , and q the number of elements in κ. • | |K denotes the absolute value of K which takes πK to q −1 . • ψ denotes a fixed non-trivial additive character of K (i.e. a continuous homomorphism K −→ { z ∈ C | |z| = 1 }). • n denotes a positive integer. ¯ of K and denote by κ • We fix an algebraic closure K ¯ the residue field ¯ of the ring of integers of K. This is an algebraic closure of κ. ¯ It is • K nr denotes the maximal unramified extension of K in K. also equal to the union of all finite unramified extensions of K in ¯ Its residue field is equal to κ K. ¯ and the canonical homomorphism nr Gal(K /K) −→ Gal(¯ κ/κ) is an isomorphism of topological groups. • ΦK ∈ Gal(¯ κ/κ) denotes the geometric Frobenius x 7→ x1/q and σK its inverse, the arithmetic Frobenius x 7→ xq . We also denote by ΦK and σK the various maps induced by ΦK resp. σK (e.g. on Gal(K nr /K)). • If G is any Hausdorff topological group we denote by Gab its maximal abelian Hausdorff quotient, i.e. Gab is the quotient of G by the closure of its commutator subgroup. • If A is an abelian category, we denote by Groth(A) its Grothendieck group. It is the quotient of the free abelian group with basis the isomorphism classes of objects in A modulo the relation [V ′ ] + [V ′′ ] = [V ] for objects V , V ′ and V ′′ in A which sit in an exact sequence 0 −→ V ′ −→ V −→ V ′′ −→ 0. For any abelian group X and any function λ which associates to isomorphism classes of objects in A an element in X and which is additive (i.e. λ(V ) = λ(V ′ ) + λ(V ′′ ) if there exists an exact sequence 0 −→ V ′ −→ V −→ V ′′ −→ 0) we denote the induced homomorphism of abelian groups Groth(A) −→ X again by λ.

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1 The local Langlands correspondence 1.1

The lo al Langlands orresponden e for GL(1)

(1.1.1) In this introductory section we state the local Langlands correspondence for GL1 which amounts to one of the main theorems of abelian local class field theory. For the sake of brevity we use Galois cohomology without explanation. Galois cohomology will not be needed in the sequel. (1.1.2) For any finite extension L of K of degree m and for α ∈ K × we denote by (α, L/K) ∈ Gal(L/K)ab the norm residue symbol of local class field theory. Using Galois cohomology it can be defined as follows (see e.g. [Se1] 2, for an alternative more elementary description see [Neu] chap. IV, V): The group H 2 (Gal(L/K), L× ) is cyclic of order m and up to a sign 1 ZZ/ZZ. We use now the sign convention of [Se1]. canonically isomorphic to m 1 . Let vL/K be the generator of H 2 (Gal(L/K), L× ) corresponding to − m By a theorem of Tate (e.g. [AW] Theorem 12) we know that the map ˆ q (Gal(L/K), ZZ) −→ H ˆ q+2 (Gal(L/K), L× ) which is given by cup-product H with vL/K is an isomorphism. Now we have ˆ −2 (Gal(L/K), ZZ) = H1 (Gal(L/K), ZZ) = Gal(L/K)ab H and ˆ 0 (Gal(L/K), L× ) = K ∗ /NL/K (L× ) H where NL/K denotes the norm of the extension L of K. Hence we get an isomorphism ∼

ϕL/K : Gal(L/K)ab −→ K ∗ /NL/K (L× ). We set (α, L/K) = ϕ−1 L/K ([α]) where [α] ∈ K × /NL/K (L× ) is the class of α ∈ K × .

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(1.1.3) If L is a finite unramified extension of K of degree m we also have the following description of the norm residue symbol (cf. [Se1] 2.5): Let ΦK ∈ Gal(L/K) be the geometric Frobenius (i.e. it induces on residue fields −1 the map σK : x 7→ x−q ). Then we have for α ∈ K × v (α)

(α, L/K) = ΦKK

.

(1.1.4) In the sequel we will only need the isomorphisms ϕL/K . Nevertheless let us give the main theorem of abelian local class field theory: Theorem: The map L 7→ φ(L) := NL/K (L× ) = Ker( , L/K) defines a bijection between finite abelian extensions L of K and closed subgroups of K × of finite index. If L and L′ are finite abelian extensions of K, we have L ⊂ L′ if and only if φ(L) ⊃ φ(L′ ). In this case L is characterized as the fixed field of (φ(L), L′ /K). Proof : See e.g. [Neu] chap. V how to deduce this theorem from the isomorphism Gal(L/K) ∼ = K × /NL/K (L× ) using Lubin Tate theory. We note that this is a purely local proof. (1.1.5) If we go to the limit over all finite extensions L of K, the norm residue symbol defines an isomorphism ∼

ab ¯ −→ lim K × /NL/K (L× ). lim Gal(L/K)ab = Gal(K/K) ←− L

←− L

The canonical homomorphism K × −→ lim K × /NL/K (L× ) is injective with ←− L

dense image and hence we get an injective continuous homomorphism with dense image, called the Artin reciprocity homomorphism ab ¯ ArtK : K × −→ Gal(K/K) .

¯ of K. (1.1.6) Let OK¯ be the ring of integers of the algebraic closure K ¯ Every element of Gal(K/K) defines an automorphism of OK¯ which reduces

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to an automorphism of the residue field κ ¯ of OK¯ . We get a surjective map ¯ π: Gal(K/K) −→ Gal(¯ κ/κ) whose kernel is by definition the inertia group IK of K. The group Gal(¯ κ/κ) is topologically generated by the arithmetic Frobenius automorphism σK which sends x ∈ κ ¯ to xq . It contains the free abelian group hσK i generated by σK as a subgroup. ¯ is K nr , the union of all unramified extensions The fixed field of IK in K ¯ By definition we have an isomorphism of topological groups of K in K. ∼

Gal(K nr /K) −→ Gal(¯ κ/κ).

(1.1.7) The reciprocity homomorphism is already characterized as follows ab ¯ (cf. [Se1] 2.8): Let f : K × −→ Gal(K/K) be a homomorphism such that: (a) The composition ¯ −→ Gal(¯ κ/κ) K × −→ Gal(K/K) f

v (α)

is the map α 7→ ΦKK . (b) For α ∈ K × and for any finite abelian extension L of K such that α ∈ NL/K (L× ), f (α) is trivial on L. Then f is equal to the reciprocity homomorphism ArtK . (1.1.8) We keep the notations of (1.1.6). The Weil group of K is the inverse image of hσK i under π. It is denoted by WK and it sits in an exact sequence 0 −→ IK −→ WK −→ hσK i −→ 0. We endow it with the unique topology of a locally compact group such that the projection WK −→ hσK i ∼ = ZZ is continuous if ZZ is endowed with the discrete topology and such that the induced topology on IK equals the the ¯ profinite topology induced by the topology of Gal(K/K). Note that this ¯ topology is different from the one which is induced by Gal(K/K) via the ¯ inclusion WK ⊂ Gal(K/K). But the inclusion is still continuous, and it has dense image. (1.1.9) There is the following alternative definition of the Weil group: As classes in H 2 correspond to extensions of groups, we get for every finite

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extension L of K an exact sequence 1 −→ L× −→ W (L/K) −→ Gal(L/K) −→ 1 corresponding to the class vL/K . For L ⊂ L′ we get a diagram Lx× NL′ /L

1 −→ 1 −→

L′×

−→

W (L/K)

−→

Gal(L/K) x

−→ 1

−→

W (L′ /K) −→ Gal(L′ /K) −→ 1

which can be commutatively completed by an arrow W (L′ /K) −→ W (L/K) such that we get a projective system (W (L/K))L where L runs through the ¯ Its projective limit is the Weil group of set of finite extensions of K in K. K and the projective limit of the homomorphisms W (L/K) −→ Gal(L/K) ¯ is the canonical injective homomorphism WK −→ Gal(K/K) with dense image. ab (1.1.10) Denote by WK the maximal abelian Hausdorff quotient of WK , i.e. the quotient of WK by the closure of its commutator subgroup. As the ¯ map WK −→ Gal(K/K) is injective with dense image, we get an induced injective map ab ¯ W ab ֒→ Gal(K/K) . K

It follows from (1.1.9) and from the definition of ab ¯ ArtK : K × −→ Gal(K/K) ab that the image of ArtK is WK . We get an isomorphism of topological groups ∼

ab ArtK : K × −→ WK . ab This isomorphism maps O × K onto the abelianization IK of the inertia group and a uniformizing element to a geometric Frobenius elements, i.e. if πK is a uniformizer, the image of ArtK (πK ) in Gal(¯ κ/κ) is ΦK .

(1.1.11) We can reformulate (1.1.10) as follows: Denote by A1 (K) the set of isomorphism classes of irreducible complex representations (π, V ) of K × = GL1 (K) such that the stabilizer of every vector in V is an open subgroup

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of K. It follows from the general theory of admissible representations that every (π, V ) in A1 (K) is one-dimensional (see paragraph 2.1 below). Hence A1 (K) is equal to the set of continuous homomorphisms K × −→ C × where we endow C with the discrete topology. On the other hand denote by G1 (K) the set of continuous homomorphisms WK −→ C × = GL1 (C) where we endow C × with its usual topology. Now a homomorphism WK −→ C × is continuous if and only if its restriction to the inertia group IK is continuous. But IK is compact and totally disconnected hence its image will be a compact and totally disconnected subgroup of C × hence it will be finite. It follows that a homomorphism WK −→ C × is continuous for the usual topology of C × if and only if it is continuous with respect to the discrete topology of C × . Therefore (1.1.10) is equivalent to: Theorem (Local Langlands for GL1 ): There is a natural bijection between the sets A1 (K) and G1 (K). The rest of these lectures will deal with a generalization of this theorem to GLn .

1.2

Formulation of the lo al Langlands orresponden e

(1.2.1) Denote by An (K) the set of equivalence classes of irreducible admissible representations of GLn (K). On the other hand denote by Gn (K) the set of equivalence classes of Frobenius semisimple n-dimensional complex Weil-Deligne representations of the Weil group WK (see chap. 2 and chap. 3 for a definition of these notions). (1.2.2) THEOREM (Local Langlands conjecture for GLn over p-adic fields): There is a unique collection of bijections recK,n = recn : An (K) −→ Gn (K) satisfying the following properties:

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T. Wedhorn

(1) For π ∈ A1 (K) we have rec1 (π) = π ◦ Art−1 K .

(2) For π1 ∈ An1 (K) and π2 ∈ An2 (K) we have L(π1 × π2 , s) = L(recn1 (π1 ) ⊗ recn2 (π2 ), s), ε(π1 × π2 , s, ψ) = ε(recn1 (π1 ) ⊗ recn2 (π2 ), s, ψ). (3) For π ∈ An (K) and χ ∈ A1 (K) we have recn (πχ) = recn (π) ⊗ rec1 (χ).

(4) For π ∈ An (K) with central character ωπ we have det ◦ recn (π) = rec1 (ωπ ).

(5) For π ∈ An (K) we have recn (π ∨ ) = recn (π)∨ where ( )∨ denotes the contragredient. This collection does not depend on the choice of the additive character ψ. (1.2.3) As the Langlands correspondence gives a bijection between representations of GLn (K) and Weil-Deligne representations of WK certain properties of and constructions with representations on the one side correspond to properties and constructions on the other side. Much of this is still an open problem. A few “entries in this dictionary” are given by the following theorem. We will prove it in chapter 4. Theorem: Let π be an irreducible admissible representation of GLn (K) and denote by ρ = (r, N ) the n-dimensional Weil-Deligne representation associated to π via the local Langlands correspondence. (1) The representation π is supercuspidal if and only if ρ is irreducible. (2) We have equivalent statements (i) π is essentially square-integrable.

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(ii) ρ is indecomposable. (iii) The image of the Weil-Deligne group WF′ (C) under ρ is not contained in any proper Levi subgroup of GLn (C). (3) The representation π is generic if and only if L(s, Ad ◦ ρ) has no pole at s = 1 (here Ad: GLn (C) −→ GL(Mn (C)) denotes the adjoint representation). (1.2.4) We are going to explain all occuring notations in the following two chapters.

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2 Explanation of the GL(n)-side 2.1

Generalities on admissible representations

(2.1.1) Throughout this chapter let G be a connected reductive group over K and set G = G(K). Then G is a locally compact Hausdorff group such that the compact open subgroups form a basis for the neighborhoods of the identity (this is equivalent to the fact that G has compact open subgroups and they are all profinite). In particular, G is totally disconnected. To understand (1.2.2) and (1.2.3) we will only need the cases where G is either a product of GLn ’s or the reductive group associated to some central skew field D over K. Nevertheless, in the first sections we will consider the general case of reductive groups to avoid case by case considerations. In fact, almost everywhere we could even work with an arbitrary locally compact totally disconnected group (see e.g. [Vi] for an exposition). (2.1.2) In the case G = GLn and hence G = GLn (K), a fundamental system of open neighborhoods of the identity is given by the open compact m subgroups Cm = 1 + πK Mn (OK ) for m ≥ 1. They are all contained in C0 = GLn (OK ), and it is not difficult to see that C0 is a maximal open compact subgroup and that any other maximal open compact subgroup is conjugated to C0 (see e.g. [Moe] 2). (2.1.3) Definition: A representation π: G −→ GL(V ) on a vector space V over the complex numbers is called admissible if it satisfies the following two conditions: (a) (V, π) is smooth, i.e. the stabilizer of each vector v ∈ V is open in G. (b) For every open subgroup H ⊂ G the space V H of H-invariants in V is finite dimensional. We denote the set of equivalence classes of irreducible admissible representations of G by A(G). For G = GLn (K) we define An (K) = A(GLn (K)).

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Note that the notions of “smoothness” and “admissibility” are purely algebraic and would make sense if we replace C by an arbitrary field. In fact for the rest of this survey article we could replace C by an arbitrary noncountable algebraically closed field of characteristic zero. We could avoid the “non-countability assumption” if we worked consequently only with admissible representations. Further, most elements of the general theory even work over algebraically closed fields of characteristic ℓ with ℓ 6= p ([Vi]). (2.1.4) As every open subgroup of G contains a compact open subgroup, a representation (π, V ) is smooth if and only if [ V = VC C

where C runs through the set of open and compact subgroups of G, and it is admissible if in addition all the V C are finite-dimensional. (2.1.5) Example: A smooth one-dimensional representation of K × is a quasi-character of K × or by abuse of language a multiplicative quasicharacter of K, i.e. a homomorphism of abelian groups K × −→ C × which is continuous for the discrete or equivalently for the usual topology of C × (cf. (1.1.11)). (2.1.6) Let H(G) be the Hecke algebra of G. Its underlying vector space is the space of locally constant, compactly supported measures φ on G with complex coefficients. It becomes an associative C-algebra (in general without unit) by the convolution product of measures. If we choose a Haar measure dg on G we can identify H(G) with the algebra of all locally constant complex-valued functions with compact support on G where the product is given by Z f1 (hg −1 )f2 (g) dg.

(f1 ∗ f2 )(h) =

G

(2.1.7) If C is any compact open subgroup of G, we denote by H(G//C) the subalgebra of H(G) consisting of those φ ∈ H(G) which are left- and

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right-invariant under C. If we choose a Haar measure of G, we can identify H(G//C) with the set of maps C\G/C −→ C with finite support. The algebra H(G//C) has a unit, given by eC := vol(C)−1 1C where 1C denotes the characteristic function of C. If C ′ ⊂ C is an open compact subgroup, H(G//C) is a C-subalgebra of H(G//C ′ ) but with a different unit element if C 6= C ′ . We have [ H(G) = H(G//C). C

(2.1.8) If (π, V ) is a smooth representation of G, the space V becomes an H(G)-module by the formula Z π(g)dφ π(φ)v = G

for φ ∈ H(G). This makes sense as the integral is essentially a finite sum by (2.1.7). S As V = C V C where C runs through the open compact subgroups of G, every vector v ∈ V satisfies v = π(eC )v for some C. In particular, V is a non-degenerate H(G)-module, i.e. H(G) · V = V . We get a functor from the category of smooth representations of G to the category of non-degenerate H(G)-modules. This functor is an equivalence of categories [Ca] 1.4. (2.1.9) Let C be an open compact subgroup of G and let (π, V ) be a smooth representation of G. Then the space of C-invariants V C is stable under H(G//C). If V is an irreducible G-module, V C is zero or an irreducible H(G//C)-module. More precisely we have Proposition: The functor V 7→ V C is an equivalence of the category of admissible representations of finite length such that every irreducible subquotient has a non-zero vector fixed by C with the category of finitedimensional H(G//C)-modules. Proof : This follows easily from [Cas1] 2.2.2, 2.2.3 and 2.2.4.

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(2.1.10) Corollary: Let (π, V ) be an admissible representation of G and let C be an open compact subgroup of G such that for every irreducible subquotient V ′ of V we have (V ′ )C 6= 0. Then the following assertions are equivalent: (1) The G-representation π is irreducible. (2) The H(G//C)-module V C is irreducible. (3) The associated homomorphisms of C-algebras H(G//C) −→ EndC (V C ) is surjective. Proof : The equivalence of (1) and (2) is immediate from (2.1.9). The equivalence of (2) and (3) is a standard fact of finite-dimensional modules of an algebra (see e.g. [BouA] chap. VIII, §13, 4, Prop. 5). (2.1.11) Corollary: Let (π, V ) be an irreducible admissible representation of G and let C ⊂ G be an open compact subgroup such that H(G//C) is commutative. Then dimC (V C ) ≤ 1. (2.1.12) For G = GLn (K) the hypothesis that H(G//C) is commutative is fulfilled for C = GLn (OK ). In this case we have H(G//C) = C[T1±1 , . . . , Tn±1 ]Sn where the symmetric group Sn acts by permuting the variables Ti . More generally, let G be unramified, which means that there exists a reductive model of G over OK , i.e. a flat affine group scheme over OK such that its special fibre is reductive and such that its generic fibre is equal to G. This is equivalent to the condition that G is quasi-split and split over an unramified extension [Ti] 1.10. If C is a hyperspecial subgroup of G (i.e. ˜ K ) for some reductive model G), ˜ the Hecke algebra it is of the form G(O H(G//C) can be identified via the Satake isomorphism with the algebra of invariants under the rational Weyl group of G of the group algebra of the cocharacter group of a maximal split torus of G [Ca] 4.1. In particular, it is commutative. (2.1.13) Under the equivalence of the categories of smooth G-representations and non-degenerate H(G)-modules the admissible representations (π, V )

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correspond to those non-degenerate H(G)-modules such that for any φ ∈ H(G) the operator π(φ) has finite rank. In particular, we may speak of the trace of π(φ) if π is admissible. We get a distribution φ 7→ Tr(π(φ)) which is denoted by χπ and called the distribution character of π. It is invariant under conjugation. (2.1.14) We keep the notations of (2.1.13). If {π1 , . . . , πn } is a set of pairwise non-isomorphic irreducible admissible representations of G, then the set of functionals {χπ1 , . . . , χπ2 } is linearly independent (cf. [JL] Lemma 7.1). In particular, two irreducible admissible representations with the same distribution character are isomorphic. (2.1.15) Let (π, V ) be an admissible representation of G = G(K). By a theorem of Harish-Chandra [HC] the distribution χπ is represented by a locally integrable function on G which is again denoted by χπ , i.e. for every φ ∈ H(G) we have Z χπ (g) dφ. Tr π(φ) = G

The function χπ is locally constant on the set of regular semisimple (see 5.1.3 for a definition in case G = GLn (K)) elements in G (loc. cit.), and it is invariant under conjugation. Therefore it defines a function χπ : {G}reg −→ C on the set {G}reg of conjugacy classes of regular semisimple elements in G. (2.1.16) Proposition (Lemma of Schur): Let (π, V ) be an irreducible smooth representation of G. Every G-endomorphism of V is a scalar. Proof : We only consider the case that π is admissible (we cannot use (2.1.17), because its proof uses Schur’s lemma hence we should not invoke (2.1.17) if we do not want to run into a circular argument; a direct proof of the general case can be found in [Ca] 1.4, it uses the fact that C is not

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countable). For sufficiently small open compact subgroups C of G we have V C 6= 0. Hence we have only to show that every H(G//C)-endomorphism f of a finite dimensional irreducible H(G//C)-module W over C is a scalar. As C is algebraically closed, f has an eigenvalue c, and Ker(f − c idW ) is a H(G//C)-submodule different from W . Therefore f = c idW . (2.1.17) Proposition: Let (π, V ) be an irreducible and smooth complex representation of G. (1) The representation π is admissible. (2) If G is commutative, it is one-dimensional. Proof : The first assertion is difficult and can be found for G = GLn (K) in [BZ1] 3.25. It follows from the fact that every smooth irreducible representation can be embedded in a representation which is induced from a smaller group and which is admissible (more precisely it is supercuspidal, see below). Given (1) the proof of (2) is easy: By (1) we can assume that π is admissible. For any compact open subgroup C of G the space V C is finitedimensional and a G-submodule. Hence V = V C for any C with V C 6= (0) and in particular V is finite-dimensional. But it is well known that every irreducible finite-dimensional representation of a commutative group H on a vector space over an algebraically closed field is one-dimensional (apply e.g. [BouA] chap. VIII, §13, Prop. 5 to the group algebra of H). (2.1.18) Proposition: Every irreducible smooth representation of GLn (K) is either one-dimensional or infinite-dimensional. If it is one-dimensional, it is of the form χ◦det where χ is a quasi-character of K × , i.e. a continuous homomorphism K × −→ C × . We leave the proof as an exercise (show e.g. that the kernel of a finitedimensional representation π: GLn (K) −→ GLn (C) is open, deduce that π is trivial on the subgroup of unipotent upper triangular matrices U , hence π is trivial on the subgroup of GLn (K) which is generated by all conjugates of U and this is nothing but SLn (K)). (2.1.19) Let Z be the center of G. As K is infinite, Z(K) is the center Z of G. In the case G = GLn (K) we have Z = K × . For (π, V ) ∈ A(G) we

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denote by ωπ : Z −→ C × its central character, defined by ωπ (z) idV = π(z) for z ∈ Z. It exists by the lemma of Schur. For G = GLn (K), ωπ is a quasi-character of K × . (2.1.20) Proposition Assume that G/Z is a compact group. Then every irreducible admissible representation (π, V ) of G is finite-dimensional. Proof : By hypothesis we can find a compact open subgroup G0 of G such that G0 Z has finite index in G (take for example the group G0 defined in 2.4.1 below). The restriction of an irreducible representation π of G to G0 Z decomposes into finitely many irreducible admissible representations. By the lemma of Schur, Z acts on each of these representation as a scalar, hence they are also irreducible representations of the compact group G0 and therefore they are finite-dimensional. (2.1.21) Let (π, V ) be a smooth representation of G and let χ be a quasicharacter of G. The twisted representation πχ is defined as g 7→ π(g)χ(g). The G-submodules of (π, V ) are the same as the G-submodules of (πχ, V ). In particular π is irreducible if and only if πχ is irreducible. Further, if C is a compact open subgroup of G, χ(C) ⊂ C × is finite, and therefore χ is trivial on a subgroup C ′ ⊂ C of finite index. This shows that π is admissible if and only if πχ is admissible. If G = GLn (K) every quasi-character χ is of the form χ′ ◦ det where χ′ is a multiplicative quasi-character of K (2.1.18), and we write πχ′ instead of πχ. (2.1.22) Let π: G −→ GL(V ) be a smooth representation of G. Denote by V ∗ the C-linear dual of V . It is a G-module via (gλ)(v) = λ(g −1 v) which is not smooth if dim(V ) = ∞. Define V ∨ = { λ ∈ V ∗ | StabG (λ) is open }.

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This is a G-submodule π ∨ which is smooth by definition. It is called the contragredient of the G-module V . Further we have: (1) π is admissible if and only if π ∨ is admissible and in this case the biduality homomorphism induces an isomorphism V −→ (V ∨ )∨ of G-modules. (2) π is irreducible if and only if π ∨ is irreducible. (3) In the case of G = GLn (K) we can describe the contragredient also in the following way: If π is smooth and irreducible, π ∨ is isomorphic to the representation g 7→ π(t g −1 ) for g ∈ GLn (K). Assertions (1) and (2) are easy (use that (V ∨ )C = (V C )∗ for every compact open subgroup C). The last assertion is a theorem of Gelfand and Kazhdan ([BZ1] 7.3).

2.2

Indu tion and the Bernstein-Zelevinsky lassifi ation for GL(n)

(2.2.1) Fix an ordered partition n = (n1 , n2 , . . . , nr ) of n. Denote by Gn the algebraic group GLn1 × · · · × GLnr considered as a Levi subgroup of G(n) = GLn . Denote by Pn ⊂ GLn the parabolic subgroup of matrices of the form A1 A2 ∗ ... 0 ... Ar for Ai ∈ GLni and by Un its unipotent radical. If (πi , Vi ) is an admissible representation of GLni (K), π1 ⊗ . . . ⊗ πr is an admissible reprentation of Gn (K) on W = V1 ⊗ · · · ⊗ Vr . By extending this representation to Pn and by normalized induction we get a representation π1 × · · · × πr of GLn (K) whose underlying complex vector space V is explicitly defined by V =

1/2 f : GLn (K) −→ W f smooth, f (umg) = δn (m)(π1 ⊗ · · · ⊗ πr )(m)f (g) . for u ∈ Un (K), m ∈ GLn (K) and g ∈ GLn (K)

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Here we call a map f : GLn (K) −→ W smooth if its stabilizer { g ∈ GLn (K) | f (gh) = f (h) for all h ∈ GLn (K) } is open in GLn (K) (or equivalently if f is fixed by some open compact of 1/2 GLn (K) acting by right translation), and δn denotes the positive square root of the modulus character δn (m) = | det(AdUn (m))|. The group GLn (K) acts on V by right translation.

(2.2.2) Definition: An irreducible smooth representation of GLn (K) is called supercuspidal if there exists no proper partition n such that π is a subquotient of a representation of the form π1 × · · · × πr where πi is an admissible representation of GLni (K). We denote by A0n (K) ⊂ An (K) the subset of equivalence classes of supercuspidal representations of GLn (K).

(2.2.3) Let πi be a smooth representation of GLni (K) for i = 1, . . . , r. Then π = π1 ×· · ·×πr is a smooth representation of GLn (K) with n = n1 +· · ·+nr . Further it follows from the compactness of GLn (K)/Pn that if the πi are admissible, π is also admissible ([BZ1] 2.26). Further, by [BZ2] we have the following Theorem: If the πi are of finite length (and hence admissible by (2.1.17)) for all i = 1, . . . , r (e.g. if all πi are irreducible), π1 ×· · ·×πr is also admissible and of finite length. Conversely, if π is an irreducible admissible representation of GLn (K), there exists a unique partition n = n1 + · · · + nr of n and unique (up to isomorphism and ordering) supercuspidal representations πi of GLni (K) such that π is a subquotient of π1 × · · · × πr .

(2.2.4) If π is an irreducible admissible representation of GLn (K) we denote the unique unordered tuple (π1 , · · · , πr ) of supercuspidal representations such that π is a subquotient of π1 × · · · × πr the supercuspidal support.

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(2.2.5) Definition: Let π: GLn (K) −→ GL(V ) be a smooth representation. For v ∈ V and λ ∈ V ∨ the map cπ,v,λ = cv,λ : G −→ C,

g 7→ λ(π(g)v)

is called the (v, λ)-matrix coefficient of π. (2.2.6) Let (π, V ) be an admissible representation. (1) For v ∈ V = (V ∨ )∨ and λ ∈ V ∨ we have cπ,v,λ(g) = cπ∨ ,λ,v (g −1 ). (2) If χ is a quasi-character of K × we have cπχ,v,λ(g) = χ(det(g))cπ,v,λ.

(2.2.7) Theorem: Let π be a smooth irreducible representation of GLn (K). Then the following statements are equivalent: (1) π is supercuspidal. (2) All the matrix coefficents of π have compact support modulo center. (3) π ∨ is supercuspidal. (4) For any quasi-character χ of K × , πχ is supercuspidal. Proof : The equivalence of (1) and (2) is a theorem of Harish-Chandra [BZ1] 3.21. The equivalence of (2), (3) and (4) follows then from (2.2.6). (2.2.8) For any complex number s and for any admissible representation we define π(s) as the twist of π with the character | |s , i.e. the representation g 7→ | det(g)|s π(g). If π is supercuspidal, π(s) is also supercuspidal. Define a partial order on A0n (K) by π ≤ π ′ iff there exists an integer n ≥ 0 such that π ′ = π(n). Hence every finite interval ∆ is of the form ∆(π, m) = [π, π(1), . . . , π(m − 1)]. The integer m is called the length of the interval and nm is called its degree. We write π(∆) for the representation π × · · · × π(m − 1) of GLnm (K).

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Two finite intervals ∆1 and ∆2 are said to be linked if ∆1 6⊂ ∆2 , ∆2 6⊂ ∆1 , and ∆1 ∪ ∆2 is an interval. We say that ∆1 precedes ∆2 if ∆1 and ∆2 are linked and if the minimal element of ∆1 is smaller than the minimal element of ∆2 . (2.2.9) Theorem (Bernstein-Zelevinsky classification ([Ze], cf. also [Ro])): (1) For any finite interval ∆ ⊂ A0n (K) of length m the representation π(∆) has length 2m−1 . It has a unique irreducible quotient Q(∆) and a unique irreducible subrepresentation Z(∆). (2) Let ∆1 ⊂ A0n1 (K), . . . , ∆r ⊂ A0nr (K) be finite intervals such that for i < j, ∆i does not precede ∆j (this is an empty condition if ni 6= nj ). Then the representation Q(∆1 ) × · · · × Q(∆r ) admits a unique irreducible quotient Q(∆1 , . . . , ∆r ), and the representation Z(∆1 ) × · · · × Z(∆r ) admits a unique irreducible subrepresentation Z(∆1 , . . . , ∆r ). (3) Let π be a smooth irreducible representation of GLn (K). Then it is isomorphic to a representation of the form Q(∆1 , . . . , ∆r ) (resp. Z(∆′1 , . . . , ∆′r′ )) for a unique (up to permutation) collection of intervals ∆1 , . . . , ∆r (resp. ∆′1 , . . . , ∆′r′ ) such that ∆i (resp. ∆′i ) does not precede ∆j (resp. ∆′j ) for i < j. (4) Under the hypothesis of (2), the representation Q(∆1 ) × · · · × Q(∆r ) is irreducible if and only if no two of the intervals ∆i and ∆j are linked. (2.2.10) For π ∈ A0n (K) the set of π ′ in A0n (K) which are comparable with π with respect to the order defined above is isomorpic (as an ordered set) to ZZ, in particular it is totally ordered. It follows that given a tuple of intervals ∆i = [πi , . . . , πi (mi − 1)], i = 1, . . . , r we can always permute them such that ∆i does not precede ∆j for i < j. Denote by Sn (K) the set of unordered tuples (∆1 , . . . , ∆r ) where ∆i is P an interval of degree ni such that ni = n. Then (2) and (3) of (2.2.9) are equivalent to the assertion that the maps Q: Sn (K) −→ An (K),

(∆1 , . . . , ∆r ) 7→ Q(∆1 , . . . , ∆r ),

Z: Sn (K) −→ An (K),

(∆1 , . . . , ∆r ) 7→ Z(∆1 , . . . , ∆r ),

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are bijections. The unordered tuple of supercuspidal representations πi (j) for i = 1, . . . , r and j = 0, . . . , mi − 1 is called the supercuspidal support. It is the unique unordered tuple of supercuspidal representations ρ1 , . . . , ρs such that π = Q(∆1 , . . . , ∆s ) and π ′ = Z(∆1 , . . . , ∆r ) is a subquotient of ρ1 × · · · × ρs ([Ze]). (2.2.11) If Rn (K) is the Grothendieck group of the category of admissible L representations of GLn (K) of finite length and R(K) = n≥0 Rn (K), then ([π1 ], [π2 ]) 7→ [π1 × π2 ]

defines a map R(K) × R(K) −→ R(K) which makes R(K) into a graded commutative ring ([Ze] 1.9) which is isomorphic to the ring of polynomials in the indeterminates ∆ for ∆ ∈ S(K) = S n≥1 Sn (K) (loc. cit. 7.5). The different descriptions of An (K) via the maps Q and Z define a map t: R(K) −→ R(K),

Q(∆) 7→ Z(∆).

We have: Proposition: (1) The map t is an involution of the graded ring R. (2) It sends irreducible representations to irreducible representations. (3) For ∆ = [π, π(1), . . . , π(m − 1)] we have t(Q(∆)) = Q(π, π(1), . . . , π(m − 1)) where on the right-hand side we consider π(i) as intervals of length 1. (4) We have t(Q(∆1 , . . . , ∆r )) = Z(∆1 , . . . , ∆r ), t(Z(∆1 , . . . , ∆r )) = Q(∆1 , . . . , ∆r ). Proof : Assertions (1) and (3) follow from [Ze] 9.15. The second assertion had been anounced by J.N. Bernstein but no proof has been published. It has been proved quite recently in [Pr] or [Au1] (see also [Au2]). Assertion (4) is proved by Rodier in [Ro] th´eor`eme 7 under the assumption of (2).

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(2.2.12) For each interval ∆ = [π, . . . , π(m − 1)] we set ∆∨ = [π(m − 1)∨ , . . . , π ∨ ] = [π ∨ (1 − m), . . . , π ∨ (−1), π ∨ ]. It follows from [Ze] 3.3 and 9.4 (cf. also [Tad] 1.15 and 5.6) that we have ∨ Q(∆1 , . . . , ∆r )∨ = Q(∆∨ 1 , . . . , ∆r ), ∨ Z(∆1 , . . . , ∆r )∨ = Z(∆∨ 1 , . . . , ∆r ).

In particular we see that the involution on R induced by [π] 7→ [π ∨ ] commutes with the involution t in (2.2.11).

(2.2.13) Example: Let ∆ ⊂ A1 (K) be the interval ∆ = (| |(1−n)/2 , | |(3−n)/2 , . . . , | |(n−1)/2 ). The associated representation of the diagonal torus T ⊂ GLn (K) is equal to −1/2 δB where δB (t) = | det AdU (t)|K is the modulus character of the adjoint action of T on the group of unipotent upper triangular matrices U and where B is the subgroup of upper triangular matrices in GLn (K). Hence we see that π(∆) = | |(1−n)/2 × | |(3−n)/2 × . . . × | |(n−1)/2 consists just of the space of smooth functions on B\G with the action of G induced by the natural action of G on the flag variety B\G. Hence Z(∆) is the trivial representation 1 of constant functions on G/B. The representation Q(∆) = t(Z(1)) is called the Steinberg representation and denoted by St(n). It is selfdual, i.e. St(n)∨ = St(n) (in fact, it is also unitary and even square integrable, see the next section). For n = 2 the length of π(∆) is 2, hence we have St(n) = π(∆)/1.

2.3

Square integrable and tempered representations

(2.3.1) We return to the general setting where G is an arbitrary connected reductive group over K. Every character α: G −→ G m defines on K-valued

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points a homomorphism α: G −→ K × . By composition with the absolute value | |K we obtain a homomorphism |α|K : G −→ IR>0 and we set \ G0 = Ker(|α|K ). α

If G = GLn then every α is a power of the determinant, hence we have GLn (K)0 = { g ∈ GLn (K) | | det(g)|K = 1 }. Let r be a positive real number. We call an admissible representation (π, V ) of G essentially Lr if for all v ∈ V and λ ∈ V ∨ the matrix coefficient cv,λ is Lr on G0 , i.e. the integral Z |cλ,v |r dg G0

exists (where dg denotes some Haar measure of G0 ). An admissible representation is called Lr if it is essentially Lr and if it has a central character (2.1.19) which is unitary. Let Z be the center of G. Then the composition G0 −→ G −→ G/Z has compact kernel and finite cokernel. Hence, if (π, V ) has a unitary central character ωπ , the integral Z |cv,λ |r dg

Z\G

makes sense, and (π, V ) is L if and only if this integral is finite. r

(2.3.2) Proposition: Let π be an admissible representation of G which is ′ Lr . Then it is Lr for all r ′ ≥ r. Proof : This follows from [Si3] 2.5. (2.3.3) Definition: An admissible representation of G is called essentially square integrable (resp. essentially tempered) if it is essentially L2 (resp. essentially L2+ε for all ε > 0). We have similar definitions by omitting “essentially”. By (2.3.2), any (essentially) square integrable representation is (essentially) tempered.

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(2.3.4) The notion of “tempered” is explained by the following proposition (which follows from [Si1] §4.5 and [Si3] 2.6): Proposition: Let π be an irreducible admissible representation of G such that its central character is unitary. Then the following assertions are equivalent: (1) π is tempered. (2) Each matrix coefficient defines a tempered distribution on G (with the usual notion of a tempered distribution: It extends from a linear form on the locally constant functions with compact support on G to a linear form on the Schwartz space of G (the “rapidly decreasing functions on G”), see [Si1] for the precise definition in the p-adic setting). (3) The distribution character of π is tempered. (2.3.5) Example: By (2.2.7) any supercuspidal representation is essentially Lr for all r > 0. In particular it is essentially square integrable. (2.3.6) If (π, V ) is any smooth representation of G which has a central character, then there exists a unique positive real valued quasi-character χ of G such that πχ has a unitary central character (for G = GLn (K) this is clear as every quasi-character factors through the determinant (2.1.18), for arbitrary reductive groups this is [Cas1] 5.2.5). Hence for G = GLn (K) the notion of “essential square-integrability” is equivalent to the notion of “quasi-square-integrability” in the sense of [Ze]. In particular it follows from [Ze] 9.3: Theorem: An irreducible admissible representation π of GLn (K) is essentially square-integrable if and only if it is of the form Q(∆) with the notations of (2.2.9). It is square integrable if and only if ∆ is of the form [ρ, ρ(1), . . . , ρ(m − 1)] where the central character of ρ((m − 1)/2) is unitary. (2.3.7) We also have the following characterization of tempered representations in the Bernstein-Zelevinsky classification (see [Kud] 2.2): Proposition: An irreducible admissible representation Q(∆1 , . . . , ∆r ) of GLn (K) is tempered if and only if the Q(∆i ) are square integrable.

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(2.3.8) If π = Q(∆1 , . . . , ∆r ) is a tempered representation no two of the intervals ∆i = [ρi , . . . , ρi (mi − 1)] are linked as cent(∆i ) = ρi ((mi − 1)/2) has unitary central character and all elements in ∆i different from cent(∆i ) have a non-unitary central character. Therefore we have π = Q(∆1 ) × · · · × Q(∆r ).

(2.3.9) Let π = Q(∆1 , . . . , ∆r ) be an arbitrary irreducible admissible representation. For each ∆i there exists a unique real number xi such that Q(∆i )(−xi ) is square integrable. We can order the ∆i ’s such that y1 := x1 = · · · = xm1 > y2 := xm1 +1 = · · · = xm2 > · · · > ys := xms−1 +1 = · · · = xr . In this order ∆i does not precede ∆j for i < j and all ∆i ’s which correspond to the same yj are not linked. For j = 1, . . . , s set πj = Q(∆mj−1 +1 )(−yj ) × · · · × Q(∆mj )(−yj ) with m0 = 0 and ms = r. Then all πj are irreducible tempered representation, and π is the unique irreducible quotient of π1 (y1 ) × · · · × πs (ys ). This is nothing but the Langlands classification which can be generalized to arbitrary reductive groups (see [Si1] or [BW]).

2.4

Generi representations

(2.4.1) Fix a non-trivial additive quasi-character ψ: F −→ C × and let n(ψ) −n be the exponent of ψ, i.e. the largest integer n such that ψ(πK OK ) = 1. (2.4.2) Let Un (K) ⊂ GLn (K) be the subgroup of unipotent upper triangular matrices and define a one-dimensional representation θψ of Un (K) by θψ ((uij )) = ψ(u12 + · · · + un−1,n ).

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If π is any representation of GLn (K) we can consider the space of homomorphisms of Un (K)-modules HomUn (K) (π|Un (K) , θψ ). If π is smooth and irreducible we call π generic if this space is non-zero. (2.4.3) In the next few sections we collect some facts about generic representations of GLn (K) which can be found in [BZ1], [BZ2] and [Ze]. Note that in loc. cit. the term “non-degenerate” is used instead of “generic”. First of all we have: Proposition: (1) The representation π is generic if and only if π ∨ is generic. (2) For all multiplicative quasicharacters χ: K × −→ C × , π is generic if and only if χπ is generic. (3) The property of π being generic does not depend of the choice of the non-trivial additive character ψ. (2.4.4) Via the Bernstein-Zelevinsky classification we have the following characterization of generic representations ([Ze] 9.7): Theorem: An irreducible admissible representation π = Q(∆1 , . . . , ∆r ) is generic if and only if no two segments ∆i are linked. In particular we have π∼ = Q(∆1 ) × · · · × Q(∆r ).

(2.4.5) Corollary: Every essentially tempered (and in particular every supercuspidal) representation is generic. (2.4.6) If (π, V ) is generic, it has a Whittaker model: Choose a 0 6= λ ∈ HomU (K) (π|U (K) , θψ ) and define a map V −→ { f : GLn (K) −→ C | f (ug) = θ(u)f (g) for all g ∈ GLn (K), u ∈ U (K) }, v 7→ (g 7→ λ(π(g)v))

.

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This is an injective homomorphism of GLn (K)-modules if GLn (K) acts on the right-hand side by right translation, and we call its image the Whittaker model of π with respect to ψ and denote it by W(π, ψ). (2.4.7) The concept of a generic representation plays a fundamental role in the theory of automorphic forms: If π is an irreducible admissible representation of the adele valued group GLn (AL ) for a number field L, it can be decomposed in a restricted tensor product O π= πv v

where v runs through the places of L and where πv is an admissible irreducible representation of GLn (Lv ) (see Flath [Fl] for the details). If π is cuspidal, all the πv are generic by Shalika [Sh].

2.5

Definition of L- and epsilon-fa tors

(2.5.1) Let π and π ′ be smooth irreducible representations of GLn (K) and of GLn′ (K) respectively. We are going to define L- and ε-factors of the pair (π, π ′ ). We first do this for supercuspidal (or more generally for generic) representation and then use the Bernstein-Zelevinsky classification to make the general definition. Assume now that our fixed non-trivial additive character ψ (2.4.1) is ¯ Let π and π ′ be generic representations of GLn (K) unitary, i.e. ψ−1 = ψ. and GLn′ (K) respectively. To define L- and ε-factors L(π × π ′ , s) and ε(π × π ′ , s, ψ) we follow [JPPS1]. Consider first the case n = n′ . Denote by S(K n ) the set of locally constant functions φ: K n −→ C with compact support. For elements W ∈ ¯ in the Whittaker models and for any φ ∈ S(K n ) W(π, ψ), W ′ ∈ W(π ′ , ψ) define Z ′ Z(W, W , φ, s) = W (g)W ′ (g)φ((0, . . . , 0, 1)g)| det(g)|s dg Un (K)\GLn (K)

where dg is a GLn (K)-invariant measure on Un (K)\GLn (K). This is absolutely convergent if Re(s) is sufficiently large and it is a rational function of

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q −s . The set ¯ and φ ∈ S(K n ) } { Z(W, W ′ , φ, s) | W ∈ W(π, ψ), W ′ ∈ W(π ′ , ψ) generates a fractional ideal in C[q s , q −s ] with a unique generator L(π ×π ′ , s) of the form P (q −s )−1 where P ∈ C[X] is a polynomial such that P (0) = 1. Further ε(π × π ′ , s, ψ) is defined by the equality ˆ ˜ ,W ˜ ′ , 1 − s, φ) Z(W, W ′ , s, φ) Z(W n ′ ′ (−1) ε(π × π , s, ψ) = ω . π L(π ∨ × π ′∨ , 1 − s) L(π × π ′ , s) ˜ by W ˜ (g) = W (wn t g −1 ) where wn ∈ GLn (K) is the Here we define W permutation matrix corresponding to the longest Weyl group element (i.e. to the permutation which sends i to n + 1 − i). Because of (2.1.22) this is an ¯ In the same way we define W ˜ ′ ∈ W(π ′∨ , ψ). Finally element of W(π ∨ , ψ). φˆ denotes the Fourier transform of φ with respect to ψ given by Z ˆ φ(y)ψ(t y x) dy φ(x) = Kn

for x ∈ K n . ¯ and Now consider the case n′ < n. For W ∈ W(π, ψ), W ′ ∈ W(π ′ , ψ) for j = 0, 1, . . . , n − n′ − 1 define g 0 0 Z Z )W ′ (g) W ( x Ij 0 Z(W, W ′ , j, s) = Un′ (K)\GLn′ (K) Mj×n′ (K) 0 0 In−n′ −j ′

· | det(g)|s−(n−n )/2 dx dg where dg is a GLn′ (K)-invariant measure on Un′ (K)\GLn′ (K) and dx is a Haar measure on the space of (j × n′ )-matrices over K. Again this is absolutely convergent if Re(s) is sufficiently large, it is a rational function of q −s and these functions generate a fractional ideal with a unique generator L(π × π ′ , s) of the form P (q −s )−1 where P ∈ C[X] is a polynomial such that P (0) = 1. In this case ε(π × π ′ , s, ψ) is defined by ˜ ,W ˜ ′ , n − n′ − 1 − j, 1 − s) Z(wn,n′ W Z(W, W ′ , j, s) n−1 ′ ′ (−1) = ω ε(π×π , ψ, s) π L(π ∨ × π ′∨ , 1 − s) L(π × π ′ , s) where wn,n′ is the matrix In0 ′ w 0 ′ ∈ GLn (K). n−n

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Finally for n′ > n we define L(π × π ′ , s) = L(π ′ × π, s),

ε(π × π ′ , ψ, s) = ε(π ′ × π, ψ, s).

In all cases L(π × π ′ , s) does not depend on the choice of ψ, and ε(π × π ′ , s, ψ) is of the form cq −f s for a non-zero complex number c and an integer f which depend only on π, π ′ and ψ. This finishes the definition of the L- and the ε-factor for the generic case (and in particular for the supercuspidal case). Note that if π and π ′ are supercuspidal, we have Y (2.5.1.1) L(π × π ′ , s) = L(χ, s) χ

∼π where χ runs over the unramified quasi-characters of K × such that χπ ′∨ = and where L(χ, s) is the L-function of a character as defined in Tate’s thesis (see also below). In particular we have L(π × π ′ , s) = 1 for π ∈ A0n (K) and π ′ ∈ A0n′ (K) with n 6= n′ . It seems that there is no such easy way to define ε(π × π ′ , ψ, s) for supercuspidal π and π ′ . However, Bushnell and Henniart [BH] prove that ε(π × π ∨ , ψ, 1/2) = ωπ (−1)n−1 for every irreducible admissible representation π of GLn (K). (2.5.2) From the definition of the L- and ε-factor in the supercuspidal case we deduce the definition of the L- and ε-factor for pairs of arbitrary smooth irreducible representations π and π ′ by the following inductive relations using (2.2.9)(cf. [Kud]): (1) We have L(π × π ′ , s) = L(π ′ × π, s) and ε(π × π ′ , ψ, s) = ε(π ′ × π, ψ, s). (2) If π is of the form Q(∆1 , . . . , ∆r ) (2.2.9) and if π ′ is arbitrary, then L(π × π ′ , s) =

r Y

L(Q(∆i ) × π ′ , s)

ε(π × π ′ , ψ, s) =

r Y

ε(Q(∆i ) × π ′ , ψ, s).

i=i

i=i

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(3) If π is of the form Q(∆), ∆ = [σ, σ(r − 1)] and π ′ = Q(∆′ ), ∆′ = [σ ′ , σ ′ (r ′ − 1)] with r ′ ≥ r, then L(π × π ′ , s) =

r Y

L(σ × σ ′ , s + r + r ′ − 1)

i=1 r r+r Y Y−2i ′

′

ε(π × π , ψ, s) =

i=1

×

ε(σ × σ ′ , ψ, s + i + j − 1)

j=0

′ −2i−1 r+rY

j=0

L(σ ∨ × σ ′∨ , 1 − s − i − j) . L(σ × σ ′ , s + i + j − 1)

(2.5.3) Let 1: K × −→ C × be the trivial multiplicative character. For any smooth irreducible representation π of GLn (K) we define L(π, s) = L(π × 1, s), ε(π, ψ, s) = ε(π × 1, ψ, s). For n = 1, L(π, s) and ε(π, ψ, s) are the local L- and ε-factors defined in Tate’s thesis. For n > 1 and π supercuspidal, we have L(π, s) = 1, while ε(π, ψ, s) is given by a generalized Gauss sum [Bu]. (2.5.4) Let (π, V ) be a smooth and irreducible representation of GLn (K). For any non-negative integer t define a b t t Kn (t) = { ∈ GLn (OK )|c ∈ M1×n−1 (πK OK ), d ≡ 1 (mod πK OK ) }. c d In particular, we have Kn (0) = GLn (OK ). The smallest non-negative integer t such that V Kn (t) 6= (0) is called the conductor of π and denoted by f (π). By [JPPS1] (cf. also [BHK]) it is also given by the equality ε(π, ψ, s) = ε(π, ψ, 0)q −s(f (π)+nn(ψ)) where n(ψ) denotes the exponent of ψ (2.4.1).

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3 Explanation of the Galois side 3.1

Weil-Deligne representations

(3.1.1) Let WK be the Weil group of K (1.1.8) and let ϕK : WK −→ ¯ Gal(K/K) be the canonical homomorphism. ¯ A representation of WK (resp. of Gal(K/K)) is a continuous homo¯ morphism WK −→ GL(V ) (resp. Gal(K/K) −→ GL(V )) where V is a finite-dimensional complex vector space. Denote by Rep(WK ) (resp. ¯ Rep(Gal(K/K))) the category of representations of the respective group. Note that a homomorphism of a locally profinite group (e.g. WK or ¯ Gal(K/K)) into GLn (C) is continuous for the usual topology of GLn (C) if and only if it is continuous for the discrete topology. (3.1.2) For w ∈ WK we set |w| = |w|K = |Art−1 K (w)|K . Then the map WK −→ C × , w 7→ |w|s is a one-dimensional representation (i.e. a quasi-character) of WK for every complex number s. All onedimensional representations of WK which are trivial on IK (i.e. which are unramified) are of this form ([Ta1] 2.3.1). ¯ (3.1.3) As ϕK is injective with dense image, we can identify Rep(Gal(K/K)) with a full subcategory of Rep(WK ). A representation in this subcategory is called of Galois-type. By [Ta2] 1.4.5 a representation r of WK is of Galoistype if and only if its image r(WK ) is finite. Conversely, by [De2] §4.10 and (3.1.2) every irreducible representation r of WK is of the form r = r ′ ⊗ | |s for some complex number s and for some representation r ′ of Galois type. (3.1.4) A representation of Galois-type of WK is irreducible if and only ¯ if it is irreducible as a representation of Gal(K/K). Further, if σ is any irreducible representation of WK , it is of Galois type if and only if the image of its determinant det ◦σ is a subgroup of finite order of C × .

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¯ Then we have a canonical (3.1.5) Let L be a finite extension of K in K. injective homomorphism WL −→ WK with finite cokernel. Hence restriction and induction of representations give functors resL/K :Rep(WK ) −→ Rep(WL ) indL/K :Rep(WL ) −→ Rep(WK ) satisfying the usual Frobenius reciprocity. More precisely, any representation of WK becomes a representation of WL by restriction r 7→ r|WK . This defines the map resL/K . Conversely, let r: WL −→ GL(V ) be a representation of WL . Then we define indL/K (r) as the representation of WK whose underlying vector space consists of the continuous maps f : WK −→ V such that f (xw) = r(x)f (w) for all x ∈ WL and w ∈ WK . Note that in the context of the cohomology of abstract groups this functor “induction” as defined above is often called “coinduction”.

(3.1.6) Definition: A Weil-Deligne representation of WK is a pair (r, N ) where r is a representation of WK and where N is a C-linear endomorphism of V such that (3.1.6.1)

r(γ) N = |Art−1 K (γ)|K N r(γ)

for γ ∈ WK . It is called Frobenius semisimple if r is semisimple.

(3.1.7) Remark: Let (r, N ) be a Weil-Deligne representation of WK . (1) Let γ ∈ WK be an element corresponding to a uniformizer πK via ArtK . Applying (3.1.6.1) we see that N is conjugate to qN , hence every eigenvalue of N must be zero which shows that N is automatically nilpotent. (2) The kernel of N is stable under WK , hence if (r, N ) is irreducible, N is equal to zero. Therefore the irreducible Weil-Deligne representations of WK are simply the irreducible continuous representations of WK .

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(3.1.8) Let ρ1 = (r1 , N1 ) and ρ2 = (r2 , N2 ) be two Weil-Deligne representations on complex vector spaces V1 and V2 respectively. Their tensor product ρ1 ⊗ ρ2 = (r, N ) is the Weil-Deligne representation on the space V1 ⊗ V2 given by r(w)(v1 ⊗ v2 ) = r1 (w)v1 ⊗ r2 (w)v2 ,

N (v1 ⊗ v2 ) = N1 v1 ⊗ v2 + v1 ⊗ N2 v2

for w ∈ WK and vi ∈ Vi , i = 1, 2. Further, HomC (V1 , V2 ) becomes the vector space of a Weil-Deligne representation Hom(ρ1 , ρ2 ) = (r, N ) by (r(w)ϕ)(v1) = r2 (w)(ϕ(r1 (w)−1 v1 )),

(N ϕ)(v1) = N2 (ϕ(v1 ))−ϕ(N1 (v1 ))

for ϕ ∈ HomC (V1 , V2 ), w ∈ WK and v1 ∈ V1 . In particular we get the contragredient ρ∨ of a Weil-Deligne representation as the representation Hom(ρ, 1) where 1 is the trivial one-dimensional representation. (3.1.9) Consider WK as a group scheme over Q (not of finite type) which is the limit of the constant group schemes associated to the discrete groups WK /J where J runs through the open normal subgroups of IK . Denote by ′ WK the semi-direct product ′ | G WK = WK × a

where WK acts on G a by the rule wxw−1 = |w|K x. This is a group scheme (neither affine nor of finite type) over Q whose R-valued points for some | R, and the Q-algebra R without non-trivial idempotents are given by WK × law of composition is given by (w1 , x1 )(w2 , x2 ) = (w1 w2 , |w2 |−1 K x1 + x2 ). A Weil-Deligne representation of WK is the same as a complex finite′ dimensional representation of the group scheme WK whose underlying WK representation is semisimple (to see this use the fact that representations of the additive group on a finite-dimensional vector space over a field in characteristic zero correspond to nilpotent endomorphisms). ′ | C) is called The group scheme WK (or also its C-valued points WK × the Weil-Deligne group.

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(3.1.10) It follows from the Jacobson-Morozov theorem that we can also interpret a Weil-Deligne representation as a continuous complex semisimple representation of the group WK × SL2 (C). If η is such a representation, we associate a Weil-Deligne representation (r, N ) by the formulas ! 1 0 |w| 2 ) r(w) = η(w, 1 0 |w|− 2 and exp(N ) = η(1,

1 1 ). 0 1

A theorem of Kostant assures that two representations of WF × SL2 (C) are isomorphic if and only if the corresponding Weil-Deligne representations are isomorphic (see [Ko] for these facts).

3.2

Definition of L- and epsilon-fa tors

(3.2.1) Let ρ = ((r, V ), N ) be a Frobenius semisimple Weil-Deligne representation. Denote by VN the kernel of N and by VNIK the space of invariants in VN for the action of the inertia group IK . The L-factor of ρ is given by L(ρ, s) = det(1 − q −s Φ|V IK )−1 N

where Φ ∈ WK is a geometric Frobenius. If ρ and ρ′ are irreducible WeilDeligne representations of dimension n, n′ respectively, we have Y (3.2.1.1) L(ρ ⊗ ρ′ , s) = L(χ, s) χ

ab where χ runs through the unramified quasi-characters of K × ∼ such = WK ∨ ′ ′ that χ ⊗ ρ = ρ (compare (2.5.1.1)). In particular L(ρ ⊗ ρ , s) = 1 for n 6= n′ . Fix a non-trivial additive character ψ of K and let n(ψ) be the largest −n integer n such that ψ(πK OK ) = 1. Further let dx be an additive Haar measure of K.

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To define ε(ρ, ψ, s) we first define the ε-factor of the Weil group representation (r, V ). Assume first that V is one-dimensional, i.e. r = χ is a quasi-character ab χ: WK −→ C × .

Let χ be unramified (i.e. χ(IK ) = (1) or equivalently χ = | |s for some complex number s). Then we set ε(χ, ψ, dx) = χ(w)q n(ψ) voldx (OK ) = q n(ψ)(1−s) voldx (OK ) where w ∈ WK is an element whose valuation is n(ψ). If χ is ramified, let f (χ) be the conductor of χ, i.e. the smallest integer f f such that χ(ArtK (1 + πK OK )) = 1, and let c ∈ K × be an element with valuation n(ψ) + f (χ). Then we set Z ε(χ, ψ, dx) = χ−1 (ArtK (x))ψ(x)dx. c−1 OK ×

The ε-factors attached to (r, V ) with dim(V ) > 1 are characterized by the following theorem of Langlands and Deligne [De2]: Theorem: There is a unique function ε which associates with each choice of a local field K, a non-trivial additive character ψ of K, an additive Haar measure dx on K and a representation r of WK a number ε(r, ψ, dx) ∈ C × such that (1) If r = χ is one-dimensional ε(χ, ψ, dx) is defined as above. (2) ε( , ψ, dx) is multiplicative in exact sequences of representations of WK (hence we get an induced homomorphism ε( , ψ, dx): Groth(Rep(WK )) −→ C × ). (3) For every tower of finite extensions L′ /L/K and for every choice of additive Haar measures µL on L and µL′ on L′ we have ε(indL′ /L [r ′ ], ψ ◦ TrL/K , µL ) = ε([r ′ ], ψ ◦ TrL′ /K , µL′ ) for [r ′ ] ∈ Groth(Rep(WL′ )) with dim([r ′ ]) = 0. Note that we have ε(χ, ψ, αdx) = αε(χ, ψ, dx) for α > 0 and hence via inductivity ε(r, ψ, αdx) = αdim(r) ε(χ, ψ, dx). In particular if [r] ∈

280

T. Wedhorn

Groth(Rep(WK )) is of dimension 0, ε([r], ψ, dx) is independent of the choice of dx. Now we can define the ε-factor of the Weil-Deligne representation ρ = (r, N ) as ε(ρ, ψ, s) = ε(| |s r, ψ, dx) det(−Φ|V IK /V IK ) N

where dx is the Haar measure on K which is self-dual with respect to the Fourier transform f 7→ fˆ defined by ψ: Z ˆ f (y) = f (x)ψ(xy) dx. In other words ([Ta1] 2.2.2) it is the Haar measure for which OK gets the volume q −d/2 where d is the valuation of the absolute different of K (if the ramification index e of K/Q p is not divided by p, we have d = e − 1, in general d can be calculated via higher ramification groups [Se2]). Note that ε(ρ, ψ, s) is not additive in exact sequences of Weil-Deligne representations as taking coinvariants is not an exact functor. (3.2.2) Let ρ be an irreducible Weil-Deligne representation of dimension n, then we can define the conductor f (ρ) of ρ by the equality ε(ρ, ψ, s) = ε(ρ, ψ, 0)q −s(f (ρ)+nn(ψ)) where n(ψ) denotes the exponent of ψ (2.4.1). This is a nonnegative integer which can be explicitly expressed in terms of higher ramification groups (e.g. [Se2] VI,§2, Ex. 2). (3.2.3) For any m ≥ 1 we define the Weil-Deligne representations Sp(m) = ((r, V ), N ) by V = Ce0 ⊕ · · · ⊕ Cem−1 with r(w)ei = |w|i ei and N ei = ei+1

(0 ≤ i < m − 1),

N em−1 = 0.

In this case we have VN = VNIK = Cem−1 and Φei = q −i ei for a geometric Frobenius Φ ∈ WK . Hence the L-factor is given by 1 L(ρ, s) = . 1−s−m 1−q

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Let ψ be an additive character such that n(ψ) = 0 and let dx be the Haar measure on K which is self-dual with respect to Fourier transform as above. Then we have ε(r, ψ, dx) = q −md/2 where d is the valuation of the absolute different of K. Hence the ε-factor is given by ε(ρ, ψ, s) = (−1)m−1q

−md−(m−2)(m−1) 2

.

(3.2.4) A Frobenius semisimple Weil-Deligne representation ρ is indecomposable if and only if it has the form ρ0 ⊗ Sp(m) for some m ≥ 1 and with ρ0 irreducible. Moreover, the isomorphism class of ρ0 and m are uniquely determined by ρ ([De1] 3.1.3(ii)). Further (as in every abelian category where all objects have finite length) every Frobenius semisimple Weil-Deligne representation is the direct sum of unique (up to order) indecomposable Frobenius semisimple Weil-Deligne representations.

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4 Construction of the correspondence 4.1

The orresponden e in the unramified ase

(4.1.1) Definition: An irreducible admissible representation (π, V ) of GLn (K) is called unramified, if the space of fixed vectors under C = GLn (OK ) is non-zero, i.e. if its conductor (2.5.4) is zero. (4.1.2) Example: A multiplicative quasi-character χ: K × −→ C × is unramified if and only if χ(O × K ) = {1}. An unramified quasi-character χ is uniquely determined by its value χ(πK ) which does not depend on the choice of the uniformizing element πK . It is of the form | |s for a unique s ∈ C/(2πi(log q)−1 )ZZ. (4.1.3) Let (χ1 , . . . , χn ) be a family of unramified quasi-characters which we can view as intervals of length zero in A01 (K). We assume that for i < j, χi does not precede χj , i.e. χ−1 i χj 6= | |K . Then Q(χ1 , . . . , χn ) is an unramified representation of GLn (K). Conversely we have [Cas2] Theorem: Every unramified representation π of GLn (K) is isomorphic to a representation of the form Q(χ1 , . . . , χn ) where the χi are unramified quasi-characters of K × . (4.1.4) An unramified representation π of GLn (K) is supercuspidal if and × only if n = 1 and π is an unramified quasi-character of K . (4.1.5) Let π be an unramified representation associated to unramified quasi-characters χ1 , . . . , χn . This tuple of unramified quasi-characters induces a homomorphism T /Tc −→ C × n where T ∼ = (K × )n denotes the diagonal torus of G and where Tc ∼ = (O × K) denotes the unique maximal compact subgroup of T of diagonal matrices with coefficients in O × K.

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Thus the set of unramified representations may be identified with the set of orbits under the Weyl group Sn of GLn in Tˆ = Hom(T /Tc , C × ) = (C × )n where the last isomorphism is given by the identification T /Tc = ZZn ,

diag(t1 , . . . , tn ) 7→ (vK (t1 ), . . . , vK (tn )).

(4.1.6) To shorten notations set C = GLn (OK ). The Hecke algebra H(GLn (K)//C) is commutative and canonically isomorphic to the Sn invariants of the group algebra ([Ca] 4.1) ±1 C[X ∗ (Tˆ)] = C[X∗ (T )] ∼ = C[t±1 1 , . . . , tn ].

If (π, V ) is an unramified representation, V C is one-dimensional (2.1.11), hence we get a canonical homomorphism λπ : H(GLn (K)//C) −→ End(V C ) = C. For every h ∈ H(GLn (K)//C) the map Tˆ/ΩGLn −→ C,

π 7→ λπ (h)

can be considered as an element in C[X ∗ (Tˆ)]Sn and this defines the isomorphism ∼ H(GLn (K)//C) −→ C[X ∗ (Tˆ)]Sn .

(4.1.7) Definition: An n-dimensional Weil-Deligne representation ρ = ((r, V ), N ) is called unramified if N = 0 and if r(IK ) = {1}. (4.1.8) Every unramified n-dimensional Weil-Deligne representation ρ = ((r, C n ), N ) is uniquely determined by the GLn (C)-conjugacy class of r(Φ) =: gρ for a geometric Frobenius Φ. By definition this element is semisimple and hence we can consider this as an Sn -orbit of the diagonal torus (C × )n of GLn (C). Hence we get a bijection recn between unramified representations of GLn (K) and unramified n-dimensional Weil-Deligne representations.

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This is normalized by the following two conditions: (i) An unramified quasi-character χ of IK× corresponds to an unramified ab quasi-character rec1 (χ) of WK via the map ArtK from local class field theory. (ii) The representation Q(χ1 , . . . , χn ) (4.1.3) corresponds to the unramified Weil-Deligne representation rec1 (χ1 ) ⊕ · · · ⊕ rec1 (χn ). By the inductive definition of L- and ε-factors it follows that the bijection recn satisfies condition (2) of (1.2.2) (see also (2.5.1.1) and (3.2.1.1)). Further condition (3) for unramified characters and condition (4) are clearly okay, and condition (5) follows from the obvious fact that if unramified elements in An (K) or in Gn (K) correspond to the Sn -orbit of diag(t1 , . . . , tn ) their contragredients correspond to the orbit of diag(t1 , . . . , tn )−1 = −1 diag(t−1 1 , . . . , tn ). (4.1.9) From the global point of view, unramified representations are the “normal” ones: If O π= πv v

is an irreducible admissible representation of the adele valued group GLn (AL ) for a number field L as in (2.4.7), all but finitely many πv are unramified.

4.2

Some redu tions

(4.2.1) In this paragraph we sketch some arguments (mostly due to Henniart) which show that it suffices to show the existence of a family of maps (recn ) satisfying all the desired properties between the set of isomorphism classes of supercuspidal representations and the set of isomorphism classes of irreducible Weil-Deligne representations. We denote by A0n (K) the subset of An (K) consisting of the supercuspidal representations of GLn (K). Further let Gn0 (K) be the set of irreducible Weil-Deligne representations in Gn (K).

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(4.2.2) Reduction to the supercuspidal case: In order to prove the local Langlands conjecture (1.2.2), it suffices to show that there exists a unique collection of bijections recn : A0n (K) −→ Gn0 (K) satisfying (1.2.2) (1) to (5). Reasoning : This follows from (2.2.9) and from (3.2.4). More precisely, for any irreducible admissible representation π ∼ = Q(∆1 , . . . , ∆r ), with ∆i = [πi , πi (mi − 1)] and πi ∈ A0ni (K) define recn1 m1 +···+nr mr (π) =

r M

recni (πi ) ⊗ Sp(mi ).

i=1

Properties (1.2.2) (1) to (5) follow then (nontrivially) from the inductive description of the L- and the ε-factors.

(4.2.3) Reduction to an existence statement: If there exists a collection of bijections (recn )n as in (4.2.2), it is unique. This follows from the fact that representations π ∈ A0n (K) are already determined inductively by their ε-factors in pairs and that by (1.2.2)(1) rec1 is given by class field theory. More precisely, we have the following theorem of Henniart [He3]: Theorem: Let n ≥ 2 be an integer and let π and π ′ be representations in A0n (K). Assume that we have an equality ε(π × τ, ψ, s) = ε(π ′ × τ, ψ, s) for all integers r = 1, . . . , n − 1 and for every τ ∈ A0r (K). Then π ∼ = π′ .

(4.2.4) Injectivity: Every collection of maps recn as in (4.2.2) is automatically injective: If χ is a quasi-character of K × , its L-function L(χ, s) is given by (1 − χ(π)q s )−1 , if χ is unramified, L(χ, s) = 1, if χ is ramified.

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In particular, it has a pole in s = 0 if and only if χ = 1. Hence by (2.5.1.1) and (3.2.1.1) we have for π, π ′ ∈ A0n (K): recn (π) = recn (π ′ ) ⇔ L(recn (π)∨ ⊗ recn (π ′ ), s) has a pole in s = 0 ⇔ L(π ∨ × π ′ , s) has a pole in s = 0 ⇔ π = π′ .

(4.2.5) Surjectivity: In order to prove the local Langlands conjecture it suffices to show that there exists a collection of maps recn : A0n (K) −→ Gn0 (K) satisfying (1.2.2) (1) to (5). Reasoning : Because of the preservation of ε-factors in pairs it follows from (3.2.2) and (2.5.4) that recn preserves conductors. But by the numerical local Langlands theorem of Henniart [He2] the sets of elements in A0n (K) and Gn0 (K) which have the same given conductor and the same central character are finite and have the same number of elements. Hence the bijectivity of recn follows from its injectivity (4.2.4).

4.3

A rudimentary di tionary of the orresponden e

(4.3.1) In this section we give some examples how certain properties of admissible representations can be detected on the corresponding Weil-Deligne representation and vice versa. Throughout (π, Vπ ) denotes an admissible irreducible representation of GLn (K), and ρ = ((r, Vr ), N ) the n-dimensional Frobenius-semisimple Weil-Deligne representation associated to it via the local Langlands correspondence (1.2.2).

(4.3.2) First of all, we have of course: Proposition: The admissible representation π is supercuspidal if and only if ρ is irreducible (or equivalently iff r is irreducible and N = 0).

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(4.3.3) Write π = Q(∆1 , . . . , ∆s ) in the Bernstein-Zelevinsky classification (2.2.9), where ∆i = [πi , . . . , πi (mi − 1)] is an interval of supercuspidal representations of GLni (K). By (4.2.2) we have ρ=

s M

(recni (πi ) ⊗ Sp(mi )).

i=1

Set ρi = ((ri , Vri ), 0) = recni (πi ). The underlying representation of the Weil group of πi ⊗ Sp(mi ) is then given by ri ⊕ ri (1) ⊕ · · · ⊕ ri (mi − 1) where r(x) denotes the representation w 7→ r(w)|w|x for any representation r of WK and any real number x. We have (ri (j), 0) = recni (πi (j)). Further, if Ni is the nilpotent endomorphism of ρi ⊗Sp(mi ), its conjugacy class (which we can consider as a non-ordered partition of ni mi by the Jordan normal form) is given by the partition

Hence we get:

ni mi = mi + · · · + mi . {z } | ni -times

Proposition: The underlying WK -representation r of ρ depends only on the supercuspidal support τ1 , . . . , τt of π (2.2.10). More precisely, we have an isomorphism of Weil-Deligne representations (r, 0) ∼ = rec(τ1 ) ⊕ . . . ⊕ rec(τt ). The conjugacy class of N is given by the degree ni of πi and the length mi of the intervals ∆i as above. In particular, we have N = 0 if and only if all intervals ∆i are of length 1. (4.3.4) Example: The Steinberg representation St(n) (2.2.13) corresponds to the Weil-Deligne representation | |(1−n)/2 Sp(n). (4.3.5) Recall from (4.1.3) that π is unramified if and only if all intervals ∆i are of length 1 and consist of an unramified quasi-character of K × . Hence

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(4.3.3) shows that π is unramified if and only if ρ is unramified. We used this already in (4.1). (4.3.6) The “arithmetic information” of WK is encoded in the inertia subgroup IK . The quotient WK /IK is the free group generated by ΦK and hence “knows” only the number q of elements in the residue field of K. Therefore Weil-Deligne representations ρ = (r, N ) with r(IK ) = 1 should be particularly simple. We call such representations IK -spherical. Then r is a semisimple representation of < ΦK >∼ = ZZ. Obviously, every finite-dimensional semisimple representation of ZZ is the direct sum of onedimensional representations. Hence r is the direct sum of quasi-characters of WK which are necessarily unramified. On the GLn (K)-side let I ⊂ GLn (K) be an Iwahori subgroup, i.e. I is an open compact subgroup of GLn (K) which is conjugated to the group of matrices (aij ) ∈ GLn (OK ) with aij ∈ πK OK for i > j. We say that π is I-spherical if the space of I-fixed vectors is non-zero. By a theorem of Casselman ([Ca] 3.8, valid for arbitrary reductive groups - with the appropriate reformulation) an irreducible admissible representation is I-spherical if and only if its supercuspidal support consists of unramified quasi-characters. Altogether we get: Proposition: We have equivalent assertions: (1) The irreducible admissible representation π is I-spherical. (2) The supercuspidal support of π consists of unramified quasicharacters. (3) The corresponding Weil-Deligne representation ρ is IK -spherical. By (2.1.9) the irreducible admissible I-spherical representations are nothing but the finite-dimensional irreducible H(GLn (K)//I)-modules. The structure of the C-algebra H(GLn (K)//I) is known in terms of generators and relations ([IM]) and depends only on the isomorphism class of GLn over some algebraically closed field (i.e. the based root datum of GLn ) and on the number q.

(4.3.7) Finally, we translate several notions which have been defined for

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admissible representations of GLn (K) into properties of Weil-Deligne representations: Proposition: Let π be an irreducible admissible representation of GLn (K) and let ρ = (r, N ) be the corresponding Weil-Deligne representation. (1) We have equivalent statements (i) π is essentially square-integrable. (ii) ρ is indecomposable. (iii) The image of the Weil-Deligne group WF′ (C) under ρ is not contained in any proper Levi subgroup of GLn (C). (2) We have equivalent statements (i) π is tempered. (ii) Let η be a representation of WK × SL2 (C) associated to ρ (unique up to isomorphism) (3.1.10). Then η(WF ) is bounded. (iii) Let η be as in (ii) and let Φ ∈ WK a geometric Frobenius. Then η(Φ) has only eigenvalues of absolute value 1. (3) The representation π is generic if and only if L(s, Ad ◦ ρ) has no pole at s = 1 (here Ad: GLn (C) −→ GL(Mn (C)) denotes the adjoint representation). Proof : (1): The equivalence of (i) and (ii) follows from (2.3.6) and (3.2.4), the equivalence of (ii) and (iii) is clear as any factorization through a Levi subgroup GLn1 (C) × GLn2 (C) ⊂ GLn (C) would induce a decomposition of ρ. (3): This is [Kud] 5.2.2. (2): The equivalence of (ii) and (iii) follows from the facts that the image of the inertia group IK under η is finite, as IK is compact and totally disconnected, and that a subgroup H of semisimple elements in GLn (C) is bounded if and only if every element of H has only eigenvalues of absolute value 1 (use the spectral norm). Now ρ is indecomposable if and only if η is indecomposable. To prove the equivalence of (i) and (iii) we can therefore assume by (2.3.7) and (4.3.3) that ρ is indecomposable, i.e. that π = Q(∆) is essentially square integrable. Let x ∈ IR be the unique real number such that Q(∆)(x) is square integrable (2.3.6). Then the description of rec(Q(∆)) in (4.3.3) shows that | det(η(w))| = | det(r(w))| = |w|nx .

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Now π is square integrable if and only if its central character is unitary. But by property (4) of the local Langlands classification the central character of π is given by det ◦r. Hence (iii) is equivalent to (i) by the following lemma whose proof we leave as an exercise: Lemma: Let π be a supercuspidal representation of GLn (K) and denote by ωπ its central character. For an integer m ≥ 1 let δ be the interval [π((1 − m)/2), π((m − 1)/2)]. Let η be a representation of WK × SL2 (C) associated to (r, N ) = rec(Q(∆)) (3.1.10). Then for w ∈ WK all absolute 1/n values of eigenvalues of η(w) are equal to |ωπ (Art−1 . In particular K (w))| all eigenvalues of η(w) have the same absolute value. Hint : First show the result for m = 1 where there is no difference between η and r. Then the general result can be checked by making explicit the Jacobson-Morozov theorem in the case of GLnm (K).

4.4

The onstru tion of the orresponden e after Harris and Taylor

¯ ℓ of (4.4.1) Fix a prime ℓ 6= p and an isomorphism of an algebraic closure Q Q ℓ with C. Denote by κ the residue field of OK and by κ ¯ an algebraic closure of κ. For m ≥ 0 and n ≥ 1 let ΣK,n,m be the unique (up to isomorphism) one-dimensional special formal OK -module of OK -height n with Drinfeld ¯ level pm K -structure over k. Its deformation functor on local Artinian OK algebras with residue field κ ¯ is prorepresented by a complete noetherian local OK -algebra RK,n,m with residue field κ ¯ . Drinfeld showed that RK,n,m is regular and that the canonical maps RK,n,m −→ RK,n,m+1 are finite and flat. The inductive limit (over m) of the formal vanishing cycle sheaves of ¯ ℓ gives a collection (Ψi Spf(RK,n,m ) with coefficients in Q K,ℓ,h ) of infinite¯ dimensional Q ℓ -vector spaces with an admissible action of the subgroup of GLh (K) × D × × WK consisting of elements (γ, δ, σ) such that K,1/n |Nrdδ|| det γ|−1 |Art−1 K σ| = 1. For any irreducible representation ρ of D × K,1/n set ΨiK,ℓ,n (ρ) = HomD×

K,1/n

i ). (ρ, ψK,ℓ,n

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This is an admissible (GLn (K) × WK )-module. Denote by [ΨK,ℓ,n (ρ)] the virtual representation (−1)

n−1

n−1 X

(−1)i [ΨiK,ℓ,n (ρ)].

i=0

Then the first step is to prove: Construction theorem: Let π be an irreducible supercuspidal representation of GLn (K). Then there is a (true) representation ¯ ℓ ) = GLn (C) rℓ (π): WK −→ GLn (Q such that in the Grothendieck group [ΨK,ℓ,n (JL(π)∨ )] = [π ⊗ rℓ (π)] where JL denotes the Jacquet-Langlands bijection between irreducible representations of D × K,1/n and essentially square integrable irreducible admissible representations of GLn (K). Using this theorem we can define recn = recK,n : A0n (K) −→ Gn (K) by the formula recn (π) = rℓ (π ∨ ⊗ (| |K ◦ det)(1−n)/2 ). That this map satisfies (1.2.2) (1) - (5) follows from compatibility of rℓ with many instances of the global Langlands correspondence. The proof of these compatibilities and also the proof of the construction theorem follow from an analysis of the bad reduction of certain Shimura varieties. I am not going into any details here and refer to [HT]. (4.4.2) In the rest of this treatise we explain the ingredients of the construction of the collection of maps (recn ).

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5 Explanation of the correspondence 5.1

Ja quet-Langlands theory

(5.1.1) We collect some facts about skew fields with center K (see e.g. [PR] as a reference). Let Br(K) be the Brauer group of K. As a set it can be identified with the set of isomorphism classes of finite-dimensional division algebras over K with center K. For D, D ′ ∈ Br(K), D ⊗ D ′ is again a central simple algebra over K, hence it is isomorphic to a matrix algebra Mr (D ′′ ) for some D ′′ ∈ Br(K). If we set D · D ′ := D ′′ , this defines the structure of an abelian group on Br(K). This group is isomorphic to Q/ZZ where the homomorphism Q/ZZ −→ Br(K) is given as follows: For a rational number λ with 0 ≤ λ < 1 we write λ = s/r for integers r, s which are prime to each other and with r > 0 (and we make the convention 0 = 0/1). Then the associated skew field Dλ is given by Dλ = Kr [Π] where Kr is the (unique up to isomorphism) unramified extension of K of degree r and where Π is an indeterminate satisfying the relations Πr = πks and Πa = σK (a)Π for a ∈ Kr . We call r the index of Dλ . It is the order of Dλ as an element in the Brauer group and we have dimK (Dλ ) = r 2 . If B is any simple finite-dimensional K-algebra with center K, it is isomorphic to Mr (D) for some skew field D with center K. Further, B ⊗K L is a simple L-algebra with center L for any extension L of K. In particular ¯ is isomorphic to an algebra of matrices over K ¯ as there do not B ⊗K K exist any finite-dimensional division algebras over algebraically closed fields ¯ except K ¯ itself. K Conversely, if D is a skew field with center K which is finite-dimensional ¯ is isomorover K we can associate the invariant inv(D) ∈ Q/ZZ: As D ⊗K K ¯ we have dimK (D) = r 2 . The valuation phic to some matrix algebra Mr (K), vK on K extends uniquely to D by the formula 1 vD (δ) = vK (NrdD/K (δ)) r

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for δ ∈ D. Moreover D is complete in the topology given by this valuation. It follows from the definition of vD that the ramification index e of D over K is smaller than r. Set OD = { δ ∈ D | vD (δ) ≥ 0 },

PD = { δ ∈ D | vD (δ) > 0 }.

Clearly PD is a maximal right and left ideal of OD and the quotient κD = OD /PD is a skew field which is a finite extension of κ, hence it is finite and has to be commutative. Let L ⊂ D be the unramified extension corresponding to the extension κD of κ. As no skew field with center K of dimension r 2 can contain a field of K-degree bigger than r we have for the inertia index f of D over K f = [κD : κ] = [L : K] ≤ r. Hence the formula r 2 = ef shows that e = f = r. Further we have seen that D contains a maximal unramified subfield. The extension L/K is Galois with cyclic Galois-group generated by the Frobenius automorphism σK . By the Skolem-Noether theorem (e.g. [BouA] VIII, §10.1), there exists an element δ ∈ D × such that σK (x) = δxδ −1 for all x ∈ L. Then inv(D) = vD (δ) ∈

1 ZZ/ZZ ⊂ Q/ZZ r

is the invariant of D. (5.1.2) For every D ∈ Br(K) we can consider its units as an algebraic group over K. More precisely, we define for every K-algebra R D × (R) = (D ⊗K R)× . This is an inner form of GLn,K if n is the index of D. (5.1.3) Let D ∈ Br(K) be a division algebra with center K of index n. Let {d} be a D × -conjugacy class of elements in D × . The image of ¯ ∼ ¯ is a GLn (K)-conjugacy ¯ {d} in D ⊗K K class {d}′ of elements in = Mn (K) ¯ which does not depend on the choice of the isomorphism D ⊗K K ¯ ∼ GLn (K) = ¯ ¯ Mn (K) as any automorphism of Mn (K) is an inner automorphism. Further, ¯ {d}′ is fixed by the natural action of Gal(K/K) on conjugacy classes of

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¯ GLn (K). Hence its similarity invariants in the sense of [BouA] chap. 7, §5 are polynomials in K[X] and it follows that there is a unique GLn (K)¯ is conjugacy class of elements α({d}) in GLn (K) whose image in GLn (K) ′ {d} . Altogether we get a canonical injective map α from the set of D × conjugacy classes {D × } in D × into the set of GLn (K)-conjugacy classes {GLn (K)} in GLn (K). The image of α consists of the set of conjugacy classes of elliptic elements in GLn (K). Recall that an element g ∈ GLn (K) is called elliptic if it is contained in a maximal torus T (K) of GLn (K) such that T (K)/K × is compact. Equivalently, g is elliptic if and only if K[g] is a field. We call a conjugacy class {g} in GLn (K) semisimple if it consists of ¯ or, equivalently, if K[g] is a prodelements which are diagonalizable over K uct of field extensions for g ∈ {g}. A conjugacy class {g} is called regular semisimple if it is semisimple and if all eigenvalues of elements in {g} in ¯ are pairwise different. Note that every elliptic element is semisimple. K We make the same definitions for conjugacy classes in D × , or equivalently we call a conjugacy class of D × semisimple (resp. regular semisimple) if its image under α: {D × } −→ {G} is semisimple (resp. regular semisimple). (5.1.4) Denote by A2 (G) the set of isomorphism classes of irreducible admissible essentially square integrable representations of G. We now have the following theorem which is due to Jacquet and Langlands in the case n = 2 and due to Rogawski and Deligne, Kazhdan and Vigneras in general ([Rog] and [DKV]): Theorem: Let D be a skew field with center K and with index n. There exists a bijection, called Jacquet-Langlands correspondence, JL: A2 (D × ) ↔ A2 (GLn (K)) which is characterized on characters by (5.1.4.1)

χπ = (−1)n−1 χJL(π) .

Further JL satisfies the following conditions: (1) We have equality of central characters ωπ = ωJL(π) .

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(2) We have an equality of L-functions and of ε-functions up to a sign L(π, s) = L(JL(π), s),

ε(π, ψ, s) = ε(JL(π), ψ), s)

(for the definition of L- and ε-function of irreducible admissible representations of D × see e.g. [GJ]). (3) The Jacquet-Langlands correspondence is compatible with twist by characters: If χ is a multiplicative quasi-character of K, we have JL(π(χ ◦ Nrd)) = JL(π)(χ ◦ det).

(4) It is compatible with contragredient: JL(π ∨ ) = JL(π)∨ .

(5.1.5) Remark: Note that for G = D × every admissible representation is essentially square integrable as D × /K × is compact.

5.2

Spe ial p-divisible O-modules

(5.2.1) Let R be a ring. A p-divisible group over R is an inductive system G = (Gn , in )n≥1 of finite locally free commutative group schemes Gn over Spec(R) and group scheme homomorphisms in : Gn → Gn+1 such that for all integers n there is an exact sequence in−1 ◦···◦i1

p

0 −→ G1 −−−−−−→ Gn −→ Gn−1 −→ 0 of group schemes over Spec(R). We have the obvious notion of a homomorphism of p-divisible groups. This way we get a ZZp -linear category. As the Gn are finite locally free, their underlying schemes are by definition of the form Spec(An ) where An is an R-algebra which is a finitely generated locally free R-module. In particular, it makes sense to speak of the rank of An which we also call the rank of Gn . From the exact sequence above it follows that Gn is of rank pnh for some non-negative locally constant function h: Spec(R) −→ ZZ which is called the height of G.

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(5.2.2) Let G = (Gn ) be a p-divisible group over some ring R and let R′ be an R-algebra. Then the inductive system of Gn ⊗R R′ defines a p-divisible group over R′ which we denote by GR′ . (5.2.3) Let G = (Gn ) be a p-divisible group over a ring R. If there exists some integer N ≥ 1 such that pN R = 0, the Lie algebra Lie(Gn ) is a locally free R-module for n ≥ N whose rank is independent of n ≥ N . We call this rank the dimension of G. More generally, if R is p-adically complete we define the dimension of G as the dimension of the p-divisible group GR/pR over R/pR. (5.2.4) Let R be an OK -algebra. A special p-divisible OK -module over R is a pair (G, ι) where G is a p-divisible group over R and where ι: OK −→ End(G) is a homomorphism of ZZp -algebras such that for all n ≥ 1 the OK action induced by ι on Lie(Gn ) is the same as the OK -action which is induced from the R-module structure of Lie(G) via the OK -module structure of R. In other words the induced homomorphism OK ⊗ZZp OK −→ End(Lie(Gn )) factorizes through the multiplication OK ⊗ZZp OK −→ OK . The height ht(G) of a special p-divisible OK -module (G, ι) is always divisible by [K : Q p ] and we call htOK (G) := [K : Q p ]−1 ht(G) the OK height of (G, ι). (5.2.5) If (G = (Gn ), ι) is a special p-divisible OK -module over an OK algebra R and if R −→ R′ is an R-algebra, we get an induced OK -action ι′ on GR′ and the pair (GR′ , ι′ ) is a special p-divisible OK -module over R′ which we denote by (G, ι)R′ . (5.2.6) Let k be a perfect extension of the residue class field κ of OK . Denote by W (k) the ring of Witt vectors of k. Recall that this is the unique (up to unique isomorphism inducing the identity on k) complete discrete valuation ring with residue class field k whose maximal ideal is generated by p. Further W (k) has the property that for any complete local noetherian ring R with residue field k there is a unique local homomorphism W (k) −→ R inducing the identity on k.

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In particular, we can consider W (κ). It can be identified with the ring of integers of the maximal unramified extension of Q p in K (use the universal property of the ring of Witt vectors). Set WK (k) = W (k) ⊗W (κ) OK . This is a complete discrete valuation ring of mixed characteristic with residue field k which is a formally unramified OK -algebra (i.e. the image of pK generates the maximal ideal of WK (k)). There exists a unique continuous automorphism σK of W (k) which induces the automorphism x 7→ xq on k. We denote the induced automorphism σK ⊗ idOK again by σK . (5.2.7) Proposition: The category of special p-divisible OK -modules (G, ι) over k and the category of triples (M, F, V ) where M is a free WK (k)-module of rank equal to the OK -height and F (resp. V ) is a σ- (resp. σ −1 -) linear map such that F V = V F = πK idM are equivalent. Via this equivalence there is a canonical functorial isomorphism M/V M ∼ = Lie(G). We call (M, F, V ) = M (G, ι) the Dieudonn´e module of (G, ι). Proof : To prove this we use covariant Dieudonn´e theory for p-divisible groups as in [Zi1] for example. Denote by σ the usual Frobenius of the ring of Witt vectors. Covariant Dieudonn´e theory tells us that there is an equivalence of the category of p-divisible groups over k with the category of triples (M ′ , F ′ , V ′ ) where M ′ is a free W (k)-module of rank equal to the height of G and with a σ-linear (resp. a σ −1 -linear) endomorphism F ′ (resp. V ′ ) such that F ′ V ′ = V ′ F ′ = p idM ′ and such that M ′ /V ′ M ′ = Lie(G). Let us call this functor M ′ . Let (G, ι) be a special p-divisible OK -module. Then the Dieudonn´e module M ′ (G) is a W (k) ⊗ZZp OK -module, the operators F ′ and V ′ commute with the OK -action and the induced homomorphism OK ⊗ZZp k −→ End(M ′ /V ′ M ′ ) factors through the multiplication OK ⊗ZZp k −→ k. We have to construct from these data a triple (M, F, V ) as in the claim of the proposition. To do this write Y W (k) ⊗ZZp OK = W (k) ⊗ZZp W (κ) ⊗W (κ) OK = WK (k). Gal(κ/IFp )

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By choosing the Frobenius σ = σQ p as a generator of Gal(κ/IFp ) we can identify this group with ZZ/rZZ where pr = q. We get an induced decomposition M ′ = ⊕i∈ZZ/rZZ Mi′ where the Mi are WK (k)-modules defined by Mi = { m ∈ M ′ | (a ⊗ 1)m = (1 ⊗ σ −i (a))m for all a ∈ W (κ) ⊂ OK }. The operator F ′ (resp. V ′ ) is homogeneous of degree −1 (resp. +1) with respect to this decomposition. By the condition on the OK -action on the Lie ′ ′ = Mi′ = M ′ /V ′ M ′ and hence that V Mi−1 algebra we know that M0′ /V Mr−1 for all i 6= 0. We set M = M0′ and V = (V ′ )r |M0′ . It follows that we have M/V M = M ′ /V ′ M ′ . Further the action of πK on M/V M = M ′ /V M ′ equals the scalar multiplication with the image of πK under the map OK −→ κ −→ k but this image is zero. It follows that V M contains πK M and hence we can define F = V −1 πK . Thus we constructed the triple (M, F, V ) and it is easy to see that this defines an equivalence of the category of triples (M ′ , F ′ , V ′ ) as above and the one of triples (M, F, V ) as in the claim. (5.2.8) Let (G, ι) be a special p-divisible OK -module over a ring R. We call it ´etale if it is an inductive system of finite ´etale group schemes. This is equivalent to the fact that its Lie algebra is zero. If p is invertible in R, (G, ι) will be always ´etale. Now assume that R = k is a perfect field of characteristic p and let (M, F, V ) be its Dieudonn´e module. Then (G, ι) is ´etale if and only if M = V M. In general there is a unique decomposition (M, F, V ) = (M´et , F, V ) ⊕ (Minf , F, V ) such that V is bijective on M´et and such that V N Minf ⊂ πK Minf for large N (define M´et (resp. Minf ) as the projective limit over n of S T m m n n m Ker(V |M/πK M ))). We call the WK (k)m V (M/πK M ) (resp. of rank of M´et the ´etale OK -height of (M, F, V ) or of (G, ι). We call (G, ι) formal or also infinitesimal if its ´etale OK -height is zero. (5.2.9) Proposition: Let k be an algebraically closed field of characteristic p. For all non-negative integers h ≤ n there exists up to isomorphism exactly one special p-divisible OK -module of OK -height n, ´etale OK -height h and of dimension one. Its Dieudonn´e module (M, F, V ) is the free WK (k)-module

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with basis (d1 , . . . , dh , e1 , . . . , en−h ) such that V is given by V di = di , V ei = ei+1 ,

i = 1, . . . , h i = 1, . . . , n − h − 1

V en−h = πK e1 . This determines also F by the equality F = V −1 πK . The key point to this proposition ist the following lemma due to Dieudonn´e: Lemma: Let M be a free finitely generated WK (k)-module and let V a be a σK -linear bijection where a is some integer different from zero. Then there exists a WK (k)-basis (e1 , . . . , en ) of M such that V ei = ei . A proof of this lemma in the case of K = Q p can be found in [Zi1] 6.26. The general case is proved word by word in the same way if one replaces everywhere p by πK . Proof of the Proposition: Let (G, ι) be a special p-divisible OK -module as in the proposition and let (M, F, V ) be its Dieudonn´e module. We use the decomposition (M, F, V ) = (M´et , F, V ) ⊕ (Minf , F, V ) and can apply the lemma to the ´etale part. Hence we can assume that h = 0 (note that Minf /V Minf = M/V M ). By definition of Minf , V acts nilpotent on M/πK M . We get a decreasing filtration M/πK M ⊃ V (M/πK M ) ⊃ · · · ⊃ V N (M/πK M ) = (0). The successive quotients have dimension 1 because dimk (M/V M ) = 1. Hence we see that V n M ⊂ πK M . On the other hand we have lengthWK (k) (M/V n M ) = n lengthWK (k) (M/V M ) = n = lengthWK (k) (M/πK M ) which implies V n M = πK M . Hence we can apply the lemma to the operator −1 n πK V and we get a basis of elements f satisfying V n f = πK f . Choose an element f of this basis which does not lie in V M . Then the images of ei := V i−1 f for i = 1, . . . , n in M/πK M form a basis of the k-vector space M/πK M . Hence the ei form a WK (k)-basis of M , and V acts in the desired form. (5.2.10) Definition: We denote the unique formal p-divisible OK -module

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of height h and dimension 1 over an algebraically closed field k of characteristic p by Σh,k .

′ (5.2.11) Denote by DK,1/h the ring of endomorphisms of Σh,k and set ′ DK,1/h = DK,1/h ⊗ZZ Q. Then this is “the” skew field with center K and invariant 1/h ∈ Q/ZZ (5.1.1). This follows from the following more general proposition:

Proposition: Denote by L the field of fractions of WK (k) and fix a rational number λ. We write λ = r/s with integers r and s which are prime to each other and with s > 0 (and with the convention 0 = 0/1). Denote by Nλ = (N, V ) the pair consisting of the vector space N = Ls and of the −1 σK -linear bijective map V which acts on the standard basis via the matrix

0 1 0

0

0 0 1 ...

... 0 ... 0

0 ... ... 1

r πK 0 0 . 0

Then End(Nλ ) = { f ∈ EndL (N ) | f ◦ V = V ◦ f } is the skew field Dλ with center K and invariant equal to the image of λ in Q/ZZ (cf. (5.1.1)). s

Proof : We identify IFqs with the subfield of k of elements x with xq = x. This contains the residue field κ of OK and we get inclusions OK ⊂ OKs := W (IFqs ) ⊗W (κ) OK ⊂ WK (k) and hence K ⊂ Ks ⊂ L. These extensions are unramified, [Ks : K] = s, and Ks can be described as s the fixed field of σK in L. To shorten notations we set Aλ = End(Nλ ). As Nλ does not have any non-trivial V -stable subspaces (cf. [Zi1] 6.27), Aλ is a skew field and its center contains K. For a matrix (uij ) ∈ End(Ls ) an easy explicit calculation

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301

shows that (uij ) ∈ Aλ if and only if we have the relations u11 ui+1,j+1 u1,j+1 ui+1,j

= = = =

−1 σK (uss ), −1 σK (uij ), r −1 πK σK (usj ), −r −1 πK σK (uis ),

1 ≤ i, j ≤ s − 1, 1 ≤ j ≤ s − 1, 1 ≤ i ≤ s − 1.

s It follows that σK (uij ) = uij for all i, j, and hence uij ∈ Ks . Further, sending a matrix (uij ) ∈ Aλ to its first column defines a Ks -linear isomorphism Aλ ∼ = Kss , hence dimKs (Aλ ) = s. The Ks -algebra homomorphism

ϕ: Ks ⊗K Aλ −→ Ms (Ks ),

α ⊗ x 7→ αx

is a homomorphism of Ms (Ks )-left modules and hence it is surjective as the identity matrix is in its image. Therefore ϕ is bijective. In particular, Ks is the center of Ks ⊗K Aλ and hence the center of Aλ is equal to K. Now define r 0 0 . . . 0 πK 1 0 ... 0 0 . . . 0 ∈ Aλ . Π = 0 1 ... 0

...

0

1

0

r Then we have the relations Πs = πK and Πd = σK (d)Π for d ∈ Aλ . We get an embedding Dλ = Ks [Π] ֒→ Aλ by

Π 7→ Π

Ks ∋ α 7→

−1 σK (α) −2 σK (α)

··· α

∈ Aλ ⊂ Ms (Ks )

which has to be an isomorphism because both sides have the same Kdimension. (5.2.12) Over a complete local noetherian ring R with perfect residue field k we have the following alternative description of a special formal p-divisible

302

T. Wedhorn

OK -module due to Zink [Zi2]. For this we need a more general definition of the Witt ring. Let R be an arbitrary commutative ring with 1. The Witt ring W (R) is characterized by the following properties: (a) As a set it is given by RIIN0 , i.e. elements of W (R) can be written as infinite tuples (x0 , x1 , . . . , xi , . . .). (b) If we associate to each ring R the ring W (R) and to each homomorphism of rings α: R −→ R′ the map W (α): (x0 , x1 , . . .) 7→ (α(x0 ), α(x1 ), . . .), then we obtain a functor from the category of rings into the category of rings. (c) For all integers n ≥ 0 the so called Witt polynomials wn : W (R) −→ R n

(x0 , x1 , . . .) 7→ xp0 + px1p

n−1

+ . . . + pn xn

are ring homomorphisms. For the existence of such a ring see e.g. [BouAC] chap. IX, §1. If we endow the product RIIN0 with the usual ring structure the map x 7→ (w0 (x), w1 (x), . . .) defines a homomorphism of rings W∗ : W (R) −→ RIIN0 . The ring W (R) is endowed with two operators τ and σ which are characterized by the property that they are functorial in R and that they make the following diagrams commutative W (R) W∗ y RIIN0

W (R) W∗ y RIIN0

τ

−−−−−→ x7→(0,px0 ,px1 ,...)

−−−−−−−−−−→ σ

−−−−−→ x7→(x1 ,x2 ,...)

−−−−−−−→

W (R) W y ∗ RIIN0 ,

W (R) W y ∗ RIIN0 .

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303

The operator τ can be written explicitly by τ (x0 , x1 , . . .) = (0, x0 , x1 , . . .) and it is called Verschiebung of W (R). It is an endomorphism of the additive group of W (R). If R is of characteristic p (i.e. pR = 0), σ can be described as (x0 , x1 , . . .) 7→ (xp0 , xp1 , . . .). For an arbitrary ring, σ is a ring endomorphism and it is called Frobenius of W (R). There are the following relations for σ and τ : (i) σ ◦ τ = p · idW (R) , (ii) τ (xσ(y)) = τ (x)y for x, y ∈ W (R), (iii) τ (x)τ (y) = pτ (xy) for x, y ∈ W (R), (iv) τ (σ(x)) = τ (1)x for x ∈ W (R), and we have τ (1) = p if R is of characteristic p. We have a surjective homomorphism of rings w0 : W (R) −→ R,

(x0 , x1 , . . .) 7→ x0 ,

n and we denote its kernel τ (W (R)) by IR . We have IR = τ n (W (R)) and W (R) is complete with respect to the IR -adic topology. If R is a local ring with maximal ideal m, W (R) is local as well with maximal ideal { (x0 , x1 , . . .) ∈ W (R) | x0 ∈ m }.

(5.2.13) Now we can use Zink’s theory of displays to give a description of special formal p-divisible groups in terms of semi-linear algebra. Let R be a complete local noetherian OK -algebra with perfect residue field k. We extend σ and τ to W (R) ⊗ZZp OK in an OK -linear way. Then we get using [Zi2]: Proposition: The category of special formal p-divisible OK -modules of height h over R is equivalent to the category of tuples (P, Q, F, V −1 ) where • P is a finitely generated W (R) ⊗ZZp OK -module which is free of rank h over W (R), • Q ⊂ P is a W (R) ⊗ZZp OK -submodule which contains IR P , and the quotient P/Q is a direct summand of the R-module P/IR P such that the induced action of R ⊗ZZp OK on P/Q factorizes through the multiplication R ⊗ OK −→ R, • F : P −→ P is a σ-linear map,

304

T. Wedhorn

• V −1 : Q −→ P is a σ-linear map whose image generates P as a W (R)module, satisfying the following two conditions: (a) For all m ∈ P and x ∈ W (R) we have the relation V −1 (τ (x)m) = xF (m). (b) The unique W (R) ⊗ZZp OK -linear map V # : P −→ W (R) ⊗σ,W (R) P satisfying the equations V # (xF m) = p · x ⊗ m V # (xV −1 n) = x ⊗ n for x ∈ W (R), m ∈ P and n ∈ Q is topologically nilpotent, i.e. the homomorphism V N # : P −→ W (R) ⊗σN ,W (R) P is zero modulo IR + pW (R) for N sufficiently large. (5.2.14) To define the notion of a Drinfeld level structure we need the following definition: Let R be a ring and let X = Spec(A) where A is finite locally free over R of rank N ≥ 1. For any R-algebra R′ we denote by X(R′ ) the set of R-algebra homomorphisms A −→ R′ (or equivalently of all R′ -algebra homomorphisms A ⊗R R′ −→ R′ ). The multiplication with an element f ∈ A ⊗R R′ defines an R′ -linear endomorphism of A ⊗R R′ . As A is finite locally free we can speak of the determinant of this endomorphism which is an element Norm(f ) in R′ . We call a finite family of elements ϕ1 , . . . , ϕN ∈ X(R′ ) a full set of sections of X over R′ if we have for every R′ -algebra T and for all f ∈ A⊗R T an equality in T N Y ϕi (f ). Norm(f ) = i=1

(5.2.15) Let R be an OK -algebra and let G be a special p-divisible OK module over R. We assume that its OK -height h is constant on Spec(R),

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305

e.g. if R is a local ring (the only case which will be used in the sequel). The m OK -action on G defines for every integer m ≥ 1 the multiplication with πK m [πK ]: G −→ G.

This is an endomorphism of p-divisible groups whose kernel is a finite locally m free group scheme G[πK ] over Spec(R) of rank q mh . ′ Let R′ be an R-algebra. A Drinfeld pm K -structure on G over R is a homomorphism of OK -modules m α: (p−m /OK )h −→ G[πK ](R′ ) h such that the finite set of α(x) for x ∈ (p−m K /OK ) forms a full set of sections.

(5.2.16) It follows from the definition (5.2.14) that if α: (p−m /OK )h −→ ′ ′ m G[πK ](R′ ) is a Drinfeld pm K -structure over R then for any R -algebra T the composition m m αT : (p−m /OK )h −→ G[πK ](R′ ) −→ G[πK ](T ), α

where the second arrow is the canonical one induced by functoriality from R′ −→ T , is again a Drinfeld pm K -structure. (5.2.17) Being a Drinfeld pm K -structure is obviously a closed property. More −m h m precisely: Let α: (p /OK ) −→ G[πK ](R′ ) be a homomorphism of abelian groups. Then there exists a (necessarily unique) finitely generated ideal a ⊂ R′ such that a homomorphism of OK -algebras R′ −→ T factorizes over R′ /a if and only if the composition αT of α with the canonical homomorphism m m G[πK ](R′ ) −→ G[πK ](T ) is a Drinfeld pm K -structure over T . It follows that for every special formal p-divisible OK -module (G, ι) over some OK -algebra R the functor on R-algebras which associates to each Ralgebra R′ the set of Drinfeld pm K -structures on (G, ι)R′ is representable by an R-algebra DLm (G, ι) which is of finite presentation as R-module. Obviously DL0 (G, ι) = R. m (5.2.18) Let α: (p−m /OK )h −→ G[πK ](R′ ) be a Drinfeld pm K -structure over ′ ′ R . As α is OK -linear, it induces for all m ≤ m a homomorphism ′

′

′

m m ](R′ ). α[πK ]: (p−m /OK )h −→ G[πK

306

T. Wedhorn

Proposition: This is a Drinfeld pm K -structure. ′

For the proof of this non-trivial fact we refer to [HT] 3.2 (the hypothesis in loc. cit. that Spec(R′ ) is noetherian with a dense set of points with residue field algebraic over κ is superfluous as we can always reduce to this case by [EGA] IV, §8 and (5.2.17)). If R′ is a complete local noetherian ring with perfect residue class field (this is the only case which we will use in the sequel) the proposition follows from the fact that we can represent (G, ι) by a formal group law and that in this case a Drinfeld level structure as defined above is the same as a Drinfeld level structure in the sense of [Dr1]. (5.2.19) Let (G, ι) be a special formal p-divisible OK -module over an OK algebra R. By (5.2.18) we get for non-negative integers m ≥ m′ canonical homomorphisms of R-algebras DLm′ (G, ι) −→ DLm (G, ι). It follows from [Dr1] 4.3 that these homomorphisms make DLm (G, ι) into a finite locally free module over DLm′ (G, ι). (5.2.20) Example: If R is an OK -algebra of characteristic p and if G is a special formal p-divisible OK -module of OK -height h and of dimension 1, then the trivial homomorphism h m αtriv : (p−m K /OK ) −→ G[pK ],

x 7→ 0

is a Drinfeld pm K -structure. If R is reduced, this is the only one.

5.3

Deformation of p-divisible O-modules

(5.3.1) In this paragraph we fix an algebraically closed field k of characteristic p together with a homomorphism OK −→ k. Further we fix integers h ≥ 1 and m ≥ 0. By (5.2.9) and by (5.2.20) there exists up to isomorphism only one special formal p-divisible OK -module Σh of height h and dimension triv 1 with Drinfeld pm over k. We denote the pair (Σh , αtriv ) by K -structure α Σh,m .

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307

Let C be the category of pairs (R, s) where R is a complete local noetherian OK -algebra and where s is an isomorphism of the residue class field of R with k. The morphisms in C are local homomorphisms of OK -algebras inducing the identity on k. (5.3.2) Definition: Let (R, s) ∈ C be a complete local noetherian OK algebra. A triple (G, α, ϕ) consisting of a formal special p-divisible OK module G over R, of a Drinfeld pm K -structure α of G over R and of an isomorphism ∼

ϕ: Σh,m −→ (G ⊗R k, αk ) is called a deformation of Σh,m over R. ˜ h,m , ϕ) consisting of a complete local noetherian ring A triple (Rh,m , Σ ˜ h,m , ϕ) of Σh,m is called Rh,m with residue field k and of a deformation (Σ universal deformation of Σh,m if for every deformation (G, α, ϕ) over some ˜ h,m , ϕ)R R ∈ C there exists a unique morphism Rh,m −→ R in C such that (Σ is isomorphic to (G, α, ϕ). A universal deformation is unique up to unique isomorphism if it exists. (5.3.3) Proposition: We keep the notations of (5.3.2). ˜ h,m , ϕ) of Σh,m exists. (1) A universal deformation (Rh,m , Σ (2) For m = 0 the complete local noetherian ring Rh,0 is isomorphic to the power series ring WK (k)[[t1 , . . . , th−1 ]]. (3) For m ≥ m′ the canonical homomorphisms Rh,m′ −→ Rh,m are finite flat. The rank of the free Rh,0 -module Rh,m is #GLh (OK /pm k ). (4) The ring Rh,m is regular for all m ≥ 0. Proof : Assertion (1) follows from a criterion of Schlessinger [Sch] using rigidity for p-divisible groups (e.g. [Zi1]) and the fact that the canonical functor from the category of special p-divisible OK -modules over Spf(Rh,m ) to the category of special p-divisible OK -modules over Spec(Rh,m ) is an equivalence of categories (cf. [Me] II, 4). The second assertion follows easily from general deformation theory of p-divisible groups (for an explicit description of the universal deformation and a proof purely in terms of linear algebra one can use [Zi2] and (5.2.13)). Finally, (3) and (4) are more involved (see [Dr1] §4, note that (3) is essentially equivalent to (5.2.19)).

308

T. Wedhorn

(5.3.4) Let D1/h be “the” skew field with center K and invariant 1/h. The ring Rh,m has a continuous action of the ring of units O × D1/h of the integral ˜ closure OD of OK in D1/h : Let Σg,m = (G, α, ϕ) be the universal special 1/h

formal p-divisible OK -module with Drinfeld pm K -structure over Rh,m . For × δ ∈ O D1/h the composition ϕ

δ

Σh,m −→ Σh,m −→ (G, α) ⊗Rh,m k is again an isomorphism if we consider δ as an automorphism of Σh,m (which is the same as an automorphism of Σh,0 ) by (5.2.11). Therefore (G, α, ϕ ◦ δ) is a deformation of Σh,m over Rh,m and by the definition of a universal deformation this defines a continuous automorphism δ: Rh,m −→ Rh,m . (5.3.5) Similarly as in (5.3.4) we also get a continuous action of GLh (OK /pm K) ˜ on Rh,m : Again let Σg,m = (G, α, ϕ) be the universal special formal pm divisible OK -module with Drinfeld pm K -structure over Rh,m . For γ ∈ GLh (OK /pK ), m α ◦ γ is again a Drinfeld pK -structure, hence (G, α ◦ γ, ϕ) is a deformation of Σh,m and defines a continuous homomorphism γ: Rh,m −→ Rh,m .

(5.3.6) By combining (5.3.4) and (5.3.5) we get a continuous left action of × −→ GLh (OK /pm GLh (OK ) × O × K ) × O D1/h D1/h −→

on Rh,m . Now we have the following lemma ([HT] p. 52) Lemma: This action can be extended to a continuous left action of GLh (K) × D × 1/h on the direct system of the Rh,m such that for m2 >> m1 and for (γ, δ) ∈ GLh (K) × D × the diagram 1/h Rh,m x 1

W (k)

commutes.

(γ,δ)

−−−−−→ v

σKK

(det(γ))−vK (Nrd(δ))

−−−−−−−−−−−−−→

Rh,m x 2

W (k)

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5.4

309

Vanishing y les

(5.4.1) Let W be a complete discrete valuation ring with maximal ideal (π), residue field k and field of fractions L. Assume that k is algebraically closed (or more generally separably closed). The example we will use later on is the ring W = WK (k) for an algebraically closed field k of characteristic ¯ and s = Spec(k). p. Set η = Spec(L), η¯ = Spec(L) We will first define vanishing and nearby cycles for an algebraic situation (cf. [SGA 7] exp. I, XIII). Then we will generalize to the situation of formal schemes. (5.4.2) Let f : X −→ Spec(W ) be a scheme of finite type over W and define Xη¯ and Xs by cartesian diagrams X s fs y s

i

−→ −→

¯ j

X ←− fy Spec(W ) ← η ←

X η¯ f y η¯ η¯.

The formalism of vanishing cycles is used to relate the cohomology of Xs and of Xη¯ together the action of the inertia group on the cohomology of Xη¯. Fix a prime ℓ different from the characteristic p of k and let Λ be a finite abelian group which is annihilated by a power of ℓ. For all integers n ≥ 0 the sheaf Ψn (Λ) = i∗ Rn ¯j∗ Λ is called the sheaf of vanishing cycles of X over W . Via functoriality it ¯ ¯ carries an action of Gal(L/L). Note that by hypothesis Gal(L/L) equals the inertia group of L. (5.4.3) If f : X −→ Spec(W ) is proper, the functor i∗ induces an isomorphism (proper base change) ∼

i∗ : H p (X, Rq ¯jΛ) −→ H p (Xs , i∗ Rq ¯j∗ Λ) and the Leray spectral sequence for ¯j can be written as H p (Xs , Ψn (Λ)) ⇒ H p+q (Xη¯, Λ).

310

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¯ This spectral sequence is Gal(L/L)-equivariant. This explains, why the vanishing cycles “measure” the difference of the cohomology of the special and the generic fibre. (5.4.4) Let Λ be as above. If X is proper and smooth over W , there is no difference between the cohomology of the generic and the special fibre. More precisely, we have ([SGA 4] XV, 2.1): Proposition: Let f : X −→ W be a smooth and proper morphism. Then we have for all n ≥ 0 a canonical isomorphism ∼

H n (Xη¯, Λ) −→ H n (Xs , Λ). ¯ This isomorphism is Gal(L/L)-equivariant where the action on the rightn hand side is trivial. Further ψ (Λ) = 0 for n ≥ 1 and ψ0 = Λ. (5.4.5) Now we define vanishing cycles for formal schemes: We keep the notations of (5.4.1). We call a topological W -algebra A special, if there exists an ideal a ⊂ A (called an ideal of definition of A) such that A is complete with respect to the a-adic topology and such that A/an is a finitely generated W -algebra for all n ≥ 1 (in fact the same condition for n = 2 is sufficient ([Ber2] 1.2)). Equivalently, A is topologically W -isomorphic to a quotient of the topological W -algebra W {T1 , . . . , Tm }[[S1 , . . . , Sn ]] = W [[S1 , . . . , Sn ]]{T1 , . . . , Tm }, in particular A is again noetherian. Here if R is a topological ring, R{T1 , . . ., Tm } denotes the subring of power series X cn T n n∈IINn 0

in R[[T1 , . . . , TM ]] such that for every neighborhood V of 0 in R there is only a finite number of coefficients cn not belonging to V . If R is complete Hausdorff with respect to the a-adic topology for an ideal a, we have a canonical isomorphism ∼

R{T1 , . . . , Tm } −→ lim(R/an )[T1 , . . . , Tm ]. ←− n

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311

Note that all the rings Rh,m defined in (5.3.3) are special WK (k)algebras. (5.4.6) Let A be a special W -algebra and let X = Spf(A). Denote by X rig its associated rigid space over L. It can be constructed as follows: For A = W {T1 , . . . , Tm }[[S1 , . . . , Sn ]] we have X rig = E m × D n where E m and D n are the closed resp. open polydiscs of radius 1 with center at zero in Lm resp. in Ln . If now A is some quotient of the special W -algebra A′ = W {T1 , . . . , Tm }[[S1 , . . . , Sn ]] with kernel a′ ⊂ A′ , X rig is the closed rigid analytic subspace of Spf(A′ )rig defined by the sheaf of ideals a′ OSpf(A′ )rig . This defines a functor from the category of special W -algebras to the category of rigid-analytic spaces over L (see [Ber2] §1 for the precise definition). (5.4.7) Let A be a special W -algebra and let a ⊂ A be its largest ideal of definition. A topological A-algebra B which is special as W -algebra (or shorter a special A-algebra - an abuse of language which is justified by [Ber2] 1.1) is called ´etale over A if B is topologically finitely generated as A-algebra (which is equivalent to the fact that for every ideal of definition a′ of A, a′ B is an ideal of definition of B) and if the morphism of commutative rings A/a −→ B/aB is ´etale in the usual sense. The assignment B 7→ B/aB defines a functor from the category of ´etale special A-algebras to the category of ´etale A/a-algebras which is an equivalence of categories and hence we get an equivalence of ´etale sites (A)´et ∼ (A/a)´et . Note that for every ideal of definition a′ of A the ´etale sites (A/a)´et and (A/a′ )´et coincide. We just chose the largest ideal of definition to fix notations. On the other hand, if we write X = Spf(A) and Y = Spf(B) for an ´etale special A-algebra B we get a (quasi-)´etale morphism of rigid analytic spaces Y rig −→ X rig .

312

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By combining this functor with a quasi-inverse of B 7→ B/aB we get a morphism of ´etale sites ∼ ¯ ´et −→ (X rig )´et −→ (X )´et −→ s: (X rig ⊗L L) (Xred )´et

with Xred = Spec(A/a). Let Λ be a finite abelian group which is annihilated by a power of ℓ. For n ≥ 0 the sheaves ψn (Λ) := Rn s∗ Λ are called vanishing cycle sheaves. (5.4.8) Let A be a special W -algebra with largest ideal of definition a, X = Spf(A). Then the group AutW (X ) of automorphisms of X over W (i.e. of continuous W -algebra automorphisms A → A) acts on ψn (Λ). Further we have the following result of Berkovich [Ber2]: Proposition: Assume that ψi (ZZ/ℓZZ) is constructible for all i. Then there exists an integer n ≥ 1 with the following property: Every element g ∈ AutW (X ) whose image in AutW (A/an ) is the identity acts trivially on ψi (ZZ/ℓm ZZ) for all integers i, m ≥ 0.

5.5

Vanishing y les on the universal deformation of spe ial p-divisible O-modules

(5.5.1) Let k be an algebraic closure of the residue field κ of OK . Then ˆ nr , the completion of the maximal W = WK (k) is the ring of integers of K unramified extension of K. Further denote by IK the inertia group and by WK the Weil group of K. (5.5.2) Consider the system P of special formal schemes . . . −→ Spf(Rm,h ) −→ Spf(Rm−1,h ) −→ . . . −→ Spf(R0,h ). By applying the functor ( )rig we get a system P rig of rigid spaces . . . −→ Spf(Rm,h )rig −→ Spf(Rm−1,h )rig −→ . . . −→ Spf(R0,h )rig .

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313

these systems P and P rig have an action by the group GLh (OK ) × O × D1/h (5.3.4) and (5.3.5). Denote by Ψim (Λ) the vanishing cycle sheaf for Spf(Rh,m ) with coefficients in some finite abelian ℓ-primary group Λ and set ¯ ℓ. Ψim = (lim Ψim (ZZ/ℓn ZZ)) ⊗ZZℓ Q ←− n

Note that we have ˆ¯ ZZ ), lim Ψim (ZZ/ℓn ZZ) = H i ((Spf Rm )rig ⊗Kˆ nr K, ℓ ←− n

ˆ¯ K ˆ nr ). Furin particular these ZZℓ -modules carry an action by IK = Gal(K/ ther write Ψi = lim Ψim . −→ m

¯ℓ ∼ Via our chosen identification Q = C we can consider Ψim and Ψi as C-vector spaces which carry an action of GLh (OK ) × O × D1/h × IK .

(5.5.3) Lemma: We have the following properties of the (GLh (OK ) × i i O× D1/h × IK )-modules Ψm and Ψ . (1) (2) (3) (4)

The Ψim are finite-dimensional C-vector spaces. We have Ψim = Ψi = 0 for all m ≥ 0 and for all i > h − 1. The action of GLh (OK ) on Ψi is admissible. i The action of O × D1/h on Ψ is smooth.

(5) The action of IK on Ψi is continuous. Proof : For the proof we refer to [HT] 3.6. We only remark that (3) – (5) follow from general results of Berkovich [Ber2] and [Ber3] if we know (1). To show (1) one uses the fact that the system of formal schemes P comes from an inverse system of proper schemes of finite type over W (cf. the introduction) and a comparison theorem of Berkovich which relates the vanishing cycles of a scheme of finite type over W with the vanishing cycle sheaves for the associated formal scheme.

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(5.5.4) Let Ah be the group of elements (γ, δ, σ) ∈ GLh (K) × D × 1/h × WK such that vK (det(γ)) = vK (Nrd(δ)) + vK (Art−1 K (σ)). The action of GLh (K) × D × 1/h on the system (Rh,m )m (5.3.6) gives rise to i an action of AK on Ψ . Moreover, if (ρ, Vρ ) is an irreducible admissible representation of D × 1/h over C (and hence necessarily finite-dimensional (2.1.20)) then we set Ψi (ρ) = HomO× (ρ, Ψi ). 1/h

This becomes naturally an admissible GLh (K) × WK -module if we define for φ ∈ Ψi (ρ) and for x ∈ Vρ ((γ, σ)φ)(x) = (γ, δ, σ)φ(ρ(δ)−1 x) −1 where δ ∈ D × 1/h is some element with vK (Nrd(δ)) = vK (det(γ))−vK (ArtK (σ)).

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Bibliography

[AT]

E. Artin, J. Tate: Class Field Theory, Benjamin (1967).

[Au1]

A.-M. Aubert: Dualit´e dans le groupe de Grothendieck de la cat´egorie des repr´esentations lisses de longueur finie d’un groupe r´eductif p-adique, Trans. Amer. Math. Soc. 347, no. 6, 2179–2189 (1995).

[Au2]

A.-M. Aubert: Erratum: Duality in the Grothendieck group of the category of finite-length smooth representations of a p-adic reductive group, Trans. Amer. Math. Soc. 348, no. 11, 4687–4690 (1996).

[AW]

M.F. Atiyah, C.T.C. Wall: Cohomology of Groups, in J.W.S Cassels, A. Fr¨ ohlich (Ed.): Algebraic Number Theory, Acadamic Press, (1967).

[Ber1]

V.G. Berkovich: Vanishing cycles for formal schemes, Inv. Math. 115, 539–571 (1994).

[Ber2]

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SCHOOL ON AUTOMORPHIC FORMS ON GL(n)

31 July - 18 August 2000

Editors

L. G¨ ottsche The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy G. Harder Mathematisches Institut der Universit¨at Bonn, Bonn, Germany M.S. Raghunathan Tata Institute of Fundamental Research, Mumbai, India

SCHOOL ON AUTOMORPHIC FORMS ON GL(n) – First edition c 2008 by The Abdus Salam International Centre for Theoretical Physics Copyright The Abdus Salam ICTP has the irrevocable and indefinite authorization to reproduce and disseminate these Lecture Notes, in printed and/or computer readable form, from each author. ISBN 92-95003-37-3

Printed in Trieste by The Abdus Salam ICTP Publications & Printing Section

iii PREFACE One of the main missions of the Abdus Salam International Centre for Theoretical Physics in Trieste, Italy, founded in 1964, is to foster the growth of advanced studies and scientific research in developing countries. To this end, the Centre organizes a number of schools and workshops in a variety of physical and mathematical disciplines. Since unpublished material presented at the meetings might prove to be of interest also to scientists who did not take part in the schools and workshops, the Centre has decided to make it available through a publication series entitled ICTP Lecture Notes. It is hoped that this formally structured pedagogical material on advanced topics will be helpful to young students and seasoned researchers alike. The Centre is grateful to all lecturers and editors who kindly authorize ICTP to publish their notes in this series. Comments and suggestions are most welcome and greatly appreciated. Information regarding this series can be obtained from the Publications Office or by e-mail to “pub− [email protected]”. The series is published in-house and is also made available on-line via the ICTP web site: “http://publications.ictp.it”.

Katepalli R. Sreenivasan, Director Abdus Salam Honorary Professor

v CONTENTS M.S. Raghunathan Automorphic Forms in GL(n) I: Decomposition of the Space of Cusp Forms and Some Finiteness Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 T.N. Venkataramana Classical Modular Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 J.W. Cogdell Notes on L-functions for GLn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 D. Prasad and A. Raghuram Representation Theory of GL(n) over Non-Archimedean Local Fields . . 159 G. Harder The Langlands Program (An Overview) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 T. Wedhorn The Local Langlands Correspondence for GL(n) over p-adic Fields . . . . 237

vii Introduction The School on Automorphic Forms on GL(n) took place at the Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste from 31 July to 18 August 2000, under the direction of G. Harder (Universit¨at Bonn) and M.S. Raghunathan (Tata Institute for Fundamental Research, Mumbai). The local organizer was Lothar G¨ottsche (ICTP). The central topics of the school were the theory of automorphic forms on GL(n) and the local theory of representations on GL(n) over p-adic fields. The programme included an introduction to automorphic forms, recent results on L-functions on GL(n), an introduction to the Langlands programme, an introduction to the theory of local representations and an outline of the proof of the local Langlands conjecture for GL(n) over local fields. The school consisted of two weeks of lecture courses followed by one week of conference. This lecture notes volume contains the notes of most of the lecture courses. The electronic version of these lecture notes is available at http://publications.ictp.it/ The school was financially supported by a grant from the European Commission. We are very thankful for this support. I take this opportunity to thank G. Harder and M.S. Raghunathan for organizing this school. I would also like to thank the lecturers of the school and the speakers at the conference for their very interesting lectures.

Lothar G¨ottsche June, 2003

Automorphic Forms in GL(n) I: Decomposition of the Space of Cusp Forms and Some Finiteness Results M.S. Raghunathan∗

School of Mathematics, Tata Institute of Fundamental Research, Mumbai, India

Lectures given at the School on Automorphic Forms on GL(n) Trieste, 31 July - 18 August 2000

LNS0821001

∗

[email protected]

Contents 0 Introduction

5

1 Fundamental Domains

6

2 Automorphic Forms

13

3 Cusp Forms

21

4 Proof of Theorem 2.13

33

References

37

Automorphic Forms in GL(n)

0

5

Introduction

These are notes of lectures given by the author at an Instructional School on “Automorphic Forms in GL(n)” held at the Abdus Salam International Centre for Theoretical Physics, Italy, in August 2000. The author would like to thank the ICTP for their hospitality and their patience over the long delay in making these notes available. In this first chapter we introduce some concepts in the theory of Automorphic Forms and prove some basic results. Although our central focus is on the general linear group GL(n), in this chapter we formulate the results in the more general context of reductive algebraic groups over number fields. This is because the techniques for handling the general case are no different from those needed to handle GL(n). We have also indicated what the notions used mean in the special case of GL(n) (or SL(n)), for readers unfamiliar or uncomfortable with the general theory of algebraic groups. All the results about algebraic groups needed here are to be found in the paper [B-T] of Borel and Tits. In §1 we describe a good fundamental domain for G(k) in G(A) where k is a number field, A is its Ad´ele ring and G is a reductive algebraic group over k. This description follows from a theorem of A. Borel [B] that describes Q a fundamental domain for an arithmetic subgroup Γ in v∈∞ G(kv ): here ∞ is the complete set of inequivalent archimedean valuations of k and for v ∈ ∞, kv is the completion of k at u. Conversely Borel’s theorem can be deduced from this description (of a fundamental domain for G(k) in G(A)). A quick and elegant proof of this (due to R. Godement and A. Weil) is to be found in [G]. In §2 the definition of automorphic forms is given and the central finite dimensionality results are formulated. These concepts and results are due to R. Langlands [L] (see also [H-C]). In §3 we introduce the space of cusp-forms and prove that the representation of G(A) in the space of L2 -cusp-forms is completely reducible, a result due to I. Gelfand and I. Piatetishi-Shapiro. This is needed in §4 where the finite dimensionality theorems formulated in §2 are proved. We need in §3 some well known results from Analysis; these can be found in R. Narasimhan’s book [N].

6

1

M.S. Raghunathan

Fundamental Domains

1.1. Notations Throughout these notes we will adopt the following notations. We denote by k a number field and by V a complete set of mutually inequivalent valuations of k. Let ∞ (resp. Vf ) denote the subset of archimedean (resp. non-archimedean) valuations in V. For v ∈ V, kv denotes the completion of k with respect to v. We denote by Ov for v ∈ Vf the ring of integers in kv and by pv the maximal ideal in Ov . The residue field Ov /pv is denoted Fv and we set qv =| Fv |; also pv = characteristic of Fv . We denote by | |v the absolute value on kv determined by v. We assume the absolute value | |v , v ∈ V chosen so that the following holds: Let Qv denote the closure of Q in kv and Nv : kv → Qv be the norm map. If v is archimedean we set | x |v =| Nv (x) | where | | is the usual absolute value on Qv ≃ R. If v ∈ Vf and Nv (x) = prv · y with r ∈ Z and y a unit in the ring of integers in Qv (≃ Zpv ), | x |v = p−r v . With this definition we have the well known 1.2. Product formula If x ∈ k∗ , | x |v = 1 for all but finitely may v ∈ V Q and v∈S | x |v = 1 where S = {v ∈ V || x |v 6= 1}.

Q 1.3. We denote by A the ring of adeles: A = {x = {xv }v∈V ∈ v∈V kv | xv ∈ Ov for all v 6∈ S, S a finite subset of Vf }: A is a ring under coordinateQ Q wise addition and multiplication. Let ∧ denote the subset v∈∞ kv × v∈Vf Ov ; then ∧ is subring of A. Under the product topology ∧ is a locally compact ring topological ring. A has a natural structure of a locally compact ring such that the inclusion of ∧ (with its product topology) in A is an isomorphism of topological rings on to an open subring. We denote by Af the subset {x ∈ A | xv = 0 for v ∈ ∞}. The group of Ideles I of k is the set of invertible elements in A. It is easy to see that I = {x ∈ A | xv 6= 0 for v ∈ V and | xv |v = 1 for all but a finite number of v ∈ Vf }. We also set If = {x ∈ I | xv = 1 for v ∈ ∞}. The group I is given a topology as Q Q follows: Consider the group v∈∞ kv∗ × v∈Vf Ov∗ where for v ∈ Vf , Ov∗ is the compact group of units in kv∗ ; this group has a natural structure of a locally compact group in the product topology. I is given the unique structure of a Q Q topological group for which the natural inclusion of v∈∞ kv∗ v∈Vf Ov∗× in I is an isomorphism (of topological groups) onto an open subgroup. If x ∈ I Q Q and S = {v ∈ Vf || xv |v 6= 1, we set | x |= v∈V | xv |v = v∈S | xv |v ; then x 7→| x | is a continuous homomorphism of I in R∗ and the kernel of this

Automorphic Forms in GL(n)

7

homomorphism is denoted I1 . Elements of I1 will be called “Ideles of norm 1”. We state without proof the following well known 1.4. Theorem The inclusion of k (resp. k∗ ) in A (resp. I1 ) imbeds k (resp. k∗ ) as a closed discrete subgroup of the (additive) group A (resp. (multiplicative) group I1 ) such that A/k (resp. I1 /k∗ ) is compact. 1.5. Let G denote a connected reductive linear algebraic group over k. (Our main interest is in the case when G = GL(n) the general linear group and we will draw attention to what our terms and definitions mean in this special case). Let S be a maximal k-split torus in G (when G = GL(n), S can be taken to be the group of diagonal matrices in G). Let X ∗ (S) denote the group of characters (all characters of S are defined over k) of S and Φ ⊂ X ∗ (S) the subset of k-roots of G with respect to S: recall that a k-root is a non-trivial character for the adjoint action of S on Lie algebra of G (when G = GL(n), Φ consists of the characters d = diagonal (d1 , · · · , dn ) 7→ di /dj for some pair (i, j), 1 ≤ i, j ≤ n, i 6= j). We fix an ordering on X ∗ (S) and denote by Φ+ (resp. ∆) the set of positive (resp. simple) roots in Φ (when G = GL(n), one usually chooses the ordering given as follows: any character χ on S (= diagonal matrices) is of the form χ(d) =

Y

n (χ)

di i

1≤i≤n

where d = diagonal (d1 , · · · , dn ) and ni (χ) are integers; we set χ > 0 if nr(χ) > 0 where r(χ) = min {i | 1 ≤ i ≤ n, ni (χ) 6= 0}. The set of simple roots for this order are the roots d 7→ di /di+1 , 1 ≤ i < n). We fix once and for all a realisation of G as an algebraic subgroup (over k) of GL(N ) for some N (when G = GL(n), we can take N = n). For v ∈ V then G(kv ) is a 2 closed subset of GL(N, kv ) hence a locally closed subset of M (N, kv ) ≃ kvN and thus acquires a locally compact topology. For v ∈ Vf we denote by Mv the compact open subgroup G(kv ) ∩ GL(N, Ov )(Mv = GL(n, Ov ) when G = GL(n)) of G(kv ). G(A) (resp. G(Af ) the A (resp. Af ) points of G has a natural identification with {g = {gv }v∈V (resp.Vf ) | gv ∈ G(kv ) and the set {v ∈ Vf | gv 6∈ Mv } is finite}. We have a natural inclusion Y

v∈Vf

Mv ֒→ G(Af )

8

M.S. Raghunathan

and G(Af ) is made into a locally compact topological group by stipulating that the above inclusion is an isomorphism of topological groups of the group on the left onto an open subgroup (the topological group G(Af ) obtained in this fashion is independent of the realisation of G as a k-algebraic subgroup Q of GL(N ) for some N ). We set G∞ = v∈∞ G(kv ). We then have a natural identification of G(A) with G∞ × G(Af ). We give to G∞ the product Q topology on v∈∞ G(kv ) and to G(A) the product topology on G∞ ×G(Af ). Let pf (resp. p∞ ) be the projection of G(A) on G(Af ) (resp. G∞ ). For g ∈ G(A), we set gf = pf (g) and g∞ = p∞ (g). Let df (resp. d∞ ) = pf ◦ d (resp. p∞ ◦ d). We have a natural inclusion d : G(k) ֒→ G(A) of the k-points of G in G(A) as a closed discrete subgroup of G(A). In the sequel we identify G(k) with d(G(k)). We will now describe certain k-algebraic (resp. closed) subgroups of G and (resp. G(A)) that will be needed in the sequel. Let Ad denote the adjoint representation of G on its Lie algebra (over k) L(G). Under the adjoint action of S, L(G) decomposes into a direct sum of eigenspaces, the eigencharacters being the roots and the trivial character. For α ∈ Φ, let L(G)(α) be the eigenspace corresponding ` to α and let L(U+ ) = α∈Φ+ L(G)(α). Then there is a unique k-subgroup U of G normalised by S and with L(U) as the Lie subalgebra of L(G) corresponding to U. Also U is normalised by the Z(S) the centraliser of S and P = Z(S) · U is a parabolic subgroup defined over k : P is a minimal kparabolic subgroup of G. We denote by o P the intersection of the kernels of all squares of characters on P defined over k; then P = S·o P and o P = B·U where B is a reductive algebraic k-subgroup of o P which is anisotropic over k (i.e. B does not admit a non-trivial k-split toral subgroup). As S is k-split we may treat it as obtained from a Q-split torus by the base change Q ֒→ kthus in the sequel we will treat S as a Q split torus. We then have for each v ∈ ∞ an inclusion S(Qv ) ֒→ S(kv ) when Qv is the closure of Qv in kv ; and since all the v ∈ ∞ induce on Q the unique archimedean topology with R Q as the completion, we obtain a diagonal inclusion S(R) ֒→ v∈∞ S(kv ). Let Q v∈∞ S(kv ) → S(A) be the inclusion s = {sv }v∈∞ 7→ {sv }v∈V where for v ∈ V\∞, sv = 1. We denote by A the image of the identity component of Q S(R) (a subgroup v∈∞ S(kv )) in G(A). The group A is a closed subgroup of G(A) isomorphic to a product of ℓ = dim S copies of the multiplicative group R+ = {x ∈ R | x > 0}. For a constant c > 0 let Ac = {x ∈ A | α(x) ≤ c for α ∈ ∆}: α is a character (defined over k) on S and as is easily seen on A takes positive real values. With these notations we have

Automorphic Forms in GL(n)

9

1.6. Theorem (R. Godement and A. Weil) There is a compact subset Ω of o P(A) such that Ω ·o P(k) = P(A). There is a maximal compact (open) subgroup K of G(A) and a constant c0 > 0 such that for all c ≥ c0 and compact subsets Ω′ of o P(A) with Ω′ ⊃ Ω, one has K.Ac .Ω′ .G(k) = G(A). 1.7. Corollary Kf .Ω′f .df G(k) = G(Af ) where Ω′f = pf (Ωf ). 1.8. Let Lf be a compact open subgroup of G(Af ) = {g ∈ G(A) | gv = 1 for all v ∈ ∞}. Let ΓLf = γ ∈ G(k) | pf (γ) ∈ Lf }. Now G∞ × Lf has a natural identification with an open subgroup of G(A) and ΓLf is in a natural fashion a closed discrete subgroup of G∞ × Lf . Since Lf is compact, the projection d∞ (ΓLf ) of ΓLf on G∞ is a discrete subgroup of G∞ . The maximal compact subgroup K above decomposes as a direct product K∞ × Kf following the product decomposition G(A) = G∞ × G(Af ) with K∞ (resp. Kf ) a maximal compact subgroup of G∞ (resp. G(Af )). We may also assume that Ω is of the form Ω∞ × Ωf , where Ω∞ (resp. Ωf ) is a compact subset of o P∞ (resp. o PAf )). It follows then that if g ∈ G∞ , there exists γ ∈ G(k) such that gγ∞ ∈ K∞ Ac · Ω∞ and γf ∈ Kf · Ωf . Now Lf being an open compact subgroup and Kf · Ωf being a compact set of G(Af ), we can find finitely many elements θ1 , · · · , θr ∈ G(Af ) such that Kf Ωf ⊂ ∪1≤i≤r θi Lf . In particular we have γf = θi ℓi for some ℓi ∈ Lf and 1 ≤ i ≤ r. Now if γf , γf′ are such that γf = θi ℓi and γf′ = θi ℓ′i with ′ ℓi , ℓ′i ∈ Lf , then γf′ = γf .ℓ−1 i ℓi = γf ζ with ζ ∈ Lf . We thus see that we may assume that the θi = ξif with ξi in G(k). Let Ξ = {ξ1 , · · · , ξr } ⊂ G(k). Then one finds that K∞ Ac · Ω∞ · Ξ∞ · d∞ (ΓLf ) = G∞ . We summarize the above discussion in 1.9. Theorem (A. Borel) Let L be a compact open subgroup of G(Af ) and Γ = d∞ (ΓL ), where ΓL is the subgroup d−1 f (L) ⊂ G(k). Then Γ a discrete subgroup of G∞ . There exists a finite subset Ξ in G(k) a constant c0 > 0 and a compact set Ω∞ ⊂◦ P∞ such that K∞ · Ac · Ω′∞ · Ξ∞ · Γ = G∞ where K∞ is a maximal compact subgroup of G∞ . Further for c′ > c if Sc′ = K∞ · Ac′ · Ω′∞ , the set {γ ∈ Γ | Sc′ ξ ∩ Sc′ ξ ′ 6= φ}

10

M.S. Raghunathan

is finite for ξ, ξ ′ ∈ Ξ∞ . (The last assertion is an additional piece of information that cannot be deduced from Theorem 1.6.) 1.10. Suppose now that the centre of G does not contain a split torus (the centre of GL(n) is a 1-dimensional split torus so the considerations of this paragraph are not applicable to GL(n); they are however applicable to the group SL(n)). In this case by using the Iwasawa decomposition in G∞ it can be shown that the Haar measure of the set Sc′ is finite. Since a left translation invariant Haar measure on G∞ is also right translation invariant, one sees that a Haar measure on G∞ defines a left translation (under G∞ ) invariant finite measure on G∞ /Γ. Y [We will elaborate on this in the case G = SL(n). In this case G∞ = SL(n, kv ). We can identify SL(n, kv ) v∈∞

with SL(n, R) or Y SL(n, C) according to kv ≃ R or C. The compact group K∞ is a product Kv where for v ∈ ∞, Kv = SO(n) or SU (n) according v∈∞

to kv ≃ R or C. The group A can be identified with the group of diagonal matrices in SL(n, R) with positive real diagonal entries - the inclusion of Q in k induces for each v ∈ ∞, an inclusion SL(n, R) in SL(n, kv ) and hence a diagonal inclusion of SL(n, R) in G∞ . For c′ > 0, Ac′ = {d = diagonal (d1 , . . . , dn ) | di > 0 for 1 ≤ i ≤ n, d1 · d2 . . . dn = 1 and di /di+1 ≤ c for 1 ≤ i < n}. The group P consists of upper triangular matrices in SL(n) and let U be the Ysubgroup of upper triangular unipotent matrices in P. The o group ◦ P∞ = P(kv ) and o P(kv ) = B(kv ). U(kv ) where B(kv ) = {d = v∈∞

diagonalY(d1 · · · dn ) | di ∈ kv∗ , | di |v = 1 for 1 ≤ i ≤ n}. One sees then that B∞ = B(kv ) is compact and contained in K∞ and as it centralises A, we v∈∞

see that taking Ω′ to be left translation invariant under B∞ , Sc = K∞ ·Ac′ ·Ω′1 with Ω′1 a suitable compact subset of U∞ . The natural map K∞ × A × U∞ → G∞

is an analytic isomorphism of manifolds. The Haar measure on G∞ pulled back to K∞ ×A×U∞ takes the form ρ2 (a)·dk·da·du where dk (resp. da, du) is a Haar measure on K∞ (resp. A, U) and ρ2 is the homomorphism of A in R+ determined as follows: Inner conjugation by a ∈ A carries the Haar 2 measure duRon U+ ∞ into ρ (a) · du (U∞ is normalised by A). It is also easy to see that A′ ρ2 (a)da < ∞. Since K∞ and Ω′1 are compact, it is immediate c

Automorphic Forms in GL(n)

11

that the Haar measure of Sc′ is finite.] We continue with the assumption that G contains no nontrivial k-split central torus (so that G∞ /Γ has finite Haar measure). We assert now that in this case the measure induced on G(A)/G(k) by the Haar measure on G(A) is finite as well. To see this we denote by G1 the closure of G(k) (imbedded diagonally in G(Af ). Now G1 = G(k) if G∞ is compact; in this case one knows (Godement Criterion) that G(A)/G(k) is compact and the finiteness of the Haar measure follows. Thus we assume that G∞ is not compact (this is the case when G = SL(n) with n ≥ 2). Then one knows that G(Af )/G1 is compact and that G1 is normal in G(Af ) (when G = SL(n) one has in fact G1 = G(Af )-this is seen easily using the fact that SL(n, kv ) is generated by upper and lower triangular unipotent matrices and the fact (by the Chinese remainder theorem) that k is dense in Af . π : G(A) → G∞ · G1 \G(A) ≃ G1 \G(Af ) be the natural map. From the definition of G1 it is clear that π factors through G(A)/G(k). Since G1 \G(Af ) is compact we conclude that in order to show that G(A)/G(k) has finite Haar measure it suffices to show that G∞ · G1 /G(k) has finite Haar measure. Now if L is any compact open subgroup of G(Af ), G∞ · G1 ⊂ G∞ · L · G(k) and the quotient L · G1 /G1 is compact. Thus it suffices to show that G∞ · L · G(k)/G(k) has finite Haar measure for a compact open subgroup L ⊂ G(Af ). Clearly one has a natural identification of G∞ · L · G(k)/G(k) with G∞ · L/(G∞ · L ∩ G(k)); since G∞ /Γ (Γ = projection of ΓL = G∞ · L ∩ G(k) on G∞ ) has finite Haar measure and L is compact G∞ · L · G(k)/G(k) has finite Haar measure proving our contention. We have thus 1.11. Theorem Assume that G has no non trivial k split torus in its centre. Then G(A)/G(k) has finite Haar measure. If G contains no non trivial k-split torus G(A)/G(k) is compact. 1.12. Suppose now that C ⊂ G is the maximal central k split torus. Then G = G/C contains no non trivial central split torus. In the case of main interest to us viz when G = GL(n), C(≃ GL(1)) is of dimension 1 and is the entire centre and G is the group P GL(n). By Theorem 1.10, G(A)/G(k) has finite Haar measure. Now it is known that if π : G → G is the natural map, π(G(A)) is a closed normal subgroup of G(A) and G(A)/π(G(A)) is compact. Also the kernel of π : G(A) → G(A) is evidently C(A). It follows that G(A)/π(G(k)) is compact so that G(A)/C(A) · G(k) is compact as well. Once again we elaborate on this in the special case G = GL(n).

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M.S. Raghunathan

Here for any field k′ ⊃ k the natural map GL(n)(k′ ) → P GL(n)(k′ ) is surjective - this is an immediate consequence of Hilbert Theorem 90. Using the realisation of P GL(n) as an algebraic k subgroup of GL(n2 ) got from the adjoint representation, one shows that GL(n, Ov ) → P GL(n)(Ov ) is surjective for all v ∈ Vf . It is then immediate that the map GL(n, A) → P GL(n)(A) is surjective. Observe next that C(A)(≃ GL(1, A)) ≃ I in a natural fashion and in the sequel we will treat continuous characters χ : I → C∗ as characters on C(A) under this isomorphism. Let F χ (G(A), G(k)) denote the vector space of all functions f : G(A) → C such that we have f (gc) = χ(c)−1 f (g) for c ∈ C(A) and f (gx) = f (g) for x ∈ G(k) Note that in order that F χ (G(A), G(k)) be non-trivial we need χ to be 1 ∗ trivial on C(k). If χ is unitary i.e., maps C(A) into S = {z ∈ C | z |= 1}, then the function g 7→| f (g) | on G(A) is invariant under C(A) · G(k) and hence defines a function f on G(A)/C(A) · G(k). We will say that f ∈ F χ (G(A), G(k)) is square integrable if f is square integrable on G(A)/C(A)· G(k). Let G′ = [G, G] be the commutator subgroup. Then the connected component S′ of the identity in S ∩ G′ is a maximal split torus in G′ . The natural map C × S′ → S is not in general an isomorphism (it is not in the case G = GL(n) where G′ = SL(n)). However the group A in G(A) breaks up into a direct product (C(A) ∩ A) × (S′ (A) ∩ A) i.e., the natural map (C(A) ∩ A) × (S′ (A) ∩ A) → A is an isomorphism. Let A0 = C(A) ∩ A and A′ = S′ (A) ∩ A so that AY ≃ A0 × A′ . We now define a subgroup A10 in | χ(av ) |v = 1 for every character χ of G A0 : A10 = {a ∈ A0 ⊂ C(A) | v∈∞

defined over k}. Since χ(av ) ∈ R+ for all a ∈ A0 and any character χ on C, and the subgroup of characters on C which are restrictions of characters 1 on G has finite Y index in the group of all characters on C, we see that A0 = {a ∈ A0 | | χ(av ) |v = 1 for all characters χ on C}. The sequence v∈∞

{1} → A10 → A0 → A0 /A10 → {1} splits and has in fact a natural splitting. To see this fix an isomorphism of C with a product GL(1)r of r copies of GL(1). Note that such an isomorphism yields compatible decompositions of A0 and A10 as products of r copies of the same groups. This means that we need to give the ‘natural splitting’ in the case when C = GL(1). Here C(A) = I and A0 is the product of

Automorphic Forms in GL(n) | ∞ | copies of R+ imbedded in

Y

13

kv∗ through the inclusions R ⊂ kv ≃ R

v∈∞ 1 or C(vY ∈ ∞). The group YA0 is then naturally isomorphic to the subgroup {x ∈ kv∗ | xv ∈ R+ , xv = 1} and a natural supplement to A10 in A is v∈∞ v∈∞ Y ∗ kv | xv = x for all v ∈ ∞ with x ∈ R+ }. Thus we have the group {x ∈ v∈∞

obtained a splitting using an isomorphism of C with (GL(1))r . It is easy to see that the splitting is independent of the choice of this isomorphism. Now let G∗ = {g ∈ G(A) | | χ(g) |= 1 for every character χ of G defined over k. Then G∗ ⊃ A10 and if ′ A0 is the natural supplement to A10 in A0 , then ′ A ∩ G∗ = {1} and the natural map 0 ′

A0 × G∗ → G(A)

is an isomorphism (of locally compact groups). The product formula (1.2) shows that G(k) ⊂ G∗ . Using the fact that I1 /k∗ is compact, it is now easy to deduce the following from Theorem 1.10. 1.13. Theorem G∗ /G(k) has finite Haar measure. 1.14. It is easy to see that any compact subgroup of G(A) is contained in G∗ . Consequently there is a c > 0 and a compact subset Ω ⊂ o P(A) ∩ G∗ = P∗ such that for all Ω ⊃ Ω′ , Ω ⊂ o P(A) ∩ G∗ , and c′ ≥ c we have G∗ = K · A10 · A′c′ · Ω′ · G(k) where A′c′ = {a ∈ A1 | α(a) ≤ c′ for all α ∈ △}. Let C(A)1 = {g ∈ C(A) | | χ(g) |= 1 for all characters χ on C}. Then C(k) ⊂ C(A)1 (product formula) and C(A)1 /C(k) is compact. Hence we can find a compact subset C of C(A)1 such that C · C(k) = C(A)1 . Clearly C · Ω′ = Ω0 is a compact subset of P∗ and one has G∗ = K · A′c · Ω0 · G(k).

2

Automorphic Forms

2.1. We continue with the notations introduced in §1. We fix a unitary character χ : C(A) → S 1 on C(A) and assume that χ is trivial on C(k). We introduce now some function spaces. Recall that we defined F χ (G(A), G(k))

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M.S. Raghunathan

as the vector space of all functions f : G(A) → C satisfying the following condition f (xgγ) = χ(x)−1 f (x) for all g ∈ G(A), x ∈ C(A) and γ ∈ G(k). In the sequel we often write F χ for F χ (G(A), G(k)) (where there is no ambiguity about the group G we are talking about). We set C χ = C χ (G(A), G(k)) = {f ∈ F χ | f

continuous}

Ccχ = Ccχ (G(A), G(k)) = {f ∈ C χ | f has compact support modulo C(A)G(k)}. A function f R: G(A) → C in F χ is square integrable iff f is Borel measurable and k f k22 = G(A)/C(A)G(k) | f (g) |2 dµ(g) < ∞− | f (xgγ) |=| f (g) | for g ∈ G(A), x ∈ C(A) and γ ∈ G(k) so that | f (g) | can be treated as a function on G(A)/C(A)G(k); µ is the Haar measure on this homogeneous space of the locally compact group G(A)/C(A). We denote by Lχ2 = Lχ2 (G(A), G(k)) the set of all square integrable functions in F χ modulo the equivalence of equality almost everywhere with respect to µ. It is a Hilbert space under the inner product Z ′ f (g)f ′ (g)dµ(g). (f, f ) 7→ G(A)/C(A)G(k)

We have (as usual) an inclusion Ccχ ֒→ Lχ2 . A function f : Ω → C, Ω an open set in G(A) is said to be C ∞ if the following conditions hold: (i) for any g0 ∈ Ω, there is a neighbourhood Ω′ of g0 in Ω and a compact open subgroup L ⊂ G(Af ) such that LΩ′ ⊂ Ω and f (xg) = f (g) for all g ∈ Ω′ and x ∈ L. (ii) For any g0 ∈ Ω, there is neighbourhood U of the identity in G∞ such that U g0 ⊂ Ω and the map x 7→ f (xg0 ) is C ∞ on U . Following the decomposition G∞ × G(Af ) ∼ = G(A) as a direct product we may regard any complex-valued function f on G(A) as a function of two variables (one in G∞ and the other in G(Af )). If f is C ∞ , then it is C ∞ in the first variable and locally constant in the second variable; also the partial derivatives of f with respect to the first variable are themselves C ∞ on G(A). We denote by C χ∞ = C χ∞ (G(A), G(k)) (resp. Ccχ∞ = Ccχ∞ (G(A), G(k)) the vector space of C ∞ functions (resp. C ∞ functions with support compact modulo C(A) · G(k)). We have evidently inclusions Ccχ∞ ֒→ Ccχ ֒→ Lχ2 .

Automorphic Forms in GL(n)

15

2.2. In the sequel E will denote any one of the function spaces F χ , C χ , Ccχ , C χ∞ , Ccχ∞ , Lχ2 introduced above and for an open compact subgroup L of G(Af ), E L will denote the subspace of functions in E which are invariant under left translations by L. We observe that a function f in E is determined by its restriction to G∗ (since ′ A0 G∗ = G(A) and ′ A0 ⊂ C(A)). On the other hand G∗ = K · A10 · A′c · Ω′ · C · G(k) with C ⊂ C(A) (cf. 1.14). Further K = K∞ · Kf with K∞ ⊂ G∞ and Kf ⊂ G(Af ) and one may assume that Ω′ = Ω∞ · Ωf with Ω∞ (resp. Ωf ) a compact subset of ◦ P∞ (resp. ◦ P(Af )). Now Kf · Ωf is a compact subset of G(Af ). It follows that if Lf ⊂ G(Af ) is a compact open subgroup of G(Af ), there exists a finite set H′ ⊂ G(Af ) such that Kf · Ωf ⊂ Lf · H′ . It follows from Corollary 1.7 that we have Lf · H′ · df (G(k)) = G(Af ). We introduce an equivalence relation on H′ as follows: for θ, θ ′ ∈ H′ , θ ∼ θ ′ iff ′ there exists ρ ∈ G(k) with θρf θ −1 ∈ Lf (it is easily checked that this is an equivalence relation). Let H ⊂ H′ be a subset containing exactly one element in each equivalence class. We then assert that any f ∈ E Lf is determined by its restriction to G∞ Lf H (and in view of Lf -invariance) by its restriction to G∞ · H. Evidently to prove this it suffices to show that G∞ · Lf · H · C(A) · G(k) = G(A). We have seen that G(A) = ′ A0 · G∗ =

′

A0 · K∞ · A10 · A′c · Ω∞ · Kf · Ωf · C(A)G(k) ⊂ G∞ · Lf · H′ · C(A) · G(k).

Thus it suffices to show that if θ ∼ θ ′ , G∞ · Lf · θ · G(k) ⊃ G∞ · Lf · θ ′ . ′

Since θ ∼ θ ′ , there exists ρ ∈ G(k) with θρf θ −1 = ℓ ∈ Lf so that θ ′ = ℓ−1 θρf . It follows that G∞ Lf θ ′ ρ−1 = G∞ Lf θρf ρ−1 = G∞ ρ−1 ∞ Lf θ = G∞ Lf θ. It follows that Lf · H · df (G(k)) = G(Af ). For θ ∈ H, let rθ (f ) : G∞ → C be the function rθ (f )(g) = f (gθ) = f (θg). Then rθ (f )(gγ∞ ) = rθ (f )(g) for all γ ∈ G(k) such that γf ∈ θ −1 Lf θ. In fact one has rθ (f )(gγ∞ ) = f (gγ∞ θ) = f (gγ∞ θγ −1 ) = f (gθγf−1 θ −1 θ) = f (gθ). −1 Let Γθ = d∞ (G(k) ∩ d−1 f (θ Lf θ)) i.e.,

Γθ = d∞ {γ | γf ∈ θ −1 Lf θ}.

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Let C χ (G∞ /Γθ ) (resp. Ccχ (G∞ /Γθ ), resp. C χ∞ (G∞ /Γθ ), resp. Ccχ∞ (G∞ /Γθ ), resp. Lχ2 (G∞ /Γθ )) be the space of all continuous functions (resp. continuous functions with compact support modulo C∞ (resp. C ∞ functions, resp. C ∞ functions with compact support modulo C∞ , resp. square integrable functions) on G∞ /Γθ satisfying the condition f (gx) = χ(x)−1 f (g) for x ∈ C∞ . Then if E = C χ , (resp. Ccχ , resp. C χ∞ , resp. Ccχ∞ , resp. Lχ2 ), and f ∈ E Lf , rθ (f ) ∈ Cχ (G∞ /Γθ ) (resp. Cχc (G∞ /Γθ ), resp. Cχ∞ (G∞ /Γθ ), χ resp. Cχ∞ c (G∞ /Γθ ), resp. L2 (G/Γθ ). We thus obtain maps a ` rθ : CχLf → θ∈H Cχ (G∞ /Γθ ) θ∈H a ` χL rθ : Cc f → θ∈H Cχc (G∞ /Γθ ) θ∈H a ` rθ : Cχ∞Lf → θ∈H Cχ∞ (G∞ /Γθ ) θ∈H a ` χ∞L rθ : Cc f → θ∈H Cχ∞ c (G∞ /Γθ ) θ∈H a

χLf

rθ : L2

θ∈H

→

`

χ θ∈H L2 (G∞ /Γθ ).

2.3. Proposition The maps above are isomorphisms of topological vector spaces. We record here for future use the following fact proved above. 2.4. Lemma

Lf · H · df (G(k)) = G(Af ).

2.5. Proof of 2.3 The topologies on these spaces are the standard ones. On CχL and Cχ (G∞ /Γθ ) (resp. Cχ∞L and Cχ∞ (G∞ /Γθ )) it is the topology of uniform convergence (resp. together with all derivatives) on compact sets. On CχL and Cχ∞ (resp. Cχc (G∞ /Γθ ) and Cχ∞ c c c (G∞ /Γθ )) it is the inductive limit of the topology of uniform convergence (resp. together with all derivatives) on closed sets of G(A) which are compact modulo C(A)·G(k) and closed subsets of G∞ which are compact modulo C∞ · Γθ respectively. The L2 spaces are of course given the Hilbert space structure. That the maps are continuous injections is easy to see. We need to show that the maps are surjective - the open mapping theorem would then guarantee that the maps are isomorphisms. Let {fθ }θ∈H be a collection of functions on G∞ /Γθ , θ ∈ H belonging to one of the above spaces. Define a function f on G∞ LH by setting f (gℓθ) = fθ (g). We extend the function f to all of

Automorphic Forms in GL(n)

17

G(A) by setting f (g) = f (gγ) where γ ∈ G(k) is an element such that gγ ∈ G∞ LH. We need only check that f is well defined i.e., if gγ ′ ∈ G∞ LH also for some γ ′ ∈ G(k), f (gγ) = f (gγ ′ ). Let gγ = hℓθ and gγ ′ = h′ ℓ′ θ ′ . Then setting γ −1 γ ′ = ζ, we have hℓθζ = h′ ℓ′ θ ′ . It follows that ℓθζf = ℓ′ θ ′ ′ leading to θζf θ −1 = ℓ−1 ℓ′ ∈ L i.e., θ ∼ θ ′ and hence θ = θ ′ ; and when θ = θ ′ , ζf ∈ θ −1 Lθ so that ζ ∈ Γθ , and fθ is Γθ -invariant. This proves that f is well defined. That f has the required properties of continuity, smoothness etc. if the {fθ }θ∈H have them is immediate from the definitions. 2.6. Let U denote the universal enveloping algebra of the real Lie algebra L(G) of G. Then U operates on the space of C ∞ functions on G(A) leaving stable the subspaces C χ∞ (G(A), G(k)) and Ccχ∞ (G(A), G(k)) as well as the subspace of L-invariants in these spaces for any compact open subgroup L of G(Af ). A function f ∈ C χ is K-finite if the C-linear span of {Lk f | k ∈ K} is finite dimensional. The group K acts continuously on this finite dimensional vector space V and hence the image of the profinite group Kf in GL(V ) is finite. Thus if ϕ is K-finite, ϕ is invariant under an open compact subgroup of G(Af ); and the C-span of {Lk ϕ | k ∈ K∞ } is finite dimensional. Conversely if ϕ satisfies these two conditions ϕ is K-finite. Let Z be the centre of U. A C ∞ function ϕ in C χ∞ is Z-finite iff the C-linear span of {zf | z ∈ Z} is finite dimensional. Functions relevant to harmonic analysis on G(A)/G(k) are those that do not grow too rapidly at infinity. To describe the kind of growth at infinity that we need to impose we need some preliminary definitions. We assume, as we may, that the imbedding of G in GL(N ) maps G into SL(N ) (when G = GL(n) we can take N = 2n and the imbedding to be g 0 g 7→ g ∈ GL(n)). 0 tg−1 This is done so that one takes care that the entries of g as well as those of g−1 are handled simultaneously.YWe define for g ∈ G(A) the height of g denoted k g k in the sequel as (max1≤i,j≤N | gvij |v ) where gvij , 1 ≤ v∈V

i, j ≤ N are the entries of the v-adic component gv ∈ G(kv ) of g. Observe that since gv ∈ GL(N, Ov ) for all v ∈ V\S for a finite subset S containing ∞, max1≤i,j≤N | gvij |= 1 for all but a finite number of v and thus the product above over all v ∈ V reduces to a finite product. It is easy to see that g 7→k g k is a continuous function of G(A) in R+ . In the sequel we will say that a right G(k)−invariant function f : G(A) 7→ C has moderate

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M.S. Raghunathan

growth if there exist constants c, r > 0 such that for all g ∈ G(A) one has | f (g) |≤ c k g kr with this notion of moderate growth we have 2.7. Definition An automorphic form (with central unitary character χ) on G(A) is a C ∞ function f : G(A) → C satisfying the following conditions (i) U f is of moderate growth for all U ∈ U. (ii) f (gx) = χ(x)−1 f (g) for all g ∈ C(A) and all x ∈ C(A). (iii) f (gγ) = f (g) for all g ∈ G(A) and γ ∈ G(k). (iv) f is K-finite (v) f is Z finite. The automorphic forms with central character form a vector space which we denote Aχ 2.8. Remarks (i) If a nonzero automorphic form is to exist, with χ as central character, evidently one must have χ(ρ) = 1 for all ρ ∈ C(k) in view of the condition (iii). Thus we will consider only characters χ on C(A) that are trivial on C(k). Also note that since χ is unitary, the growth condition does not lead to any contradiction. (ii) We have assumed that the group G is reductive. This means that the Lie algebra L(G∞ ) of G∞ is a direct product of an abelian Lie algebra h and a semisimple Lie algebra s. The Lie algebra s has a Cartan - decomposition s = k ⊕ p with the Killing form of s restricted to k (resp. p) negative (resp. positive) definite. Let Xi , 1 ≤ i ≤ r, Yj , 1 ≤ j ≤ s and Zk , 1 ≤ k ≤ t ′ be bases of h, k and p respectively so chosen j , Yj ′ >= −δ Xjj and Xthat < YX 2 2 Yj2 is Xi + Zk − < Zk , Zk′ >= δkk′ . Then the element C = 1≤i≤r

a central element of the enveloping algebra U. Let

1≤p≤t C′ =

X

1≤j≤s Yj2 . Then

1≤j≤s

C + 2C ′ is an element of U and the corresponding differential operator △ on G∞ is an elliptic operator. Let B be the subalgebra of End C χ (G(A), G(k))

Automorphic Forms in GL(n)

19

generated by the centre Z of U and {Lk | k ∈ K} (Lk is the left translation by k). Then the C-span of {T (f ) | T ∈ B} is finite dimensional for the automorphic form f . Now the span of the Lk f, k ∈ K contains U f for all U in the subalgebra of U generated by k. Thus C and C ′ belong to B and hence so does △. It follows that {△n f | n ∈ N} spans a finite dimensional vector space over C. We conclude that there is a monic polynomial P in one variable such that P (△)f = 0. Now P (△) is an elliptic operator on G∞ with analytic coefficients. By the regularity theorem for elliptic operators with analytic coefficients we conclude that f is analytic (in the variable in G∞ ). (iii) The third comment is that if U ∈ U and f is an automorphic form then so is U f . It is clear that it suffices to check this when U ∈ L(G). Now the map L(G∞ ) ⊗ C χ∞ given by (U, f ) 7→ U f is compatible with the action of G(A) on the two sides. G(A) acts on L(G) by the adjoint representations of the factor G∞ and on C χ∞ by left translation; we take the tensor product representation on the left-hand side. Suppose now V is a finite dimensional subspace of C χ∞ which is K as well as Z stable. Then the image of L(G)⊗V in C χ∞ under the above map is finite dimensional Z-stable as well as Kstable. As this space contains U f , for U ∈ L(G∞ ), we see that U f is Kfinite and Z-finite. That U ′ (U f ) = (U ′ U )(f ) has moderate growth is clear from the definitions. (iv) The vector space of all automorphic forms Aχ with central character χ is stable under left translations by elements of G(Af ). This is seen as follows: for g ∈ G(Af ), gKg−1 ∩ K has finite index in K and Lg f is (g−1 Kg ∩ K)finite and hence K-finite as well. Since the action of G(Af ) on the left and the action of U on the space of C ∞ functions commute, we see that (Lg f ) is Z-finite for g ∈ G(Af ). The growth condition is immediate from the fact that for g0 ∈ G(Af ) one has a constant C > 0 such that for all g ∈ G(A), k g0 g k≤ C k g k. 2.9. Lemma (Harish-Chandra: Acta Math. 116 p.118). If f is a K∞ ∞ funcfinite Z-finite C ∞ function on G∞ , there is an α ∈ C∞ c (G∞ ) (= C tions with compact support in G∞ ) such that α ∗ f = f . This lemma shows that the growth condition (i) in (2.6) can be replaced by the weaker condition: (i′ )·f is of moderate growth. In fact fix α ∈ Cc∞ (G) such that α ∗ f = f . Then one has for U ∈ U, U f = U (α ∗ f ) = U α ∗ f ; and

20

M.S. Raghunathan

it is easy to see that for any β ∈ Cc∞ , β ∗ f has moderate growth. In the sequel we will make use of Lemma 2.8 in other contexts too. 2.10. Let χ be a unitary character on C(A)/C(k). Let σ : K → GL(W ) be a finite dimensional representation of the compact subgroup K of G(A) and λ := Z → EndC W ′ a finite dimensional representation of Z. Let Aχ (G(A), G(k), σ, λ) denote the space of complex values C ∞ functions f on G(A) satisfying the following conditions (cf. 2.8). (i) f has moderate growth. (ii) f (gγ) = f (g) for g ∈ G(A), γ ∈ G(k). (iii) f (gh) = χ(h−1 )f (g) for g ∈ G(A), h ∈ C(A). (iv) The K-span of f (in F χ ) is a quotient of W as a K-module. (v) The Z-span of f (in F χ ) is a quotient of W ′ as a Z-module. With this notation our central result is 2.11. Theorem

dim Aχ (G(A), G(k), σ, λ) < ∞.

2.12. The representation σ of K when restricted to the totally disconnected subgroup Kf has a kernel Lf of finite index. Then it is easily seen that Aχ (G(A), G(k), σ, λ) is contained in the space of C ∞ functions f on G(A)/G(k) such that f (gx) = χ(x)−1 f (g) for g ∈ G(A), x ∈ C(A), f (kg) = f (g) for k ∈ Lf , the K∞ -span of f is a quotient of σ∞ as a K∞ -module and the Z-span of f is a quotient of λ. We can now appeal to Proposition 2.4 to conclude that Theorem 2.11 is equivalent to: 2.13. Theorem Let Γ ⊂ G(k) be an arithmetic subgroup. Let Aχ (G∞ , Γ, σ∞ , λ) be the space of C ∞ functions f on G such that (i) f (gγ) = f (g) for all g ∈ G∞ and γ ∈ Γ. (ii) f (gc) = χ(c)f (g) for all g ∈ G∞ and c ∈ C∞ . (iii) The K∞ -span of f is a quotient of σ∞ . (iv) The Z-span of f is a quotient of λ.

Automorphic Forms in GL(n)

21

(v) there is a constant c > 0 and integer r > 0 such that | f (g) |≤ c k g kr for g ∈ G∞ . Then Aχ (G∞ , Γ, σ∞ , λ) is finite dimensional.

3

Cusp Forms

3.1. We need a number of facts from analysis which we collect together in 3.2 - 3.11 below for convenient later use. We refer to R. Narasimhan [N] for proofs of results not given here. We begin with the following 3.2. Proposition Let X be a locally compact space and µ a Borel probability measure on X. Let Cb (X) denote the space of bounded continuous functions on X and for ϕ ∈ Cb (X), let k ϕ k∞ = lim Sup{ϕ(x) | x ∈ X}. Suppose T : L2 (X, µ) → L2 (X, µ) is a linear map such that T (ϕ) ∈ Cb (X) for all ϕ ∈ L2 (X, = µ) and there is a constant C > 0 such that k T (ϕ) k∞ ≤ C k ϕ k2 . Then T is an operator of the Hilbert Schmidt type (and hence compact). Proof. For x ∈ X, the linear form ϕ 7→ T (ϕ)(x) on L2 (= L2 (X, µ)) is bounded. It follows that there exists kx ∈ L2 such that k kx k2 ≤ C and T (ϕ)(x) =< kx , ϕ > for all ϕ ∈ L2 . Let k(x, y) = kx (y) for x, y ∈ X : k(x, y) is defined for all most all y for each x. Clearly for each fixed x, kx (y) is measurable so that k(x, y) is measurable on X × X. Now R R R |2 dµ(x) · dµ(y) = X dµ(x) Y | kx (y) |2 dµ(y) X×X R | k(x, y) ≤ X dµ(x)C 2 = C (since µ(X) = 1) Thus k(x, y) ∈ L2 (X × X, µ × µ) and Z Z k(x, y)ϕ(y)dµ(y). kx (y)ϕ(y)dµ(y) = T ϕ(x) = X

X

Thus T is a Hilbert-Schmidt operator. 3.3. The next result we want to state is the Sobolev inequality. We need to introduce some preliminary notation to state the result. Let M be be a smooth manifold. Let Ω, be a relatively compact subset of M ; the compact closure of Ω is denoted Ω. Let {Ui , | 1 ≤ i ≤ m} be an open covering of

22

M.S. Raghunathan

the closure Ω of Ω by coordinate open sets; let Vi , 1 ≤ i ≤ m, be open subsets of Ui such that the closure V i of Vi is compact and contained in Ui and further ∪1≤i≤r Vi ⊃ Ω. We fix a C ∞ volume form on M and denote the corresponding measure by µ. If x1 , · · · , xn are the coordinates in Ui , we 1 ∂α ∂1 denote by D β the operator ∂x 1 · · · ∂xn for a multi index β = (β1 , · · · , βn ) of n

n

non-negative integers. We introduce on the space of C ∞ functions on Ω (i.e. C ∞ functions defined a neighbourhood of Ω) the following norms k kr,p where r is an integer ≥ 0 and p > 1. For a C ∞ function f on Ω, k f kr,p = Sup {k D β f kpV i | | β |≤ r,

1 ≤ i ≤ m}.

Here for h a C ∞ function on Ω, k h kpV i =

Z

| h |p dµ. Vi

The norm defined above depends on the covering and the shrinking chosen, the coordinates chosen in the covering open sets, and the volume form on M . However the equivalence class of the norm k kr,p depends only on r and p and not on the choices made above. With these definitions we can now state Sobolev’s inequality. 3.4. Theorem There is a constant C, C(p, Ω) > 0 such that for all f, C ∞ on Ω, k f k∞ ≤ C k f k[n/p]∗ ,p where [n/p]∗ is the minimal integer ≥ n/p (n = dim M ). 3.5. Remark In the special case when M = Rn and we take the coordinates to be standard coordinates, the volume form as the standard one, and Ω is a disc of radius ρ centred at a point x0 , we have for any f, C ∞ on Ω, k f k∞ ≤ C k f k[n/p]∗ ,p with C = C(ρ) independent of the point x0 . This is because the measure as well as the vector fields ∂/∂xi are translation invariant. 3.6. Our next result is a very special case of a general theorem about elliptic operators on a compact manifold. We consider a connected real nilpotent Lie group N with a discrete subgroup Φ such that N/Φ is compact. Let L(N ) denote the Lie algebra of N and X1 , · · · , Xn be a basis of L(N ) over R. We consider the Xi to be right translation - invariant vector fields on N , hence they define vector fields on N/Φ which we continue to denote Xi .

Automorphic Forms in GL(n)

23

P Then ∆ = − Xi2 is a non-negative self adjoint elliptic operator on the space of RC ∞ - functions on N/Φ with respect to the inner product : (f, g) 7→ N/Φ f gdn where dn is the Haar measure on N/Φ. One has in fact for X ∈ L(N ), < Xf, g >=< f, −Xg > so that − < X 2 f, f >=< Xf, Xf >≥ 0 and hence < ∆f, f >≥ 0. In fact one has X < ∆f, f >= < Xi f, Xi f > so that ∆f = 0 if and only if Xi f = 0 for all i; and since the Xi , 1 ≤ i ≤ r give a basis for the tangent space at every point this means that f is a constant. Now the general theory of self adjoint elliptic operators applied here shows that L2 (N/Φ), dn) decomposes as an orthogonal direct sum a C⊕ H(λn ) 1≤n 0 such that for any α with | α |≤ 2r and any ϕ ∈ C ∞ (N/Φ), k X α ϕ k2 ≤ C k ∆r ϕ k2 +C ′ k ϕ k2 . Note The norm Sup {k X β ϕ k2 | | β |≤ r} is equivalent to the norm k kr,2 introduced earlier; this is easily seen from the fact that the Xi can be expressed as a linear combination of the ∂/∂xj of the local coordinates with coefficients that are C ∞ . 3.10. Corollary If ϕ ∈ C ∞ (N/Φ) is such that | α |≤ 2r, one has k X β ϕ k2 ≤ C k ∆r ϕ k2

R

N/Φ

ϕdn = 0, then for

for a constant C > 0 (independent of ϕ). 3.11. Corollary Let (N, Φ) be asRin 3.6. Then there is a constant C > 0 such that for ϕ ∈ C ∞ (N/Φ) with N/Φ ϕ(n)dn = 0, we have | ϕ(x) |p ≤ R C N/Φ | ∆r ϕ(x) |p dx for r ≥ [n/p]∗ . This is immediate from Theorem 3.4 and Corollary 3.10. From now on we go back to the notations of §1 and §2. 3.12. A function ϕ ∈ C χ (cf. 2.1) is cuspidal if the following holds: let P′ be a proper parabolic subgroup of G defined over k one U′ its unipotent radical. Then for almost all g ∈ G(A) the function u R7→ ϕ(gu), u ∈ U′ (A) is integrable for the Haar measure on U′ (A)/U′ (k)) and U′ (A)/U′ (k) ϕ(gu)du = 0 for almost all g ∈ G(A). 3.13. Remarks (i) ‘Almost all g’ above means for all g ∈ G(A) outside a set of Haar measure zero. (ii) It is known that every parabolic subgroup defined over k is conjugate to a parabolic subgroup of G defined over k containing a fixed minimal parabolic subgroup P of G defined over k. Using the G(k) invariance χ of elements R of F , it is easily seen that for ϕ to be cuspidal, it′ suffices that U′ (A)/U′ (k) ϕ(gu) vanish for almost all g ∈ G(A) for U the unipotent radical of any of the (finitely many) k-parabolic subgroups P′ containing P.

Automorphic Forms in GL(n)

25

(iii) We observe next that for ϕ to be cuspidal, it sufficient that the integral vanish for maximal parabolic subgroups defined over k and containing ′′ P. In fact if P′ is a k-parabolic subgroup and P is a maximal kparabolic subgroup containing P′ , then the unipotent radical U′ of P′ ′′ ′′ contains the unipotent radical U of P as a normal subgroup. We ′′ ′′ have thus a fibration U′ (A)/U′ (k) → U′ (A)/U (A)U (k) whose fibres ′′ ′′ are U (A)/U (k); our contention now follows from Fubinis theorem which is valid for this fibration. 3.14. Notation If E denotes one of the spaces in E = {F χ , F χ , C χc , C χ∞ and Lχ2 } we denote by o E the subspace {ϕ ∈ E | ϕ cuspidal}. 3.15. If E is one of the spaces in E, then o E is invariant under the left action of G(A) on G(A)/G(k). This is clear from the definition of cuspidal functions. Also if E ∈ E and E is not F χ , for ϕ ∈ E and α ∈ Fc∞ (G(A)), α ∗ ϕ ∈ E as is easily seen from the definition of the convolution operation R ((α ∗ ϕ)(g) = G(A) α(h)ϕ(h−1 g)dh (dh Haar measure on G(A))). It is further easy to see that ϕ 7→ α∗ϕ is a continuous linear operator on E which leaves the subspace o E stable (note that o E is closed in E (for E 6= F χ )). By the definition of Cc∞ (G(A)), there is a compact open subgroup K(α) of G(Af ) such that α is invariant under left translations by K(α). From the definition of α ∗ ϕ, it is immediate that it is K(α)-invariant on the left. Let AR∗ : E → E(E ∈ E, E 6= F χ ) be the averaging over K(α) : (A∗ ϕ)(g) = K(α) ϕ(kg)dk (dk Haar measure on K(α), g ∈ G(A)). Then A∗ is a continuous projection of E on E K(α) and of o E on o E K(α) . We note that for all ϕ ∈ E, α ∗ A∗ (ϕ) = A∗ (α ∗ ϕ)α ∗ ϕ. In particular α ∗ ϕ = A(α ∗ Aϕ) on E as well as o E. We now state the central result of this chapter. 3.16. Theorem operator.

The operator ϕ 7→ α ∗ ϕ of o Lχ2 into itself is a compact

3.17. We will reformulate the theorem for a space of functions on G∞ . In the light of the remarks made at the end of 3.15, it is clear that it suffices to χK(α) show that convolution with α is a compact operator on o L2 . Consider ` now the isomorphism r = θ∈H rθ (defined in 2.2: we take for Lf of 2.2 the ` χK(α) group K(α)) of L2 K(α) on θ∈H Lχ2 (Gα /Γθ ).

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M.S. Raghunathan

3.18. Claim For θ ∈ H, let o Lχ2 (G∞ R /Γθ ) be the subspace of all those χ functions ϕ in L2 (G∞ , Γθ ) for which U′ ∩Γ ϕ(gu)du = 0 for almost all ∞ g ∈ G∞ and for all U′ , the unipotent radical of proper k parabolic subgroup ` P′ of G. Then r maps o Lχ2 isomorphically onto θ∈H o Lχ2 (G∞ /Γθ ). 3.19. Proof It is known that the group df (U′ (k)) is dense in U′ (Af ) (this is known as the strong approximation property). Let Mθ = U′ (Af ) ∩ θ −1 K(α)θ; then Mθ is an open compact subgroup of U′ (Af ). Consequently U′ (Af ) = Mθ · df (U′ (k)). It follows that U′ (A) = U′∞ · Mθ · U′ (k) and hence the natural map ≃ U′ (A)/U′ (k) (U′∞ · Mθ )/((U′∞ · Mθ ) ∩ U′ (k)) −→

(∗)

R is a homeomorphism. For ϕ ∈o Lχ2 , we have U(A)/U′ (k) ϕ(θgu)du = 0 for θ ∈ H and g ∈ G∞ . In view of the identification (*) described above, we see that Z ϕ(θgu)du = 0. U′∞ ·Mθ /U′∞ ·Mθ ∩U′ (k)

Now if we set u = u∞ · uf with u∞ ∈ U′∞ and uf ∈ U′ (Af ), we have ϕ(θgu) = ϕ(θgu∞ uf ) = ϕ(θuf gu∞ ) = ϕ(θuf θ −1 · θgu∞ ) = ϕ(θgu∞ ) (since by definition of Mθ , uf ∈ Mθ implies that θuf θ −1 ∈ K(α) and ϕ is left K(α)-invariant). Thus integrating over Mθ first we find that Z Z ϕ(θgu)du ϕ(θgu)du = 0= (U′∞ ·Mθ )/((U′∞ ·Mθ )∩U′ (k))

proving that r maps o Lχ2 into

`o

χ θ∈H L2 (G∞

U′∞ /U′∞ ∩Γθ

| Γθ ). We need to show that χK(α)

the map is onto this subspace. To see this let ϕ ∈ L2 be such that rθ (ϕ) is in o Lχ2 (G∞ /Γθ ) for all θ-we know that such a ϕ exists and is unique. We need to show that ϕ ∈o Lχ2 i.e. we have to show that Z ϕ(gu)du = 0 (∗∗) U′ (A)/U′ (k)

for almost all g ∈ G(A) for the unipotent radical U′ of any proper k-parabolic subgroup P′ of G. Now g = g∞ · gf where g∞ ∈ G∞ and gf ∈ G(Af ). By Lemma 2.4, gf = kf ·θ·df (γ) with kf ∈ K(α), θ ∈ H and γ ∈ G(k). Since ϕ is left K(α)-invariant, ϕ(gu) = ϕ(g∞ gf u) = ϕ(g∞ kf θdf (γ)u) = ϕ(g∞ θdf (γ)u) −1 γ γ u) = ϕ(g ′ θγu). (since ϕ is left K(α)-invariant) = ϕ(g∞ θγ∞ ∞ f ∞

Automorphic Forms in GL(n)

27

′′

Set U = γuγ −1 . Then one has R

′ ′ (A)/U′ (k) ϕ(g∞ θγu)du UR ′ u′ )du′ = U′′ (A)/U′′ (k) ϕ(θg∞

R ′ θ · γuγ −1 )du = U′ (A)/U′ (k) ϕ(g∞ R ′ u′ )du′ = (U′′ ·M ′ )/(U′′ ·M ′ ∩U′′ (k)) ϕ(θg∞ ∞

θ

θ

′′

where Mθ′ = (U (Af ) ∩ θ −1 K(α)θ); and the integral in the last integral ′′ ′ )(U′′ · M ′ ∩ U ′′ (k)). It follows on (U∞ · M∞ is an Mθ′ -invariant function ∞ ∞ R ′ u)du and is thus zero since r ϕ ∈ that integral equals U′′ /U′′ ∩Γθ ϕ(θg∞ θ ∞ ∞ o Lχ (G /Γ ). Thus Claim 3.15 is established. ∞ θ 2 3.20. Lemma ForR α ∈ Cc (G(A)), let α∞ ∈ Cc (G∞ ) be the function defined by α∞ (g) = G(Af ) α(gh)dh. Then rθ (α ∗ ϕ) = α∞ ∗ ϕ for all ϕ ∈ χK(α)

L2 . This is immediate from the definition of rθ and the K(α)-invariance of α. The lemma shows that Theorem 3.13 is equivalent to the following:

3.21. Theorem Let Lf be a compact open subgroup of G(A) and set Γ = d∞ (ΓLf ). Let χ be a unitary character on C(A)(C = centre of G) which is trivial on C(k). Let α ∈ Cc (G∞ ). Then ϕ 7→ α ∗ ϕ is a compact operator on o Lχ2 (G∞ /Γ). 3.22. We begin with the observation that there is no loss of generality in assuming that the centre of G is finite. To see this observe first that if G′ is the commutator subgroup of the identity connected component of G, then C∞ · G′∞ = H has finite index in G∞ . Let ξ1 . . . ξr be a complete set of inequivalent representatives for G∞ /H in G∞ . For 1 ≤ i ≤ r and ϕ ∈ Lχ2 , let ϕi : C∞ · G∞ → C be the function defined by ϕi (g) = ϕ(ξi · g). Then ϕi (gγ) = ϕi (g) for g ∈ C∞ ·G∞ ′ = H and γ ∈ H∩Γ. Also ϕi (gc) = χ(c)ϕi (g) for c ∈ C∞ . Now every unipotent element of G∞ is contained in G∞ ′ . From this we see immediately that if ϕ is a cuspidal function so is ϕi restricted to G∞ ′ . Further ϕi is completely determined by its restriction ϕ′i to G∞ ′ . Evidently the map o χ L2 (G∞ /Γ)

−→

Y

r copies

o

′

Lχ2 (G∞ ′ /Γ′ )

28

M.S. Raghunathan

where χ′ = χ |(G′ ∩C)∞ and Γ′ = G∞ ′ ∩ Γ is an isomorphism onto a closed subset. Moreover for 1 ≤ i ≤ r R R (α ∗ ϕ)i (g) = G∞ α(h)ϕ(h−1 ξi g)dh = G∞ α(ξi−1 h)ϕ(h−1 g)dh X R X R −1 −1 ξ −1 g)dh αij (h)ϕj (h−1 = ξ h)ϕ(h ( α(ξ j g )dh j i H H 1≤j≤r

=

X1≤j≤r αij ∗ ϕj

1≤j≤r

where αij ∈ Cc∞ (H) is the function αij (h) = α(ξi−1 ξh h). The natural map π : C∞ × G∞ → H has for kernel the finite group C∞ ∩ G∞ whose order we denote by q. We regard functions on H as functions on C∞ × G∞ by composing with π. With this convention we have Rfor α ∈ Cc∞ (H) and ϕ ∈ R Lχ2 (H/H ∩ Γ), (α ∗ ϕ)(g) = H α(h)ϕ(h−1 g)dh = 1q C∞ ×G∞ α(h)ϕ(h−1 g)dh and setting h = c.x with c ∈ C∞ and x ∈ G∞ we have for g ∈ G∞ (α ∗ ϕ)′ (g) = =

1 q 1 q

R R α(xc)ϕ(c−1 x−1 g)dc ′ dx C G R ∞ R ∞ −1 ′ ′ G′ dx C∞ α(xc)χ(c g)dc = α ∗ ϕ (g) ∞

R where α′ in Cc∞ (G∞ ) is defined by α′ (x) = 1q C∞ α(xc)χ(c)dc. This discussion shows that we need to prove the theorem only in the case when G∞ = G∞ ′ has a finite centre. Now the group Γ admits a torsion free subgroup Γ′ of finite index and Γ′ evidently intersects the centre trivially. We have a natural inclusion Rof o Lχ2 (G∞ ′ (Γ) in o L2 (G∞ ′ /Γ′ ) (where o L2 (G′∞ /Γ′ ) = {ϕ ∈ L2 (G∞ ′ /Γ′ ) | U/U∩Γ′ ϕ(gu)du = 0 for all unipotent radicals of proper k-parabolic subgroups of G∞ ′ ) compatible with convolution by elements of Cc∞ (G∞ ′ ) on both the spaces. We have thus to prove the following 3.23. Theorem Let G be a connected semisimple algebraic k-group and Γ an arithmetic subgroup of G∞ . Let o L2 (G∞ /Γ) be the space of all Γunvariant functions ϕ on G∞ which are square summable on G∞ /Γ and in addition are cuspidal i.e. satisfy the following condition: If U is the R unipotent radical of a proper k-parabolic subgroup P of G, U∞∩Γ ϕ(gu)du = 0 for almost all g ∈ G∞ . Then the operator ϕ 7→ α ∗ ϕ on o L2 (G/Γ) is a compact operator for all α ∈ Cc∞ (G∞ ). Since G∞ is semisimple, G∞ /Γ has finite Haar measure. We now see from Proposition 3.2 that Theorem 3.20 is a consequence of the following:

Automorphic Forms in GL(n)

29

3.24. Proposition Let G∞ , Γ, α be as above. Then there is a constant C = C(α) > 0 such that for ϕ ∈ L2 (G∞ /Γ), | α ∗ ϕ(x) |≤ C k ϕ k2 for all x ∈ G∞ (note that α ∗ ϕ is C ∞ for ϕ ∈ L2 (G∞ /Γ)). 3.25. For c > 0 and a compact subset Ω ⊂o P∞ (notation introduced in §1.5 - 1.9) we define S(c, Ω) to be the subset K∞ ·Ac ·Ω of G∞ . According to Theorem 1.9, there is a finite set Ξ ⊂ G(k), a constant c0 > and a compact set Ω0 ⊂ o P∞ with the following properties: (i) (ξΩ0 ξ −1 )(ξ o P∞ ξ −1 ∩ Γ) = ξ o P∞ ξ −1 . (ii) If c > c0 and Ω ⊃ Ω0 is a compact subset of o P , then S(c, Ω)ΞΓ = G∞ . (iii) If c, Ω are as above the set {γ ∈ Γ | S(c, Ω)Ξγ ∩ S(c, Ω)Ξ 6= φ is finite. Condition (iii) implies in particular the following. There is a constant M = M (c, Ω) > 0 such that for ξ ∈ Ξ and all ϕ ∈ L2 (G∞ /Γ), we have Z | ϕ(g) |2 dg ≤ M k ϕ k22 (∗) S(c,Ω)ξ

3.26. We have seen that o P is a semi-direct product B · U and we have correspondingly a semi-direct product decomposition o P∞ = B∞ · U∞ . The group B∞ is reductive and K∞ ∩ B∞ is a maximal compact subgroup of B∞ . We then have an Iwasawa decomposition of B∞ as (K∞ ∩ B∞ ) · F where F is a connected solvable closed Lie subgroup of B∞ . Moreover as B∞ commutes with A, we see that for ξ ∈ Ξ the map K∞ × F × A × U∞ → G∞

(∗)

given by (k, f, a, u) 7→ k.f.a.u.ξ is an analytic diffeomorphism. Suppose now that P′ is a proper maximal k-parabolic subgroup of G containing P. Then P′ is the semidirect product Z(S′ ) · U′ where Z(S′ ) is the centraliser of a suitable 1-dimensional sub-torus of S and U′ is the unipotent radical of P′ . We have correspondingly a semidirect product decomposition Z(S′ )∞ ·U′∞ of P′∞ . Now U′∞ is a normal subgroup of U∞ and (hence) U∞ is the semidirect ′′ product (Z(S′ )∞ ∩ U∞ ) · U′∞ . Let Z(S′ )∞ ∩ U∞ = U∞ . Then we have a further refinement of the product decomposition (*) above: ′′

Φξ : K∞ × F × a × U∞ × U′∞ → G

30

M.S. Raghunathan ′′

′′

given by (k, f, a, u , u′ ) 7→ k, f, a, u , u′ , ξ. It is convenient to denote by ′′ ′′ H1 , H2 , H3 , H4 and H5 the groups K∞ , F, A, U∞ and U∞ respectively. We assume as we may say that the compact set Ω0 (resp. Ω) in 3.22 is chosen to be of the form (K∞ ∩ o P∞ ).Ω02 .Ω04 .Ω05 (resp. (K∞ ∩ o P∞ ).Ω02 .Ω04 .Ω05 with Ω0i (resp. Ωi ) a compact subset of Hi , 1 = 2, 4, 5 with Ω0i contained in the interior of Ωi . We also assume Ω05 so chosen that ξ −1 Ω05 .ξ.(ξ −1 H5 ξ ∩ Γ) = ξ −1 H5 ξ. The Lie algebra L(G∞ ) of G∞ is identified with the Lie algebra of right translation invariant vector fields on G∞ . The Lie algebras L(Hi ) of the Hi , 1 ≤ i ≤ 5 are identified with subalgebras of L(G∞ ) and hence their elements will also be regarded as right translation invariant vector fields on G∞ . On the other hand an element X of L(Hi ) determines a right translation invariant vector field on the group Hi . We denote this vector field by X ′ (to distinguish it from the vector field X on G∞ ). For e on G∞ as follows: X e is the X ∈ L(Hi ), 1 ≤ i ≤ 5 we define a vector field X image under the analytic diffeomorphism Φξ of the vector field on the product Q 1≤i≤5 Hi whose component in Hi is X while all the other components are zero. If g ∈ G∞ is such that g = h1 .h2 .h3 .h4 .h5 .ξ with hi ∈ H then we have - as is easily checked - for Xi ∈ L(Hj ), Y e (∗) X(g) = (Ad( hi )(X))(g) i<j

Q

Here when j = 1, i<j hi is the identity element while for j > 1, the product is taken in the order of increasing i : h1 h2 · · · hj−1 . 3.27. We now fix a basis B = {Xi }1≤i≤p ∪ {Yi }1≤i≤q ∪ {Zi }1≤i≤r of L(G∞ ) with the following properties: (i) Xi and Yj are eigenvectors for Ad(S), the corresponding character on S being denoted ηi and ζj respectively: Adt(Xi ) = ηi (t)Xi and Adt(Yi ) = ζi (t)Yi . (ii) {Zi | 1 ≤ i ≤ r} ∩ L(Hi ) is a basis for L(Hi ) for 1 ≤ i ≤ 3. (iii) {Xi | 1 ≤ i ≤ p} (resp. {Yi | 1 ≤ i ≤ q}) is a basis for L(H5 ) (resp. L(H4 )). From our choice of ordering on the character group of S, ηi , 1 ≤ i ≤ p and Q Q ζi , 1 ≤ i ≤ q, are positive roots so that ηi = α∈∆ αmiα , ζi = α∈∆ αni iα with miα , niα ≥ 0. For a multi-index β = (β1 , · · · , βp ) (resp. γ = (γ1 , · · · , γ1 ), resp. δ = (δ1 , · · · δr ), (β1 , γj , δk non-negative integers) we set

Automorphic Forms in GL(n)

Xβ = Yγ = Zδ =

X1β1 X2β2 Y1γ1 Y2γ2 Z δ1 Z2δ2

··· ··· ···

With this notation we have

β e 1β Xp p , X γq e 1γ Yq , Y Zrδr , Ze1δ

= = =

31

e 1β1 X e 1β2 X 2 1γ Ye1 1 Ye11γ2 Ze1δ1 · Ze1δ2 1

2

··· ··· ···

1β

Xp p 1γ Yeq q , er1δr . Z

3.28. Lemma Let g = h1 , h2 , h3 , h4 , h5 , ξ ∈ G∞ with hi ∈ Hi . Then for multi-indices β, γ, δ as above we have, setting m =| β + | γ | + | δ |, X e′ δ Ye ′ γ X e ′ β = η β (h3 )ζ γ (h3 ) Z Cβ ′ ,γ ′ ,δ′ (h1 , h2 , h3 , h4 , h−1 3 ) |β ′ |+|γ ′ |+|δ′ |≤m

with Cβ ′ γ ′ δ′ , C ∞ -functions on H1 × H2 × H4 (note that H3 normalises H4 ). Q Q Here η β = 1≤i≤p ηiβi and ζ γ = 1≤i≤q = ζiγi . Proof. One argues by induction on | β | + | γ | + | δ |. When | β | + | γ | + | δ |= 1, this follows from (*) of 3.23 and the fact that the Xi and Yj are eigenvectors with eigencharacter ηi and ζj respectively. Suppose that | β | + | γ | + | δ |> 1; then we can find β ′ , γ ′ , δ′ and with | β ′ | + e′ δ Ye ′ β X e ′ β = Te′ Ze′ δ Ye ′ γ X e ′ β ′ with | γ ′ | + | δ′ |=| β | + | γ | + | δ | −1 and Z ′δ e Ye ′ γ ′ X e ′β′ T ∈ B. Using the induction hypothesis we have an expansion for Z e′ δ Ye ′ δ X e ′ β now in terms of the X, Y, Z of the desired kind. The result for Z follows from the following observations: Te′ has the desired kind of expansion; secondly for a character χ on H3 treated as a function on G∞ , (Te′ , χ)(h3 ) = 0 ·

·

unless T ∈ LH3 and if T ∈ LH3 , (Te′ χ)(h3 ) = χ(T ) χ (h3 ) where χ is the tangent map L(H3 ) → R induced by χ. This proves the lemma.

3.29. We fix an element g0 = h01 , h02 , h03 , h05 , ξ in G∞ with hi ∈ Hi , 1 ≤ i ≤ 5. We take P′ to be the parabolic subgroup defined as follows: let α0 ∈ ∆ be such that α0 (h0 ) ≤ α(h′3 ) for all α ∈ ∆; then P′ is the proper maximal k-parabolic subgroup determined by α0 . The Lie algebra L(H5 ) is the sum of the eigenspace in L(G∞ ) for AdH3 corresponding to the eigencharacters Q (i.e. k-roots) ψ of the form α∈∆ αmα with mα integers ≥ 0 and mα0 > 0- in Q other words in the notation of Lemma 3.25, ηi = α∈∆ αmi α with miα0 > 0. Let (c′0 , Ω0 ) and (c, Ω) be as in 3.22. Then there is neighbourhood V of the identity in A such that for g0 ξ −1 = h01 , h02 , h03 , h04 , h05 ∈ S(c0 , Ω0 ) and x ∈ V, h01 , h02 , h03 x, h04 , h05 ∈ S(c, Ω). Fix such a neighbourhood V of 1 in

32

M.S. Raghunathan

A once and for all (for a fixed (c, Ω)). Let ∆ denote the elliptic operator P P ′2 2 e′ e ′2 1≤i≤p Xi on H5 and let ∆ = 1≤i≤p Xi . Then we have the L -Sobolev inequality (Theorem 3.4) if 2p > r/2, Z 0 2 e ′ p ϕ(h0 .h0 .h3 .h0 .h0 .ξ) |2 dh5 . . . |∆ (I) | ϕ(g ) | ≤ C 1 2 4 5 Ω5

where ϕ ∈ Cc∞ (G∞ ). Treating the right-hand side as a function on H1 ×H2 × H3 × H4 and applying the L1 Sobolev inequality combined with Schwarz’s inequality we have Z X e ′ δ Ye ′ γ ∆ e ′ p ϕ |2 |X | ϕ(g0 ) |2 ≤ C ′ K∞ ×Ω2 ×h03 V ×Ω4 ×Ω5 |δ|+|γ|≤q+4

dh1 , dh2 , dh3 , dh4 , dh5 e ′p = Now ∆

X

1≤i1 ,i2 ,···ip ≤p

e′ X e′ e′ ai1 · · · io X i1 i2 · · · Xip . It is now immediate from

Lemma 3.25 that we have Z ′′ 0 2 | ϕ(g ) | ≤ C E

(II)

X

| ηi1 ηi2 · · · ηip (h3 ) |2 | Z δ Y γ X β ϕ |2

1≤i1 ,i2 ,···ip ≤p |β|+|γ|+|δ|≤2p+q+r

dh1 , dh2 , dh3 , dh4 , dh5

(III)

where E = K∞ ×Ω2 ×h03 V ×Ω4 ×Ω5 . Now for h′3 , h3 V one has (ηi1 ηi2 · · · ηip )2 (h′3 ) ≤ λp · αp0 (h′3 ) for a suitable constant λp > 0. This is because we have Q for any i, with 1 ≤ i ≤ r, ηi = α∈∆ αmiα with miα ≥ 0, miα0 ≥ 1 integers; also α(h′3 ) ≤ c for α ∈ ∆ so that (ηi1 ηi2 · · · ηip · · · ηip )(h′3 ) ≤ cN α0 (h′3 )p P P where N = ( 1≤r≤p α∈∆ mir α − p). Next observe that if p is sufficiently Q Q large we have α(h3 )p < C1 δ2 (h3 ) where δ2 = 1≤i≤p ηi 1≤j≤q ζj . This is seen as follows: since α0 (h03 ) ≤ α(h03 ) for all α ∈ ∆, we see that there is a constant b > 0 such that α0 (h3 ) ≤ bα(h3 ) for all α ∈ ∆ and h3 ∈ Q h03 V . Now δ2 (h3 ) = α∈∆ α(h3 )να with να integers ≥ 1 so that δ2 (h3 ) ≥ P Q −1 να ≥ C −1 α (h )ν where ν = 0 3 1 α∈∆ να . Now if p ≥ ν, we α∈∆ (b α0 (h3 )) have α0 (h3 )ν ≤ α0 (h3 )p · cν−p ≤ cν−p C1 δ2 (h3 ). We see from (III) above now that we have Z X ′′ 2 | (Z δ Y γ X β )ϕ |2 dg (IV) | ϕ(g0 ) | ≤ C C1 S(c,Ω)·ξ |β|+|γ|+|δ|≤2p+q+r

Automorphic Forms in GL(n)

33

Note that E ⊂ S(c, Ω) and dh1 .dh2 .dh3 .dh4 .dh5 .δ2 (h3 ) = dg. Proposition 3.21 now follows for the inequality IV above: We have for α ∈ A∞ c (G∞ ) and D in the enveloping algebra of L(G∞ ), D(α ∗ ϕ) = Dα ∗ ϕ so that by (IV) R P ′′ (α ∗ ϕ)(g0 ) |2 ≤ C C1 S(c,Ω)ξ |β|+|γ|+|δ|≤2p+q+r | (Z δ Y γ X β α) ∗ ϕ |2 dg ≤ C(α) k ϕ k22 P ′′ where C(α) = C C1 M (α) with M (α) = |β|+|γ|+|δ|≤2p+q+r k Z δ Y γ X β α k01 . Note that we have for α ∈ Cc∞ (G∞ ) and ϕ ∈ L2 (G∞ /Γ), k α ∗ ϕ k2 ≤k α k2 . This proves the proposition and hence the Theorem.

4

Proof of Theorem 2.13 We begin by proving the following:

4.1. Theorem Let o Aχ (G∞ , Γ, σ∞ , λ) = {f ∈ Aχ (G∞ , Γ, σ∞ , λ) | f cuspidal }. Then o Aχ (G∞ , Γ, σ∞ , λ) is finite dimensional. 4.2. We will first show that any ϕ ∈o Aχ (G∞ , Γ, σ∞ , λ) decreases to zero rapidly at ∞ so that in particular ϕ ∈ L2 (G∞ /Γ) (and hence in o Lχ (G /Γ)). By Lemma 3.25 combined with (I) of 3.26, we have for p′ ≥ p ∞ 2 and g0 = h01 .h02 .h03 .h04 .h05 .ξ, 2

o

| ϕ(g ) | ≤

Z

X

| (ηi1 ηi1 ηi2 · · · ηip (h03 ) |2

Ω5 1≤ .i ···i ≤p′ i 1 p′

X

| cβγδ Z δ Y γ X β ϕ(g0 ξ −1 )h5 ξ) |2 dh5

|β|+|α|+|δ|≤2p′

Let I be the set of p′ -tuples i1 , . . . , ip′ with 1 ≤ i1 ≤ i2 · · · ≤ ip′ ≤ p′ and for I ∈ I, let ηI ηi1 ηi2 · · · ηip . Since ϕ and its derivatives have moderate growth there is constant B > 0 and an integer r > 0 such that | (Z δ Y γ X β ϕ(g0 ) |2 ≤ B k g0 kr for | α | + | β | + | γ |≤ 2p′ . Thus, we have any p′ ≥ p | ϕ(g0 ) |2 ≤ C ≤ C

′′

′′′

R

Ω5

X (ηI (h03 ))2

I∈I ′ α0 (h03 )p

k

g0

kr

X

|δ|+|γ|+|β|≤2p′

| (Z δ Y γ X β ϕ)(g0 ξ −1 h5 ξ) |2 dh5

34

M.S. Raghunathan

Now k g0 kr ≤ b′ Supα∈∆ α(h03 )−s for some integer s for g0 ∈ S(c0 , Ω0 ). It follows that we have ′′

′

′′

′

| ϕ(g0 ) |2 ≤ b α0 (h03 )p · α0 (h03 )−s = b α(h03 )p −s . Since p′ is at our choice we see that | ϕ(g0 ) |2 ≤ is bounded in S(c0 , Ω0 ). Hence ϕ ∈o L2 (G∞ /Γ). 4.3. Lemma (Godement) Let X be a locally compact space and µ a probability measure. Let V ⊂ L2 (X, µ) be a closed subspace. Suppose that every ϕ ∈ V is essentially bounded. Then dim H < ∞. Proof V is a closed subspace of L∞ as well since L∞ -convergence implies L2 convergence. It follows also that the identity map of V is a continuous homomorphism of V with L∞ topology on V with the L2 topology. By the open mapping theorem there is a constant c > 0 such that k ϕ k∞ < c k ϕ k2 for all ϕ ∈ V . Suppose ϕ1 , · · · , ϕn is an orthonormal set in V , and a = (a1 , · · · , an ) ∈ Cn ; then we have for almost all x ∈ X, |

n n X X X 1 k2 = C( | ai ϕi |2 ) 2 ai ϕi (x) |≤ C k i=1

i=1

It follows that if D is a dense countable subset of Cn , there is a set Y of measure zero in X such that we have for all x ∈ X\Y and all a ∈ D, n n X X 1 | ai |2 ) 2 ai ϕi (x) |≤ c( | i=1

i=1

Since D is dense in Cn , the inequality holds for all a ∈ Cn . Taking ai = fi (x), we conclude that n X | fi (x) |2 ≤ c2 i=1

Integrating over X, we have n ≤ c2 . Thus dim V < ∞. 4.4. Proof of 4.1. We have seen that we may assume that G has no central split torus so that G∞ /Γ has finite measure and further that χi trivial. We claim that o A(G∞ , Γ, σ∞ , λ) is a closed subspace of o L2 (G∞ /Γ). In fact if ϕn ∈ o A(G∞ , Γ, σ∞ , λ) converges to ϕ in o L2 , then ϕn → ϕ in

Automorphic Forms in GL(n)

35

the sense of distributions. All the ϕn are in the kernel of a fixed elliptic operator with C ∞ coefficients on G∞ /Γ hence the distribution ϕ is also in this kernel. But any distribution in the kernel of an elliptic operator with C ∞ coefficients is a C ∞ function. Thus ϕ is a C ∞ function in o L2 . Since it is the limit of the ϕn as a distribution, the K∞ -span of ϕ is a quotient of σ∞ and the Z-span a quotient of λ. Further as ϕ is k-finite and Z finite there is an α ∈ Zc∞ (G) such that α ∗ ϕ = ϕ and one has thus k ϕ k∞ ≤k α ∗ ϕ k∞ ≤ C k α k2G∞ k ϕ kG∞ /Γ . It follows that ϕ has moderate growth so that ϕ ∈ o A((G∞ , Γ, σ∞ , λ). Lemma 4.3 now clearly implies Theorem 4.1. 4.5. Clearly Theorem 4.1 is equivalent to the assertion that o (A)χ (G(A), G(k), σ, λ) is finite dimensional. To prove Theorem 2, we will argue by induction on the k-rank of [G, G]. Let {Pi }icI be the collection of all the maximal parabolic subgroups of G containing P. Let Ui be the cusp radical of Pi and Gi ⊃ S. Let supplement to Ui , Pi . Let ri : C ∞ (G(A)/G(k)) → C ∞ (Gi (A)/Gi (k)) be the map defined as follows: Z ϕ(xu)du. γi (ϕi )(x) = Ui (A)/Ui (k)

We assert that for a suitable χi and λi and σi∞ , λi (ϕ) ∈ Aχ (Gi∞ (A), Gi∞ (k), λi , σi∞ ) if ϕ ∈ Aχ (G(A), G(k), λ, σ∞ ). Assume that this is true. Observe that k-rank ([Gi , Gi ]) = k rank G − 1. Thus by induction hypothesis Aχi (Gi (A), Gi (k), σ, λ∞ ) is finite dimensional for all i ∈ I. On the other hand if ϕ is in ∩i∈I ker λi , ϕ is a cuspidal automorphic form in o Aχ (G(A), G(k), σ, λ ); and this last space is finite dimensional by Theo∞ rem 4.1. This completes the proof of Theorem 2 assuming the following: 4.6. Lemma Given χ, σ, λ there exists for every i ∈ I, χi , σi , λi , χi = χ (note that C ⊂ Gi (σi )), representation of K∞ ∩ Gi and λi : Zi → EndWi′ , (Zi the centre of the enveloping algebra Ui of L(Gi )) such that ri (Aχ (G(Af ), G(k), σ, λ)) ⊂ Aχi (Gi (Af ), Gi (k), σi , λi ) Proof Let U− be the unipotent algebraic k-subgroup of G normalised by XS and whose Lie algebra is spanned by {L(G)(β) | β a root, β = − mα α, mαi < 0}. Let P− i be the normaliser of Ui . One then has α∈∆

36

M.S. Raghunathan

− Pi ∩ P− i = Gi and L(G) = L(Ui ) ⊗ L(Gi ) ⊗ L(Ui ). Correspondingly the enveloping algebra U of L(G) can be written as

U(L(U− i )) · Ui · U(L(Ui )) − − where Ui is the enveloping algebra of L(Gi ). Now U(L(U− i )) = L(Ui )(U(L(Ui ))⊕ C and similarly U(L(Ui )) = U(L(Ui ))L(Ui ) ⊕ C. Consequently we have

U

− = L(U− i )U(L(Ui )) · Ui · U(L(Ui ))L(Ui ) − ⊕ L(Ui )U(L(U− i )) · Ui ⊕ Ui · U(L(Ui ))L(Ui ) ⊕ Ui

Let hi denote the projection of U on Ui following this direct sum decomposition. Then according to a theorem of Harish-Chandra hi restrict to U maps Z injectively into Zi and Zi is finitely generated as a Z-module. − Also the component of z ∈ CU in the direct factors L(U− i )(U(L(Ui )) · Ui and Ui U(L(Ui ))L(Ui ) are trivial. From this it is easy to see that ri (zϕ) = hi (z)ri (ϕ) for ϕ ∈ (Aχ (G(A), G(k), σ, λ). It follows that if we set Wi′ = W ⊗Z Zi , Wi′ is a finite dimensional vector space and we have a homomorphism λi : Zi → End Wi′ . Evidently Ui (ri (ϕ)) is a quotient of Wi′ as a Zi -module. Also, if Ki = K ∩ Gi (Af ), it is clear that the Ki (K-span of ϕ) and thus is a quotient of the Ki -module W (obtained by restriction of the K-module structure to K0 ). Lemma 4.5 is immediate from this.

Automorphic Forms in GL(n)

37

References [B] A. Borel, Introduction aux groupes arithm´etiques, Hermann (1969), Paris. [B-T] A. Borel and J. Tits, Groupes r´eductifs, Publ. Math. de ℓ’IHES, 27 (1965), 55–150. [G] R. Godement, Domaines fondamentaux de groupes arithm´etiques eminaire Bourbaki, (1962/63), Fas 3, No.257, Paris. [H-C] Harish-Chandra, Automorphic forms on semisimple Lie groups, LN 62, Springer-Verlag (1968). [L] R.P. Langlands, On the functional equations satisfied by Eisenstein series, LN 544, Springer-Verlag (1976). [N] R. Narashimhan, Analysis on real and complex manifolds, Advanced Studies in Pure Mathematics, Vol. 1, North Holland, Amsterdam, (1968).

Classical Modular Forms T.N. Venkataramana∗

School of Mathematics, Tata Institute of Fundamental Research, Colaba, Mumbai, India

Lectures given at the School on Automorphic Forms on GL(n) Trieste, 31 July - 18 August 2000

LNS0821002

∗

[email protected]

Contents 1 Introduction

43

2 A Fundamental Domain for SL(2, Z)

43

3 Modular Forms; Definition and Examples

49

4 Modular Forms and Representation Theory

54

5 Modular Forms and Hecke Operators

62

6 L-functions of Modular Forms

70

Classical Modular Forms

1

43

Introduction

The simplest kind of automorphic forms (apart from Gr¨ossencharacters, which will also be discussed in this conference) are the “elliptic modular forms”. We will study modular forms and their connection with automorphic forms on GL(2), in the sense of representation theory. Modular forms arise in many contexts in number theory, e.g. in questions involving representations of integers by quadratic forms, and in expressing elliptic curves over Q as quotients Jacobians of modular curves (this was the crucial step in the proof of Fermat’s Last Theorem by Andrew Wiles), etc. The simplest modular forms are those on the modular group SL(2, Z) and we will first define modular forms on SL(2, Z). To begin with, in section 2, we will describe a fundamental domain for the action of SL(2, Z) on the upper half plane h. The fundamental domain will be seen to parametrise isomorphism classes of elliptic curves. In section 3, we will define modular forms for SL(2, Z) and construct some modular forms, by using the functions E4 and E6 which we encounter already in the section on elliptic curves. In section 4, a representation theoretic interpretation of modular forms will be given, which will enable us to think of them as automorphic forms on GL(2, R). In section 5, we will give an adelic interpretation of modular forms. This will enable us to think of Hecke operators as convolution operators in the Hecke algebra; using this, we show the commutativity of the Hecke operators. We will also prove a special case of the Multiplicity One theorem.

2

A Fundamental Domain for SL(2, Z)

Notation 2.1 Denote by h the “Poincar´e upper half-plane” i.e. the space of complex numbers whose imaginary part is positive: h = {z ∈ C; z = x + iy, x, y ∈ R, y > 0}. If z ∈ C, denote respectively by Re(z) and Im(z) the real and imaginary parts of z. On the upper half plane h, the group GL(2, R)+ of real2 × 2 matrices a b with positive determinant operates as follows: let g = ∈ GL(2, R)+ , c d and let z ∈ h. Set g(z) = (az + b)/(cz + d). Notice that if cz + d = 0

44

T.N. Venkataramana

and c 6= 0 then, z = −d/c is real, which is impossible since z has positive imaginary part. Thus, the formula for g(z) makes sense. Observe that Im(g(z)) = Im(z)(det(g))/ | cz + d |2 .

(1)

The equation (1) shows that the map (g, z) 7→ g(z) takes GL(2, R)+ × h into h. One checks immediately that this map gives an action of GL(2, R)+ on the upper half plane h. Note also that | cz + d |2 = c2 y 2 + (cx + d)2 .

(2)

Therefore, | cz + d |2 ≥ y 2 or 1 according as | c |= 0 or nonzero. Therefore, Im(γ(z)) ≤ y/ min{1, y 2 } ∀γ ∈ Γ0 ⊂ SL(2, Z),

(3)

where min{1, y 2 } denotes the minimum of 1 and y 2 and Γ0 ⊂ SL(2, Z) is the group generated by the elements T = ( 10 11 ) and S = ( −10 01 ). Later we will see that Γ0 is actually SL(2, Z). The element T acts on the upper half plane h by translation by 1: 1 1 T (z) = (z) = z + 1 ∀z ∈ h. (4) 0 1 Similarly, the element S acts by inversion: 0 1 S(z) = (z) = −1/z ∀z ∈ h. −1 0

(5)

Consider the set F = {z ∈ h; −1/2 < Re(z) ≤ 1/2,

| z |≥ 1, and 0 ≤ Re(x)} if | z |= 1.

Theorem 2.2 Given z ∈ h there is a unique point z0 ∈ F and an element γ ∈ SL(2, Z) such that γ(z) = z0 . Moreover, given γ ∈ SL(2, Z), we have γ(F ) ∩ F = φ unless γ lies in a finite set ( of elements of SL(2, Z) which fix the point ω = 1/2 + i31/2 /2 ∈ h or i ∈ h). [one then says that F is a fundamental domain for the action of SL(2, Z) on the upper half plane h]. Proof We will first show that any point z on the upper half plane can be translated by an element of the subgroup Γ0 of SL(2, Z) (generated by 0 1 )) into a point in the “fundamental domain” F . T ( 10 11 ) and S = ( −1 0

Classical Modular Forms

45

Now, given a real number x, there exists an integer k such that −1/2 < x + k ≤ 1/2. Therefore, equation (4) shows that given z ∈ h there exists an integer k such that the real part x′ of T k (z) satisfies the inequalities −1/2 < x′ ≤ 1/2. Let y denote the imaginary part of z and denote by Sz the set Sz = {γ(z); γ ∈ Γ0 , Im(γ(z)) ≥ y , −1/2 < Re(γ(z)) ≤ 1/2} . We will first show that Sz is nonempty and finite. Let k be as in the previous paragraph. Then −1/2 < Re(T k (z)) ≤ 1/2 and Im(T k (z) = Im(z); therefore, T k (z) lies in Sz and Sz is nonempty. Now, equation (3) shows that the imaginary parts of elements of the set Sz are all bounded from above by y/ min 1, y 2 . By definition, the imaginary parts of points on Sz are bounded from below by y. The definition of Sz shows that Sz is a relatively compact subset of h. We get from (3) that | cz + d |2 ≤ 1; now (2) shows that | c |≤ 1/y 2 . Suppose γ ∈ γ0 = ( ac db ) is such that γ(z) ∈ Sz then, c is bounded by 1/y 2 and is in a finite set. The fact that cz + d is bounded now shows that d also lies in a finite set. Since Sz is relatively compact in h, it follows that γ(z) = (az + b)/(cz + d) is bounded for all γ(z) ∈ Sz ; therefore, az + b is bounded as well, and hence a and b run through a finite set. We have therefore proved that Sz is finite. Let y0 be the supremum of the imaginary parts of the elements of the finite set Sz ; let S1 = {z ′ ∈ Sz ; Im(z ′ ) = y0 } and let z0 ∈ S1 be an element whose real part is maximal among elements of S1 . We claim that z0 ∈ F . First observe that if z ′ ∈ Sz then S(z ′ ) = −1/z ′ has imaginary part y0 / | z |2 = Im(z ′ )/ | z ′ |2 ≤ y0 whence | z ′ |2 ≥ 1. If | z0 |> 1, then it is immediate from the definitions of F and Sz that z0 ∈ F . Suppose that | z0 |= 1. Then, S(z0 ) = −1/z0 also has absolute value 1, its imaginary part is y0 and its real part is the negative of Re(z0 ); hence S(z0 ) ∈ S1 . The maximality of the real part of z0 among elements of S1 now implies that Re(z0 ) ≥ 0. Therefore, z0 ∈ F . We have proved that every element z0 may be translated by an element of Γ0 into a point in the fundamental domain F . Suppose now that z ∈ γ −1 (F )∩F for some γ ∈ SL(2, Z). Write γ = ( ac db ) with a, b, c, d ∈ Z and ad − bc = 1. Suppose that Im(γ(z)) ≥ Im(z) = y (otherwise, replace z by γ(z)). Then, by (3) one gets (cx + d)2 + c2 y 2 ≤ 1.

(6)

Since z ∈ F , we have x2 + y 2 ≥ 1 and 0 ≤ x ≤ 1/2. Therefore y 2 ≥ 3/4 and (1 ≥)c2 y 2 ≥ c2 4/3.

(7)

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T.N. Venkataramana

This shows that c2 ≤ 1 since c is an integer. Suppose c = 0. Then, ad = 1, a, d ∈ Z and we may assume (by multiplying by the matrix −Id [minus identity] if necessary) that d = 1. Hence γ = ( 10 1b ). Then, γ(z) = z + b ∈ D which means that 0 ≤ x + b ≤ 1/2 and 0 ≤ x ≤ 1/2. Thus, −1/2 ≤ b ≤ 1/2, i.e. b = 0 and γ is the identity matrix. The other possibility is c2 = 1, and by multiplying by the matrix −Id (minus identity) we may assume that c = 1. Suppose first that d = 0. Then, bc = −1 whence b = −1. Now, (7) shows that x2 + y 2 ≤ 1. Moreover, γ(z) = az + b/z = a + bz/ | z |2 = a − z whence its real part is a − x which lies between 0 and 1/2. Since 0 ≤ x ≤ 1, it follows that 0 ≤ a ≤ 1. If a = 0 then γ = ( 01 −10 ) and lies in the isotropy of the point i ∈ h. If a = 1 then, γ = ( 11 −10 ) which lies in the isotropy of the point ω = 1/2 + i31/2 /2. We now examine the remaining case of c = 1 and d 6= 0. From (6) we get (x + d)2 + y 2 ≤ 1. If d ≥ 1 then the inequality 0 ≤ x ≤ 1/2 shows that 1 ≤ d ≤ x + d which contradicts the inequality (x + d)2 + y 2 ≤ 1, which is impossible. Thus, d ≤ −1; then the inequality 0 ≤ x ≤ 1/2 implies that x + d ≤ 1/2 + (−1) = −1/2 whence (x + d)2 ≥ 1/4. Since y 2 ≥ 3/4 the inequality (x+d)2 +y 2 ≤ 1 implies that equalities hold everywhere: y 2 = 3/4, x = 1/2 and d = −1. Thus, z = ω and z −1 = z 2 . Since 1 = ad−bc = −a−b (d = −1 and c = 1), and γ(z) = (az+b)/(z−1) = (az+b)/z 2 = −(az+b)z = a+(−a−b)z = a+z ∈ D, the real part of γ(z) is a + x = a + 1/2 and is between 0 and 1/2, i.e. −1 −1/2 ≤ a ≤ 0 i.e. a = 0 and b = −1. Therefore, γ = ( 01 −1 ) lies in the isotropy of ω. This completes the proof of Theorem (2.2). Corollary 2.3 The group SL(2, Z) is generated by the matrices T = ( 10 11 ) 0 1 ). and S = ( −1 0 Proof In the proof of Theorem (2.2), a point on the upper half plane is brought into the fundamental domain F by applying only the transformations generated by S and T . The fact that the points on the fundamental domain are inequivalent under the action of SL(2, Z) now implies that SL(2, Z) is generated by S and T .

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(The Corollary can also be proved directly by observing that ST −1 S −1 = ( 11 01 ). Now, the usual row-column reduction of matrices with integral entries implies that T and ST S −1 generate SL(2, Z)). Notation 2.4 Elliptic Functions. We recall briefly some facts on elliptic functions (for a reference to this subsection, see Ahlfors’ book on Complex Analysis). Given a point τ on the upper half plane h, the space Γτ = Z ⊕ Zτ of integral linear combinations of 1 and τ forms a discrete subgroup of C with compact quotient. The quotient Eτ = C/Γτ may be realised as the curve in P2 (C) whose intersection with the complement of the plane at infinity is given by y 2 = 4x3 − g2 x − g3

(8)

The curve Eτ = C/Γτ is called an “elliptic curve”. The map of C/Γτ to P2 is given by z 7→ (℘′ (z), ℘(z), 1) for z ∈ C. Recall the definition of ℘: if z ∈ C and does not lie in the lattice Γτ , then write ′

X ℘(z) = 1/z + (1/(z + w)2 − 1/w2 ), 2

P′ is the sum over all the non-zero points w in the lattice Γτ . The where derivative ℘′ (z) of ℘(z) is then given by X ℘′ (z) = 1/(z + w)3 , where the sum is over all the points of the lattice Γτ . One has the equation (cf. equation (8)) ℘′ (z)2 = 4℘(z)3 − g2 (τ )℘(z) − g3 (τ ).

(9)

a b ∈ SL(2, Z) and τ ∈ h, then the elliptic curve Eγ(τ ) is c d isomorphic as an algebraic group (which is also a projective variety) to the elliptic curve Eτ . The explicit isomorphism on C is given by z 7→ z/(cτ + d). It is also possible to show that if Eτ and Eτ ′ are isomorphic elliptic curves, then τ ′ is a translate of τ by an element of SL(2, Z). Thus the fundamental domain F which was constructed in Theorem (2.2) parametrises isomorphism classes of elliptic curves. If γ =

48

T.N. Venkataramana In equation (9), recall that the coefficients g2 and g3 are given by ′

g2 (τ ) = 60G4 (τ ) = 60 and

X

(mτ + n)−4

′

X g3 (τ ) = 140G6 (τ ) = 140 (mτ + n)−6

P′ is the sum over all the pairs of integers (m, n) such that not both where m and n are zero. The discriminant of the cubic equation in (9) is given by 1/(16)∆(τ ) where ∆(τ ) = g23 − 27g32 . (10) It is well known and easily proved that ℘′ (z) has a simple zero at all the 2-division points 1/2,τ /2 and (1+τ )/2 and that ℘(1/2),℘(τ /2) and ℘((1+ τ )/2) are all distinct. Thus equation (9) transforms to ℘′ (z)2 = 4(℘(z) − ℘(1/2))(℘(z) − ℘(τ /2))(℘(z) − ℘((1 + τ )/2))

(11)

Thus the discriminant of the (nonsingular) cubic in equation (9) is non-zero and so we obtain that ∆(τ ) 6= 0 (12) for all τ ∈ h. Notation 2.5 On the upper half plane h, there is a measure denoted y −2 dxdy, as z = x + iy varies in h. This measure is easily seen to be invariant under the action of elements of the group GL(2, R)+ of nonsingular matrices with positive determinant. Lemma 2.6 With respect to this measure, the fundamental domain F has finite volume. Proof We compute the volume of F . Note that if z = x + iy lies in F , then, −1/2 ≤ x ≤ 1/2 and 1/(1 − x2 )1/2 ≤ y < ∞. Thus the volume of F is the integral Z ∞ Z 1/2 Z 2 dy/y 2 ) dx( dxdy/y = F

−1/2

(1−x2 )1/2

which is easily seen to be π/3. In particular, F has finite volume.

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Notation 2.7 Let S denote the inverse image of the fundamental domain F ⊂ h under the quotient map GL(2, R) → GL(2, R)/O(2)Z = h. Then, we have proved that GL(2, Z)S = GL(2, R). The set S is called a Siegel Fundamental Domain.

3

Modular Forms; Definition and Examples

Notation 3.1 Given z ∈ h (h is the upper half plane) and an element g = ( ac db ), write j(g, z) = cz + d. Note that if j(g, z) = 0, then by comparing the real parts and imaginary parts we get c = 0 and d = 0 which is impossible since ad − bc 6= 0. Thus, j(g, z) is never zero. Definition 3.2 A function f : h → C is weakly modular of weight w if the following two conditions hold. (1) f is holomorphic on the upper half plane. (2) for all γ ∈ SL(2, Z), with γ = ( ac db ), we have the equation f ((az + b)/(cz + d)) = (cz + d)w f (z).

(13)

Given g = ( ac db ) and a function f on the upper half plane h, define g−1 ∗ f (z) = (cz + d)−w f (g(z)) ∀z ∈ h. Then, it is easily checked that the map (g, f ) → g−1 ∗ f defines an action of GL(2, R) on the space of functions on h. Thus, the condition (2) above is that the function f there is invariant under this action by SL(2, Z). Now 0 1 ) and by Corollary (2.3), SL(2, Z) is generated by the matrices S = ( −1 0 −1 T = ( 10 11 ). Thus condition (2) is equivalent to saying that γ ∗ f = f for γ = S, T . This amounts to saying that f (−1/z) = z w f (z)

(14)

f (z + 1) = f (z).

(15)

and Note that the invariance of f under the action of −1 where 1 is the identity matrix in SL(2, Z) implies that f is zero of w is odd: f (z) = (−1)w f (z). Therefore, we assume from now on (while considering modular forms for the group SL(2, Z) ) that w = 2k where k is an integer.

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Definition 3.3 The map exp : h → D ∗ given by z 7→ e2πiz = q is easily seen to be a covering map of the upper half plane h onto the set D ∗ of non-zero complex numbers of modulus less than one. The covering transformations are generated by T (z) = z + 1. A weakly modular function f is invariant under T and therefore yields a holomorphic map f ∗ : D ∗ → C given by f ∗ (q) = f (z) for all z ∈ h. We say that a weakly modular function of weight w is a modular function of weight w if f ∗ extends to a holomorphic function of D (the set of complex numbers of modulus less than one) i.e. f ∗ extends to 0 ∈ D. Let f be a weakly modular function on h. Then, f is a modular function if and only if the function f ∗ has the “Fourier expansion” (or the “qexpansion”) X f ∗ (q) = an q n , (16) n≥0

where an are complex numbers and the summation is over all non-negative integers n. Observe that a weakly modular function is modular if and only if it is bounded in the fundamental domain F . We will say that a modular form is a cusp form if the constant term of its q-expansion is zero: i.e. a0 = 0 in the notation of equation (15). Notation 3.4 Examples of modular forms. First we note that if f and g are modular forms of weights w and w′ then, the product function f g is a modular form of weight ww′ . We will first prove that for the modular group SL(2, Z), there are no non-constant “weight zero” modular forms. First note that if f is a weight zero modular form, then the function f ∗ extends to 0 and hence is bounded in a disc of radius r < 1. Its inverse image under exp : F → D ∗ is precisely the set A = {z = x + iy ∈ F ; y > − log r} and f is bounded on the set A. The complement of the set A in the fundamental domain F is compact, and f is bounded there as well, whence f is bounded on all of the fundamental domain F as well as at “infinity”. By the maximum principle, f is constant. We will now show that there are no modular forms of weight two on SL(2, Z). Suppose f is one and let F (z) be its integral from z0 to z for some fixed z0 ∈ h. The modularity of f shows that γ 7→ F (γ(z0 )) gives a homomorphism from SL(2, Z) to C. But, SL(2, Z) is generated by the finite order elements S and ST whence, this homomorphism is identically zero. This and the modularity of f shows that the integral F is invariant under SL(2, Z). It is easy to show that F ∗ is holomorphic at 0 (integrate both

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51

sides of equation (15)), and use the invariance of F under T ). Hence F is a modular form of weight zero. By the foregoing paragraph, f is a constant, i.e. f = 0. Fix an even positive integer 2k, with k ≥ 2. We will construct a modular form of degree k as follows. Let τ ∈ h and write (compare the definition of g2 and g4 in section (2.4)) X ′ G2k (τ ) = (mτ + n)−2k , (17) P′ is the sum over all the pairs of integers (m, n) not both of which are where zero. Then, G2k is easily shown to be a weakly modular function of weight 2k on the upper-half plane. If τ is varying in the fundamental domain and its imaginary part tends to infinity, then it is clear from the formula for G2k P′ −2k n = 2ζ(2k) where that G2k (τ ) tends to X ζ(s) = n−s

is the Riemann zeta function (the sum is over all the positive integers n and in the sum, the real part of s exceeds 1). Consequently, G2k is a modular form of weight 2k. We will now outline a derivation of the q-expansion of G2k . Start with the partial fraction expansion X πcot(πz) = z −1 + (z + n)−1 + (z − n)−1 (18)

where the sum is over all positive integers n. This series converges uniformly on compact subsets of the complement of Z in C. Write q = e2πiz (where i ∈ h and i2 = −1). Then one has the q-expansion X πcot(πz) = πi(q + 1)/(q − 1) = −πi − 2πi qn (19) n≥1

Differentiate 2k- times, the right-hand sides of equations (17) and (18) with respect to z. We then get the equality X X n2k−1 q n (20) (z + n)−2k = ((2k − 1)!)−1 (2πi)2k n∈Z

n≥1

Fix m and in equation (19) take for z the complex number mτ . Then sum over all m. We obtain by equations (16) and (18), the q-expansion X σ2k−1 q n (21) G2k (τ ) = 2ζ(2k) + ((2k − 1)!)−1 (2πi)2k n≥1

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where for an integer r and n ≥ 1, σr (n) is defined to be the sum d runs over the positive divisors of n. By using the power series expansion X (1 + x)−2 = nxn−1

P

dr where

n≥1

and equation (17) one has the power series identity XX X πcot(πz) = z −1 + 2 n2j z 2j−1 = z −1 + 2 ζ(2j)z 2j−1 n≥1 j≥1

(22)

j≥1

P By comparing the power series expansions cos(x) = m≥0 ((2m)!)−1 x2m P and sin(x) = m≥0 ((2m + 1)!)−1 x2m+1 with the right-hand side of equation (21) one obtains ζ(2) = π 2 /6, ζ(4) = π 4 /90 andζ(6) = π 6 /(33 .5.7).

(23)

Using (20) and (22) we get g2 = 60G4 = (4/3)π 4 + 160π 4 (q + · · · )

(24)

where the expression q + · · · is a power series in q with integral coefficients with the coefficient of q being 1. Similarly, we get (again from (20) and (22)) g3 = 140G6 = (8/27)π 6 − 25 .7π 6 /3(q + · · · ) Therefore, we get, after some calculation, that for all z ∈ h, X ∆(z) = g2 (z)3 − 27g3 (z)2 = 211 π 12 (q + τ (n)q n )

(25)

(26)

n≥2

where the τ (n) are integers. We recall that ∆(z) is never zero on the upperhalf plane (section (2.4)). The equation (26) shows that the coefficient of q in q-expansion of ∆ is non-zero, (and that its constant term is zero). Lemma 3.5 There are no modular forms of negative weight. Proof Suppose that f is a modular form of weight −l with l > 0. Form the product g = f 12 ∆l . Since f and ∆ are modular forms, so is the product. Since its weight is zero, g is a constant (see the beginning of this subsection). But, (26) shows that the q-expansion of g has no constant term. Hence g = 0 whence, f = 0.

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Lemma 3.6 Suppose that f is a cusp form. Then ∆ divides f i.e., there is a modular form g such that f = ∆g. In particular, the weight of f is at least 12. Proof Consider the quotient g = f /∆. Since ∆ has no zero in h, it follows that g is holomorphic in h. Clearly, g is weakly modular of weight =weight of f -12. Now the q-expansion of f (and also ∆), has no constant term; and the coefficient of q in the q-expansion of ∆ is non-zero. Therefore, g∗ extends to a holomorphic function in a neighbourhood of 0. That is, g is a modular function. Since the weight of g is non-negative (by Lemma (3.5)), it follows that the weight of f must be at least that of ∆, namely, 12. Corollary 3.7 The space of cusp forms of weight 12 (for SL(2, Z)), is one dimensional. Proof If f is a cusp form of weight 12, then f /∆ is a modular form of weight zero, hence is a constant. That is, the space of cusp forms of weight 12 is spanned by ∆. Theorem 3.8 The space of modular forms of weight 2k with k ≥ 0 is n spanned by the modular forms Gm 4 G6 with 4m + 6n = 2k. Proof Argue by induction on k. We have already excluded the possibilities k < 0 and k = 0 and k = 1. Suppose that k ≥ 2 and that f is modular of weight 2k. First observe that any integer k ≥ 2 may be written as 2m + 3n for non-negative integers m and n. Now, the q-expansion of G4 and G6 have non-zero constant term. n Hence h = f − λGm 4 G6 ) for a suitable constant λ, has no constant term in its q-expansion, and is a cusp form. Now, Lemma (3.6) shows that g = h/∆ is a modular form of weight 2k−12. By induction, g is a linear combination of the modular forms Ga4 Gb6 with k − 6 = 2a + 3b whence, h is a sum of monomials of the form Gp4 Gq6 with 2p + 3q = k (recall that ∆ is (60G4 )3 − 27(140G6 )2 ). Therefore, so is f . Notation 3.9 Define E2k (z) = G2k /2ζ(2k). Then, it follows from the Fourier expansion of G2 and G6 that the modular forms E4 and E6 have inP tegral Fourier coefficients. One sometimes writes ∆(z) = q + n≥2 τ (n)q n . Then, ∆ has integral Fourier coefficients as well. We now consider the Zmodule spanned by E4m E6n with 4m + 6n = 2k. We get an integral lattice

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T.N. Venkataramana

of modular forms of weight 2k. We will see later that this integral lattice is stable under the Hecke operators.

4

Modular Forms and Representation Theory

Notation 4.1 We will begin with some calculations on the Lie the group GL(2, R). Write, 0 1 1 0 1 0 0 X= , Y = , Z= , and H = 0 0 0 0 0 1 −1

algebra g of 1 . 0

(27)

The complexified Lie algebra of GL(2, R) is M2 (C) the space of 2 × 2 matrices with complex entries; the Lie algebra structure is given by (a, b) 7→ [a, b] = ab − ba; M2 (C) is spanned by X, Y, Z and A. Write A = −iH (where i ∈ h is the unique element whose square is -1). Then, A acts semisimply (under the adjoint action) on g with real eigenvalues. Write g = CE + ⊕ CE − ⊕ CZ ⊕ CA

(28)

where E − = X + iY − (i/2)A − (i/2)Z and E + = X − iY − (i/2)A + (i/2)Z. (29) Then E − and E + are eigenvectors for A with eigenvalues −2 and 2 respectively. Of course, on A and Z, A acts by 0. Thus, the complex Lie algebra spanned by E + , E − and A is isomorphic to sl2 (C). Definition 4.2 Fix the subgroup K∞ = O(2) of GL(2, R). This is the group generated by cosθ sinθ SO(2) = Rθ = :θ∈R (30) −sinθ cosθ and ι=

−1 0 . 0 1

(31)

Then, O(2) is a maximal compact subgroup of GL(2, R). Suppose that (π, V ) is a module for g as well as for O(2) such that the module structures are compatible. That is, suppose that v ∈ V and ξ ∈ g, and σ ∈ O(2). Then, π(σ)π(ξ)(v) = π(σ(ξ))(v)

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55

where σ(ξ) is the inner conjugation action of O(2) on the Lie algebra g. One then says that (π, V ) is a (g, K∞ )-module. If, as a K∞ -module, (π, V ) is a direct sum of irreducible representations of K∞ with each irreducible representation occurring only finitely many times, then one says that the (g, K∞ )module is admissible. One then sees at once that a (g, K∞ )-submodule (or a quotient module) of an admissible module is also admissible. One says that a vector v ∈ V generates (π, V ) as a (g, O(2))-module, if the smallest submodule of V containing v is all of V . Notation 4.3 The Tensor Algebra. Given a (complex) vector space V , denote by T n (V ) = V ⊗n the n-th tensor power of V . This is of course, spanned by the pure tensors, i.e. vectors of the form v1 ⊗ · · · ⊗ vn with vi ∈ V . By definition, T 0 (V ) = C and T 1 (V ) = V . Denote by T (V ) = ⊕T n (V ) where the direct sum is over all the non-negative integers n. Given non-negative integers m and n, there exists a linear map T m (V ) ⊗ T n (V ) → T m+n (V ) which on pure tensors is the map (v1 ⊗ · · · ⊗ vm ) ⊗ (w1 ⊗ · · · ⊗ wn ) 7→ (v1 ⊗ · · · ⊗ vm ⊗ w1 ⊗ · · · wn ). This extends by linearity to all of T m (V ) ⊗ T n (V ) and thence to all of the direct sum T (V ). Under this “multiplication”, T (V ) becomes an associative algebra, and is called the tensor algebra of the vector space V . The subspace T 0 (V ) = C acts simply by scalar multiplication. Notation 4.4 The Universal Enveloping Algebra. Given now a Lie algebra g, let u(g) denote the quotient of the tensor algebra T (g) of g, by the two sided ideal generated by the elements x⊗y −y ⊗x−[x, y], as x and y vary over the elements of the Lie algebra g. Here, ⊗ denotes the multiplication in the tensor algebra T (g) and the bracket [x, y] denotes the Lie bracket in g. The algebra u(g) is called the universal enveloping algebra of g. Note that g is a subspace of u(g), with [x, y] = xy − yx. Here, x, y are the images of x, y ∈ g = T 1 (g) under the quotient map T (g) → u(g). Suppose that u is some algebra over C and f : g → u a linear map such that f ([x, y]) = f (x)f (y) − f (y)f (x) for all elements x, y ∈ g. Here f (x)f (y) refers to the product of the two elements in the algebra u. Then, there exists

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a unique algebra map F : u(g) → u which extends f . This is why u(g) is called the universal enveloping algebra of g. In particular, if V is a module over g, we have a map f : g → End(V ) of Lie algebras, and we have f ([x, y]) = [f (x), f (y)] where the bracket structure on End(V ) is simply the commutator in the algebra End(V ). Therefore, by the last paragraph, we get a unique extension F : u(g) → End(V ) , with F an algebra map. In other words, the g module V is naturally a module over u(g) as well. Theorem 4.5 The Poincar´ e-Birkhoff-Witt Theorem: Let a, b and c be subalgebras of a Lie algebra g such that g = a ⊕ b ⊕ c. Then, one has the decomposition u(g) = u(a) ⊗ u(b) ⊗ u(c) Proof We must prove that every element of u(g) lies in the subspace of the right-hand side of the above equation. Argue by an induction on the degree of an element ξ ∈ T n (g). If we have an element yx for example, with y ∈ c and x ∈ c, then, we may write it as xy − [x, y]. Now xy is in the above subspace, and since g is by assumption a direct sum of a, b and c, the element [x, y] also lies in the relevant subspace. We omit the details, since this would be rather technical, and the reader can easily supply the details. We now return to the group G = GL(2, R) and its Lie algebra g. We will now prove the basic fact from representation theory which we will use. Theorem 4.6 Let (π, V ) be a (g, K∞ )-module. Suppose that v ∈ V has the following properties: (1) v generates V . (2) The connected component SO(2) of O(2) acts on v by the character determined by Rθ (v) = e2πiθm v, for some positive integer m (i.e. v is an eigenvector for A with eigenvalue m). (3) E − (v) = 0 and Z(v) = 0. Then the (π, V ) is admissible and irreducible. Proof Let u(g) denote the universal enveloping algebra of the Lie-algebra g. One has the decomposition (the Poincar´e-Birkhoff-Witt Theorem) u(g) = u(g).[E − ] + u(g).[Z] ⊕ C[E + ] ⊗ C[A]

(32)

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57

where C[ξ] denotes the algebra generated by the operator ξ. Therefore, if (as in (31)) −1 0 ι= 0 1 then by assumptions (1) and (2) of the Theorem, V = C[E + ](v) ⊕ ιC[E + ](v).

(33)

On E + the element A acts by the eigenvalue 2. Therefore, for an integer p ≥ 0, the element (E + )p (v) is an eigenvector for A with eigenvalue (2p + m), and ι(E + )p (v) is an eigenvector with eigenvalue (−2p − m) (note that under the conjugation action of ι, the element A goes to −A, hence ι takes an r-eigenspace for A into the −r-eigenspace). Since all these weights are different, equation (33) shows that V is admissible as an SO(2) module (A generates the complexified Lie algebra of SO(2)). In fact, equation (33) shows that the multiplicity of an irreducible representation of SO(2) in V is at most one, i.e. V is admissible. Suppose that W ⊂ V is a submodule. In the last paragraph, we saw that the action of A on V is completely reducible; hence the same holds for W . Suppose that w is a weight vector in W of weight j, say. By replacing w by ι(w) if necessary, we may assume that j > 0. The last paragraph shows that j = 2p + m for some p ≥ 0 and also that (E + )p (v) = w (up to scalar multiples). We may assume that p is the smallest non-negative integer such that W contains the eigenvector (E + )p (v) = w with eigenvalue 2p + m. The minimality of p implies that E − (w) = 0. Let W ′ be the submodule of W generated by the vector w. To prove the irreducibility, it is enough to show that W ′ = V . We may assume then that W = W ′ . Since v generates V and Z annihilates v, it follows that Z acts by zero on all of V . Therefore, the vector w satisfies all the properties that v does in the assumptions of the Theorem (except that in (2) the eigencharacter is 2p + m). Therefore, cf. equation (33), we have W = C[E + ](w) ⊕ ιC[E + ](w) = C[E + ](E + )p (v) ⊕ ιC[E + ](E + )p (v).

(34)

Now the equations (33) and (34) show that the codimension of W in V is finite: dim(V /W ) < ∞. Hence V /W also satisfies the assumptions of the Theorem (with v ∈ V replaced by its image v ∈ V /W ), but is finite dimensional. This is impossible by the finite dimensional representation theory of sl(2, C): a lowest weight vector (i.e. one killed by E − of sl(2))

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cannot have positive weight for A. But v has exactly this property in V /W . This shows that V /W = 0 i.e. W = V . Proposition 4.7 Given m > 0, there is a unique irreducible (g, K∞ )module ρm which satisfies the properties of 4.6. Proof The uniqueness follows easily from the above proof of Theorem (4.6). Let χm denote the one dimensional complex vector space on which the group SO(2) acts by the character Rθ 7→ e2πimθ (where Rθ is, as in (30), the rotation by θ in 2-space). Consider the space u(g) ⊗ χm . This is a representation for SO(2) (as well as for the universal enveloping algebra u(g)). Let ρm be the O(2)-module induced from this SO(2)-module. Then, ρm satisfies the properties of Theorem (4.6) and is therefore irreducible. Moreover, it is clear that any module V of the type considered in 4.6 is a quotient of ρm . By irreducibility, V = ρm . Remark 4.8 The modules ρ2k are called the discrete series representations of weight 2k of (g, O(2)). This means the following. Suppose there exists an irreducible unitary representation of the group GL(2, R), call it ρ. Suppose that this occurs discretely (i.e. is a closed subspace of ) in a space of functions L2 (G, ω) which transform according to the unitary character ω of the centre Z of GL(2, R) and which are square summable with respect to the Haar measure on the quotient GL(2, R)/Z. Given such a unitary module ρ, consider the space of vectors whose translates under the compact group O(2) form a finite dimensional vector space. This is the Harish-Chandra module of the unitary representation ρ and is a (g, O(2))-module. The representations ρ2k are the Harish-Chandra modules of discrete series representations of even weight. We will show in the next section, that these are closely related to modular forms of weight 2k. There are also the discrete series representations of odd weight, which we will not discuss, since we are dealing with the group SL(2, Z) and it has no modular forms of odd weight. We are now in a position to state the precise relationship of modular forms with representation theory.

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Notation 4.9 Let f be a modular form of weight 2k with k > 0. We will now construct a function on the group G+ = GL(2, R)+ as follows. Set Ff (g) = j(g, i)−2k f (g(i))det(g)k where i ∈ h is the point whose isotropy is the group SO(2) as in equation (30). As before, j(g, i) = cz + d, where g = ( ac db ). By using the modularity of f and the equation j(gh, z) = j(g, h(z))j(h, z) for the “automorphy factor” j(g, z), it is easy to see that Ff is invariant under left translation by elements of SL(2, Z) and also under the centre Z∞ of GL(2, R). We will now check that the (g, O(2))-module generated by Ff is isomorphic to ρ2k , with ρ2k as in 4.7. Note that Ff is contained in the space C ∞ (Z∞ GL(2, Z)\GL(2, R), the space of smooth functions on the relevant space and that the latter is a (g, O(2))-module under right translation by elements of GL(2, R). Moreover, for all y > 0 and x ∈ R we have y x = y k f (x + iy) (35) Ff 0 1 The function g 7→ f (g(i)) is right invariant under the action of SO(2) since i is the isotropy of SO(2). Using the fact that j(Rθ , i) = e−iθ (where Rθ is as in (30)) one checks that j(gRθ , i) = j(g, i)(e−2iθ ). Therefore, it follows that Ff (gRθ ) = Ff (g)e2ikθ . (36) 0 1 ) (A This equation implies that under the action of the element A = i( −1 0 generates the Lie algebra of SO(2)), Ff is an eigenvector with eigenvalue 2k. Compute the action of E − (E − as in (29)) on Ff . Using the invariance of of Ff under Z∞ and that it is an eigenvector of A with eigenvalue 4k, one sees that E − (Ff ) = (X + iY )Ff − ikFf .

Now use equation (35) to conclude that E − Ff = y 2k (∂f /∂z). Since f is holomorphic, one obtains that E − Ff = 0. The (g, O(2)) module generated by Ff satisfies the conditions of 4.6.

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Notation 4.10 Growth Properties of Ff . Consider now the growth properties of Ff . We have the quotient map GL(2, R)+ → h(⊃ F ) where F is the fundamental domain constructed in section 2. Let S be the pre-image of F under this quotient map. Then, we have the inclusion y x 2 : y > 3/4 and − 1/2 < x < 1/2 S ⊂ Z∞ O(2) 0 1 and the latter is a “Siegel set”. Now, y x Ff = y k f (x + iy). 0 1 From the modularity property of f , it follows that f is “bounded at infinity”, which means that there exists a constant C > 0 such that on the fundamental domain F of SL(2, Z), the function z 7→ f (z) = f (x + iy) is bounded by C: | f (z) |≤ C ∀z ∈ F. Therefore, on the Siegel set S, we have y x Ff ≤ Cy k , 0 1 i.e. Ff has moderate growth on the Siegel Set. Suppose now that f is a cusp form. Then, the Fourier expansion at infinity of f is of the form f (z) =

n=∞ X

an exp(2πinz)

n=1

where a(n) are the Fourier coefficients. The function X

an q n−1

is clearly bounded in a neighbourhood of infinity in the fundamental domain F and the complement of a neighbourhood of infinity being compact, it is bounded on all of F too, by a constant C.

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This shows the existence of a constant C > 0 such that for all z ∈ F , one has f (x + iy) ≤ Cexp(−y). The Haar measure on the group GL(2, R) is the product of the Haar measure on the group O(2) and the invariant measure dxdy/y 2 on the upper half plane h = GL(2, R)/O(2) constructed at the end of section 2. This is an easy exercise. Thus, the square of the absolute value of Ff integrated on S = π −1 (F ) (where π is the quotient map GL(2, R) → h ) is simply the integral of the square of the absolute value of f (z) on the domain F . The above estimate for f shows that this integral over F is finite: ! Z Z 1/2

∞

dx

−1/2

(dy/y 2 )exp(−y)

3/4. Suppose that f is a cusp form for SL(2, Z) and Ff be as in section (4.5). Given g ∈ GL(2, A) = GL(2, R) × GL(2, Af ), write g = (g∞ , gf ) accordingly. Define the function Φf on GL(2, Q)\GL(2, A) as follows. Set Φf (g∞ , gf ) = Ff (g∞ ) if gf ∈ K0 and extend to G(A) by demanding that Φf be GL(2, Q)-invariant. The SL(2, Z)-invariance of Ff implies that Φf is well defined. Now, 4.12 shows that Φf is an automorphic form on GL(2, A). By 4.13, Ff is rapidly decreasing on S0 ; moreover, Ff is a cuspidal automorphic form in the sense that for all g ∈ G(A), the following holds. Z Φf (ng)dn = 0 (43) U (Q)\U (A)

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65

where U is the group of unipotent upper triangular matrices in GL(2) with ones on the diagonal and dn is the Haar measure on U (A). To prove this, we note that the vanishing of the integral is unaltered by changing g on the (1) right by an element of Z∞ O(2) × K0 , since Φf is an eigenvector for the right translation action by K∞ Z∞ × K0 . We may hence assume that b an element of G(Af ) g = (g∞ , gf ). Now, up to elements of K0 = GL(2, Z) is upper triangular; (2) left by an element of B(Q) since G(Q) normalises U (A) and U (Q), and preserves the Haar measure on U (A). Note that (the Iwasawa decomposition) the double coset B(Q)\G(Af )/K0 Z(A) is a singleton. Hence, we may assume that g = g∞ = ( y0 x1 ). Then the above integral is the same as Z f (x + n + iy)dn U (Z)\U (R)

which is nothing but the zero-th Fourier coefficient of f , and by the cuspidality of f , this is zero. On the Siegel domain, the modular function f satisfies an estimate of the form |f (x + iy)| < Cexp(−y) where C is some constant. This implies that on the Siegel set S, the function Φf satisfies an estimate of the form Φf (g) = O(|g|−N ) for some positive integer N . This can be shown to imply that the function Φf is square summable on the quotient Z(A)GL(2, Q)\GL(2, A) with respect to the Haar measure. Further, one has the L2 -metric < , > on the space of cuspidal automorphic forms which translates to the “Petersson” metric Z f (z)g(z)y 2k (y −2 dxdy) < f, g >= F

for cusp forms f and g of weight k. As before, F is the fundamental domain for SL(2, Z). Notation 5.5 From now on, we will fix our attention on cusp forms. We have the natural inclusion of GL(2, Q) in GL(2, Af ). Let p > 0 be a prime and let gp = ( p0 10 ) be thought of as an element in GL(2, Af ) under the

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foregoing inclusion. Let Xp denote the characteristic function of the double coset of K0 through the element gp . If χp denotes the characteristic function Xp , and f is a cuspidal modular form of weight 2k, then Φ′ = Φf ∗ R(χp ) (where R(φ) denotes the right convolution by the function φ) is a smooth function on the quotient GL(2, Q)Z(A)\GL(2, A) whose “infinite” component is still ρ2k (since χp commutes with the right action of GL(2, R) on the above quotient). Since χp is K0 invariant, it follows that Φ′ is also right K0 invariant. Therefore, it corresponds to a modular form g, i.e. Φ′ = Φg . It is easy to show that Φ′ is cuspidal (the space of cusp forms is stable under right convolutions). Therefore, g is a cusp form of weight 2k as well. Denote g = T (p)(f ). Then, T (p) is called the Hecke operator corresponding to the prime p. By noting that convolution by χp is self-adjoint for the L2 metric on cuspidal automorphic functions on GL(2) one immediately sees that the operators T (p) are self-adjoint for the Petersson metric on the space of cusp forms of weight 2k. The commutativity of the unramified Hecke algebra implies that the operators T (p) (as p varies) commute as well. Definition 5.6 Now a commuting family of self-adjoint operators on a finite dimensional complex vector space can be simultaneously diagonalised. Consequently, there exists a basis of cusp forms of weight 2k which are simultaneous eigenfunctions for all the Hecke operators T (p); these are called Hecke eigenforms. If f is a Hecke eigenform for SL(2, Z) and has constant term 1, then it is called a normalised Hecke eigenform. Theorem 5.7 The Iwasawa Decomposition: Any matrix in GL(2, Af ) may be written as a product bk with b ∈ B(Af ) (the group of upper triangular b matrices), and k ∈ K0 = GL(2, Z).

Proof This is an easy application of the elementary divisors theorem. By identifying B\G with the projective line P1 , we see that the Iwasawa decomb on P1 (Af ). position amounts to the transitivity of the action of GL(2, Z) But, any element of P1 (Af ) may be written as a vector (x, y) ∈ A2f where for every prime p, the p-th components (xp , yp ) are not both zero. By changing (x, y) by an element of A∗f if necessary, (x, y) may be asb 2 . Further, x, y may be assumed to be coprime, in the sense sumed to be in Z

that for every prime p, the p-adic components xp , yp of x, y are coprime. Now, by writing everything in the notation of row vectors, we want to solve

Classical Modular Forms b the equation for g ∈ GL(2, Z)

67

(x, y) = (0, 1)g

(note that the isotropy at (0, 1) is precisely B(Af )). This amounts to finding b 2 to a basis of Z b 2 , which can be done precisely because x, y are (z, t) ∈ Z coprime. Notation 5.8 This implies that for a prime p, we have Xp = ∪( p0 x1 )K0 ∪ ( 10 0p )K0 b where the union is a disjoint union, and 0 ≤ x ≤ p − 1. Here K0 = GL(2, Z), as before. Notation 5.9 We will now state without proof the computation of T (p) for a prime p. Note that by strong approximation, the K0 invariant function Φf on the quotient Z(A)GL(2, Q)\GL(2, A) is completely determined by its values on elements of the form ( y0 x1 ) with y > 0, in the quotient. We compute (using the description of Xp in the previous section) Φf ∗ R(χp )( y0 x1 ) and find that this is equal, to p2k−1 Φg ( y0 x1 ) where g(z) = (1/p)

X

f ((z + m)/p) + p2k−1 f (pz) = T (p)(f )(z).

0≤m≤p−1

The Fourier coefficients of g at infinity are given by g(m) = a(mp) if m is coprime to p and g(m) = a(mp) + p2k−1 a(m/p) if p divides m.

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Notation 5.10 In particular, if f is an eigenfunction for all the T (p) with eigenvalue λ(p) say, the equation T (p)f = λ(p)f implies, by comparing the p-th Fourier coefficients, that a(p) = λ(p)a(1) for each p. Remark 5.11 In particular, we get from the last two paragraphs, that if f is an eigenform whose first Fourier coefficient a(1) is zero, then, a(m) = 0 for all positive m. This easily follows from induction and the formula a(mp) = λ(p)a(p) if m and p are coprime, and a(m)λ(p) = a(mp) + p2k−1 a(m/p) if p divides m. Hence f = 0. Thus, we have proved that every Hecke eigenform is a nonzero multiple of a normalised Hecke eigenform. Theorem 5.12 (The Multiplicity 1 Theorem): Let f1 and f2 be two normalised Hecke eigenforms for the action of the Hecke operators T (p) with the same eigenvalues λ(p) for every prime p. Then, f1 = f2 . Proof We will prove this by showing that the Fourier coefficients of f1 and f2 are the same. This will imply, by the Fourier expansion for modular forms, that f1 = f2 . Write f = f1 − f2 . Now, the first Fourier coefficient of f is zero, since f1 and f2 are normalised. Further, f is also a Hecke eigenform, since f1 and f2 are so, and with the same eigenvalues. Therefore, by the previous remark, f = 0. Recall that we have identified [representations π of GL(2, A) whose infinite component π∞ is ρ2k and whose finite component πf contains a non-zero GL(2, Af ) invariant vector], with [normalised eigenforms f of weight 2k for the group GL(2, Z)]. Therefore, we have proved that the multiplicity of such a π in the space of cusp forms on GL(2, A) is one. Remark 5.13 Later, Cogdell will prove that the multiplicity of a cuspidal automorphic representation of GL(n) is always 1. This is the famous multiplicity 1 theorem due to Jacquet-Langlands for GL(2) and to PiatetskiiShapiro and Shalika in general. What we have proved is therefore a very special case when n = 2, the infinite component is ρ2k , and the representation is unramified at all the local places.

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Definition 5.14 We now define inductively, the operators T (n) as follows. If m and n are coprime, we define T (mn) = T (m)T (n). This reduces us to defining T (pm ) where p is a prime. Define recursively the operators T (pm ) by the formula T (p)T (pm ) = T (pm+1 ) + pT (pm−1 ). This now implies (by the similarity of the recursive formulae for T (n) P and the Fourier coefficients a(n)), that if f = a(n)q n is a normalised Hecke eigenform, then T (n)f = a(n)f for all n. These T (n)’s are the classical Hecke operators. By construction, they commute, one has T (mn) = T (m)T (n) if m, n are coprime, and they are self-adjoint for the Petersson inner product on modular forms of weight 2k. Theorem 5.15 If f is a normalised Hecke eigenform for SL(2, Z), then, all its Fourier coefficients are algebraic integers. Proof We consider the action of the Hecke operators T (n) on the space 0 of cups forms. Note that the space of cusp forms contains the (adM2k ditive) subgroup L of those cusp forms whose Fourier coefficients are rational integers. This subgroup is stable under the action of the operators T (n). To see this, first suppose that n = p is a prime. By the formula aT (p)f (m) = a(pm) + p2k−1 a(m/p) if p divides m and aT (p)f (m) = a(pm) otherwise, we see that T (p) stabilises the subgroup L. Since the T (p) generate T (n), the operator T (n) also stabilises L. By induction on k, we see that the space M2k of modular forms of weight 2k has a basis whose Fourier coefficients are integral: M2k = CE2k ⊕ M2k−12 ∆, and ∆ and E2k have integral Fourier coefficients. This shows that the subgroup L of the last paragraph contains a basis of the space of cusp 0 . forms M2k Since the operator T (n) is self adjoint with respect to a suitable metric 0 , it follows that the eigenvalues of T (n) are all real and are on the space M2k all algebraic integers. Now the Fourier coefficients a(n) of the eigenform f are nothing but the eigenvalue λ(n) of T (n) corresponding to the eigenvector f , by the last paragraph of the previous section. Consequently, the Fourier coefficients of f are all real algebraic integers.

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6

L-functions of Modular Forms

In this section, we define the L-function of a cusp form for SL(2, Z) and prove that it has analytic continuation to the entire plane and has a nice functional equation. Later, Cogdell will prove analogous statements for cuspidal automorphic representations for GL(n). To begin with, we prove an estimate –due to Hecke– for the n-th Fourier coefficient of a cusp form of weight 2k. Pn=∞ Lemma 6.1 (Hecke) Let f = n=1 a(n)q n be a cusp form of weight 2k. Then, there exists a constant C > 0 such that a(n) ≤ Cnk/2

∀n ≥ 1.

Proof Consider the function φ defined and continuous on the upper half plane h, given by φ(z) = y k/2 |f (z)|. If γ = ( ac db ) ∈ SL(2, Z) and z ′ = γ(z) = x′ + iy ′ , then recall that y ′ = y/(|cz + d|2 ). Therefore, we obtain from the modularity property of f , that φ(γ(z)) = φ(z), i.e. φ is invariant under SL(2, Z). Hence, φ is determined by its restriction to the fundamental domain F . As z tends to infinity in F , the cuspidality condition of f shows that f (z) = O(exp(−2πy)). Therefore, φ is bounded on F and hence on all of the upper half plane h. Thus, there exists a constant C1 such that |f (z)| ≤ C1 y −k/2

∀z ∈ h.

Now consider the n-th Fourier coefficient a(n) of f . Clearly, 2πny

a(n) = e

Z

1

f (x + iy)e−2iπnx dx.

0

By applying the foregoing estimate for f to this equation, we obtain |a(n)| ≤ C1 e2πny y k/2 for all y > 0. Take y = 1/n. We then get a(n) ≤ (C1 e2π )nk/2 .

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Remark 6.2 Deligne has proved that for all primes p, |a(p)| = O(p((k−1)/2)) , by linking these estimates with the “Weil Conjectures” for the number of rational points of algebraic varieties over finite fields. P Definition 6.3 If f (z) = a(n)q n is a cusp form of weight 2k, define for a complex variable s, the Dirichlet Series L(f, s) by the formula L(f, s) =

∞ X

a(n)/ns .

n=1

From Lemma (6.1), it follows that the series converges and is holomorphic in the region Re(s) > 1 + (k/2). The function L(f, s) is called the L-function of the cusp form f . Notation 6.4 an integral expression for L(f, s)). We will now write an integral formula for the L-function of f . First consider the integral Z ∞ f (iy)y s (dy/y). 0

Since f (iy) = O(exp(−2πy)) for all y > 0 (cf. the proof of Lemma (6.1)), it follows that if Re(s) > 0, then the integral converges. Let σ be the real part of s. Then, for each n ≥ 1 the integral Z ∞ |a(n)|e−2πny y σ (dy/y) 0

converges, and is equal to (|a(n)|/nσ )(2π)−σ Γ(σ) where Γ is the classical Γ-function: Z ∞ e−t tz (dt/t). Γ(z) = 0

From the Hecke estimate a(n) = O(nk/2 ) of Lemma (6.1), it follows that the infinite sum of these integrals also converges, provided σ > k/2 + 1. Thus, by the Dominated Convergence Theorem (to justify the interchange of sum and integral), we obtain the equation Z ∞ ∞ X a(n)n−s . f (iy)y s (dy/y) = (2π)−s Γ(s) 0

n=1

We finally obtain the integral expression: Z ∞ f (iy)y s (dy/y) = (2π)−s Γ(s)L(f, s). 0

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T.N. Venkataramana

Remark 6.5 Notice that the integral expression says that the function L(f, s) can be analytically continued to the region Re(s) > 0, since the left-hand side is analytic there, and the function (2π)−s Γ(s) has no zero’s on the complex plane. Notation 6.6 The functional Equation. The L-Function and the integral expression could have been defined for any holomorphic function P f (z) = a(n)q n (q = e2πiz ), provided a(n) satisfy a Hecke estimate. We will now prove a functional equation for L(f, s), by using the modularity property, especially that f (−1/z) = (−1)k z 2k f (z). R∞ Consider now the integral I(s) = 0 f (iy)y s (dy/y), which converges for Re(s) > 0. We write the integral as a sum of the integral from 1 to ∞ and the integral from from 0 to 1. By making a change of variable y 7→ 1/y we get, Z ∞ Z 1 f (−1/(iy))y −s (dy/y). f (iy)y s (dy/y) = 1

0

Note that as f (1/(iy)) is bounded on the interval [1, ∞], and Re(s) > 0, the integral on the right side converges. Therefore, we get, using the functional equation f (−1/iy) = (−1)k y 2k f (iy), that Z

1

s

k

f (iy)y (dy/y) = (−1)

0

We then get I(s) =

Z

∞

Z

∞

f (iy)y 2k−s (dy/y).

1

f (iy)(y s + (−1)k y 2k−s )(dy/y).

1

This holds for all s, with Re(s) > 0. We now make the change of variable s 7→ 2k − s, for s in the region 0 < Re(s) < 2k. Then the above expression for I(s) shows that I(2k − s) = (−1)k I(s). This is the functional equation for L(f, s): (2π)−s Γ(s)L(f, s) = (−1)k (2π)2k−s Γ(2k − s)L(f, 2k − s) for all s in the region 0 < Re(s) < 2k. The left side of this equation is analytic in the region Re(s) > 0 and the right side is analytic in the region Re(s) < 2k. Using the functional equation (and the fact the (2π)−s Γ(s) never vanishes on the complex plane), we now see that L(f, s) has an analytic continuation over the entire complex plane.

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73

Notation 6.7 Euler Factors. In the last few sections, we derived the functional equation and analytic continuation of a cusp form for SL(2, Z). We now derive an Euler product, for the L-function of a normalised Hecke eigenform f . P n Let then f = ∞ n=1 a(n)q be a normalised Hecke eigenform of weight 2k. Fix a prime p and consider the infinite sum Lp (f, s) =

∞ X

a(pm )/pms

m=0

where Re(s) > k/2 + 1. It converges, by the Hecke estimate a(n) = O(nk/2 ). We now use the relations a(pm )a(p) = a(pm+1 ) + p2k−1 a(pm−1 ). Multiplying these equations by 1/p(m+1)s and then summing over all m ≥ 0 we get a(p)Lp (f, s) = (Lp (f, s) − 1)ps + (p2k−1 /p2s )Lp (f, s). That is, Lp (f, s)−1 = 1 − a(p)/ps + p2k−1 /p2s . We now use the fact that a(mn) = a(m)a(n) if m, n are coprime, since f is a normalised Hecke eigenform. Form the product over all primes p of these Lp (f, s). We then get (by using the Dominated Convergence Theorem Q P to justify interchanges) the equation Lp (f, s) = a(n)/ns = L(f, s). Thus we have the infinite product expansion (the product being over all primes p) Y 1/(1 − a(p)/ps + p2k−1 /p2s ). L(f, s) = p

From now on we consider the function L∗ (f, s) = (2π)−s Γ(s)L(f, s) and refer to this as the L-function of f . We have thus proved the following Theorem. P n Theorem 6.8 Let f = ∞ of n=1 a(n)q be a normalised Hecke eigenform P weight 2k for SL(2, Z). Then, the L-function L∗ (f, s) = (2π)−s Γ(s) a(n)/ns converges for Re(s) > k/2, has an analytic continuation to the entire complex plane and satisfies the functional equation L∗ (f, s) = (−1)k L∗ (f, 2k − s). Moreover, in the region Re(s) > k/2, one has the Euler product Y L∗ (f, s) = (2π)−s Γ(s) 1/(1 − a(p)/ps + p2k−1 /p2s ). p

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Remark 6.9 Recall that to each Hecke eigenform f of weight 2k, there corresponds an irreducible cuspidal automorphic representation π(f ) = π = π∞ ⊗p πp of the (restricted direct product) group GL(2, A) = GL(2, R) × Q p GL(2, Qp ) such that π∞ is the discrete series representation ρ2k and each πp is an irreducible unramified representation of GL(2, Qp ). In the lectures of Cogdell, you will see that each cuspidal automorphic representation π of GL(n, A) has an L-function attached to it –denoted L(π, s)– which satisfies a functional equation, has an analytic continuation to the entire plane, and has an Euler product comprising of terms which are monic polynomials in p−s of degree n. It turns out that for the representation π(f ) = π attached to the Hecke eigenform f of weight 2k, the L-function is nothing but L(π, s) = L∗ (f, s + (k − 1/2)), which can easily be seen to satisfy the equation L(π, s) = (−1)k L(π, 1 − s). Moreover, the local factors are of the form L(πp , s)−1 = (1 − (a(p)/pk−1/2 )/ps + 1/p2s , a monic polynomial in in p−s of degree two.

Notes on L-functions for GLn J.W. Cogdell

∗

Oklahoma State University, Stillwater, USA

Lectures given at the School on Automorphic Forms on GL(n) Trieste, 31 July - 18 August 2000

LNS0821003

∗

[email protected]

Abstract The theory of L-functions of automorphic forms (or modular forms) via integral representations has its origin in the paper of Riemann on the ζ-function. However the theory was really developed in the classical context of L-functions of modular forms for congruence subgroups of SL2 (Z) by Hecke and his school. Much of our current theory is a direct outgrowth of Hecke’s. L-functions of automorphic representations were first developed by Jacquet and Langlands for GL2 . Their approach followed Hecke combined with the local-global techniques of Tate’s thesis. The theory for GLn was then developed along the same lines in a long series of papers by various combinations of Jacquet, Piatetski-Shapiro, and Shalika. In addition to associating an L-function to an automorphic form, Hecke also gave a criterion for a Dirichlet series to come from a modular form, the so-called Converse Theorem of Hecke. In the context of automorphic representations, the Converse Theorem for GL2 was developed by Jacquet and Langlands, extended and significantly strengthened to GL3 by Jacquet, Piatetski-Shapiro, and Shalika, and then extended to GLn . What we have attempted to present here is a synopsis of this work and in doing so present the paradigm for the analysis of automorphic L-functions via integral representations. We begin with the Fourier expansion of a cusp form and results on Whittaker models since these are essential for defining Eulerian integrals. We then develop integral representations for L-functions for GLn × GLm which have nice analytic properties (meromorphic continuation, finite order of growth, functional equations) and have Eulerian factorization into products of local integrals. We next turn to the local theory of L-functions for GLn , in both the archimedean and non-archimedean local contexts, which comes out of the Euler factors of the global integrals. We finally combine the global Eulerian integrals with the definition and analysis of the local L-functions to define the global L-function of an automorphic representation and derive their major analytic properties. We next turn to the various Converse Theorems for GLn and their applications to Langlands liftings.

Contents Introduction

79

1 Fourier Expansions and Multiplicity One Theorems 80 1.1 Fourier Expansions . . . . . . . . . . . . . . . . . . . . . . . . 81 1.2 Whittaker Models and the Multiplicity One Theorem . . . . . 86 1.3 Kirillov Models and the Strong Multiplicity One Theorem . . 89 2 Eulerian Integrals for GLn 2.1 Eulerian Integrals for GL2 . . . . . . . 2.2 Eulerian Integrals for GLn × GLm with 2.2.1 The projection operator . . . . 2.2.2 The global integrals . . . . . . 2.3 Eulerian Integrals for GLn × GLn . . . 2.3.1 The mirabolic Eisenstein series 2.3.2 The global integrals . . . . . .

. . . . m> 0 after the unfolding. As we have seen in Lecture 1, if Wϕ ∈ W(π, ψ) corresponds to a decomposable vector ϕ ∈ Vπ ≃ ⊗′ Vπv then the Whittaker function factors into a product of local Whittaker functions Wϕ (g) =

Y

Wϕv (gv ).

v

Since the character χ and the adelic absolute value factor into local components and the domain of integration A× also factors we find that our global integral naturally factors into a product of local integrals Z

A×

Wϕ

a

1

×

s−1/2

d a=

χ(a)|a|

YZ v

kv×

W ϕv

av

1

χv (av )|av |s−1/2 d× av ,

with the infinite product still convergent for Re(s) >> 0, or I(s; ϕ, χ) =

Y

Ψv (s; Wϕv , χv )

v

with the obvious definition of the local integrals Ψv (s; Wϕv , χv ) =

Z

kv×

W ϕv

av 1

χv (av )|av |s−1/2 d× av .

Thus each of our global integrals is Eulerian. In this way, to π and χ we have associated a family of global Eulerian integrals with nice analytic properties as well as for each place v a family of local integrals convergent for Re(s) >> 0.

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Eulerian Integrals for GLn × GLm with m < n

Now let (π, Vπ ) be a cuspidal representation of GLn (A) and (π ′ , Vπ′ ) a cuspidal representation of GLm (A) with m < n. Take ϕ ∈ Vπ and ϕ′ ∈ Vπ′ . At first blush, a natural analogue of the integrals we considered for GL2 with GL1 twists would be Z h ϕ ϕ′ (h)| det(h)|s−(n−m)/2 dh. I n−m GLm (k)\ GLm (A) This family of integrals would have all the nice analytic properties as before (entire functions of finite order satisfying a functional equation), but they would not be Eulerian except in the case m = n − 1, which proceeds exactly as in the GL2 case. The problem is that the restriction of the form ϕ to GLm is too brutal to allow a nice unfolding when the Fourier expansion of ϕ is inserted. Instead we will introduce projection operators from cusp forms on GLn (A) to cuspidal functions on on Pm+1 (A) which are given by part of the unipotent integration through which the Whittaker function is defined. 2.2.1

The projection operator

In GLn , let Y n,m be the unipotent radical of the standard parabolic subgroup attached to the partition (m + 1, 1, . . . , 1). If ψ is our standard additive character of k\A, then ψ defines a character of Yn,m (A) trivial on Yn,m (k) since Yn,m ⊂ Nn . The group Yn,m is normalized by GLm+1 ⊂ GLn and the mirabolic subgroup Pm+1 ⊂ GLm+1 is the stabilizer in GLm+1 of the character ψ. Definition If ϕ(g) is a cusp form on GLn (A) define the projection operator Pnm from cusp forms on GLn (A) to cuspidal functions on Pm+1 (A) by “ ”Z n−m−1 p − 2 Pnm ϕ(p) = | det(p)| ϕ y ψ −1 (y) dy I n−m−1 Yn,m (k)\ Yn,m (A) for p ∈ Pm+1 (A). As the integration is over a compact domain, the integral is absolutely convergent. We first analyze the behavior on Pm+1 (A). Lemma The function Pnm ϕ(p) is a cuspidal function on Pm+1 (A).

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95

Proof: Let us let ϕ′ (p) denote the non-normalized projection, i.e., for p ∈ Pm+1 (A) set “ ” ϕ′ (p) = | det(p)|

n−m−1 2

Pnm ϕ(p).

It suffices to show this function is cuspidal. Since ϕ(g) was a smooth function on GLn (A), ϕ′ (p) will remain smooth on Pm+1 (A). To see that ϕ′ (p) is automorphic, let γ ∈ Pm+1 (k). Then Z γ 0 p 0 ′ ϕ y ϕ (γp) = ψ −1 (y) dy. 0 1 0 1 Yn,m (k)\ Y n,m (A) Since γ ∈ Pm+1 (k) and Pm+1 normalizes Yn,m and stabilizes ψ we may make −1 γ 0 γ 0 the change of variable y 7→ y in this integral to obtain 0 1 0 1 Z γ 0 p 0 ′ ϕ y ϕ (γp) = ψ −1 (y) dy. 0 1 0 1 Yn,m (k)\ Y n,m (A) Since ϕ(g) is automorphic on GLn (A) it is left invariant under GLn (k) and we find that ϕ′ (γp) = ϕ′ (p) so that ϕ′ is indeed automorphic on Pm+1 (A). We next need to see that ϕ′ is cuspidal on Pm+1 (A). To this end, let U ⊂ Pm+1 be the standard unipotent subgroup associated to the partition (n1 , . . . , nr ) of m + 1. Then we must compute the integral Z ϕ′ (up) du. U(k)\ U(A)

Inserting the definition of ϕ′ we find Z ϕ′ (up) du U(k)\ U(A)

=

Z

U(k)\ U(A)

Z

u 0 p 0 ϕ y ψ −1 (y) dy du. 0 1 0 1 Yn,m (k)\ Yn,m (A)

The group U′ = U ⋉ Y n,m is the standard unipotent subgroup of GLn associated to the partition (n1 , . . . , nr , 1, . . . , 1) of n. We may decompose this group in a second manner. If we let U′′ be the standard unipotent subgroup e n−m−1 of GLn associated to the partition (n1 , . . . , nr , n−m−1) of n and let N be the subgroup of GLn obtained by embedding Nn−m−1 into GLn by I 0 n 7→ m+1 0 n

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e n−m−1 ⋉ U′′ . If we extend the character ψ of Ym,n to U′ by then U′ = N e n−m−1 ⋉ U′′ , ψ making it trivial on U, then in the decomposition U′ = N e n−m−1 component and there it is the standard is dependent only on the N character ψ on Nn−m−1 . Hence we may rearrange the integration to give Z ϕ′ (up) du U(k)\ U(A)

=

Z

Nn−m−1 (k)\ Nn−m−1 (A)

Z

′′

ϕ u U′′ (k)\ U′′ (A)

1 0 0 n

p 0 du′′ ψ −1 (n) dy. 0 1

But since ϕ is cuspidal on GLn and U′′ is a standard unipotent subgroup of GLn then Z p 0 ′′ 1 0 du′′ ≡ 0 ϕ u 0 n 0 1 U′′ (k)\ U′′ (A) from which it follows that Z

ϕ′ (up) du ≡ 0

U(k)\ U(A)

so that ϕ′ is a cuspidal function on Pm+1 (A).

2

From Lecture 1, we know that cuspidal functions on Pm+1 (A) have a Fourier expansion summed over Nm (k)\ GLm (A). Applying this expansion to our projected cusp form on GLn (A) we are led to the following result. h n Lemma Let ϕ be a cusp form on GLn (A). Then for h ∈ GLm (A), Pm ϕ 1 has the Fourier expansion “ ” n−m−1 X h γ 0 h − 2 Pnm ϕ = | det(h)| Wϕ 1 0 In−m In−m γ∈Nm (k)\ GLm (k)

with convergence absolute and uniform on compact subsets. Proof: Once again let ′

“

ϕ (p) = | det(p)|

n−m−1 2

”

Pnm ϕ(p)

with p ∈ Pm+1 (A). Since we have verified that ϕ′ (p) is a cuspidal function on Pm+1 (A) we know that it has a Fourier expansion of the form X γ 0 ′ ϕ (p) = W ϕ′ p 0 1 γ∈Nm (k)\ GLm (k)

L-functions for GLn where Wϕ′ (p) =

Z

ϕ′ (np)ψ −1 (n) dn.

Nm+1 (k)\ Nm+1 (A) To obtain our expansion for Pnm ϕ we need terms of ϕ rather than ϕ′ .

We have Wϕ′ (p) = =

Z

Z

97

to express the right-hand side in

ϕ′ (n′ p)ψ −1 (n′ ) dn′ Nm+1 (k)\ Nm+1 (A)

Nm+1 (k)\ Nm+1 (A)

′ np 0 ϕ y g · 0 1 Y n,m (k)\ Y n,m (A)

Z

· ψ −1 (y) dy ψ −1 (n′ ) dn′ . It is elementary to see that the maximal unipotent subgroup Nn of GLn can be factored as Nn = Nm+1 ⋉ Y n,m and if we write n = n′ y with n′ ∈ Nm+1 and y ∈ Yn,m then ψ(n) = ψ(n′ )ψ(y). Hence the above integral may be written as Z p 0 p 0 −1 ′ ϕ n ψ (n) dn = Wϕ Wϕ (p) = . 0 In−m−1 0 In−m−1 Nn (k)\ Nn (A) Substituting this expression into the above we find that h n Pm ϕ 1 ” “ n−m−1 X γ 0 h − 2 Wϕ = | det(h)| 0 In−m In−m γ∈Nm (k)\ GLm (k)

and the convergence is absolute and uniform for h in compact subsets of GLm (A). 2 2.2.2

The global integrals

We now have the prerequisites for writing down a family of Eulerian integrals for cusp forms ϕ on GLn twisted by automorphic forms on GLm for m < n. Let ϕ ∈ Vπ be a cusp form on GLn (A) and ϕ′ ∈ Vπ′ a cusp form on GLm (A). (Actually, we could take ϕ′ to be an arbitrary automorphic form on GLm (A).) Consider the integrals Z h 0 n ′ Pm ϕ I(s; ϕ, ϕ ) = ϕ′ (h)| det(h)|s−1/2 dh. 0 1 GLm (k)\ GLm (A)

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The integral I(s; ϕ, ϕ′ ) is absolutely convergent for all values of the complex parameter s, uniformly in compact subsets, since the cusp forms are rapidly decreasing. Hence it is entire and bounded in any vertical strip as before. Let us now investigate the Eulerian properties of these integrals. We first replace Pnm ϕ by its Fourier expansion. Z h 0 n ′ ϕ′ (h)| det(h)|s−1/2 dh Pm ϕ I(s; ϕ, ϕ ) = 0 In−m GLm (k)\ GLm (A) Z X γ 0 h 0 Wϕ · = 0 In−m 0 In−m GLm (k)\ GLm (A) γ∈Nm (k)\ GLm (k)

· ϕ′ (h)| det(h)|s−(n−m)/2 dh. Since ϕ′ (h) is automorphic on GLm (A) and | det(γ)| = 1 for γ ∈ GLm (k) we may interchange the order of summation and integration for Re(s) >> 0 and then recombine to obtain Z h 0 ′ Wϕ I(s; ϕ, ϕ ) = ϕ′ (h)| det(h)|s−(n−m)/2 dh. 0 I n−m Nm (k)\ GLm (A) This integral is absolutely convergent for Re(s) >> 0 by the gauge estimates of [31, Section 13] and this justifies the interchange. Let us now integrate first over Nm (k)\ Nm (A). Recall that for n ∈ Nm (A) ⊂ Nn (A) we have Wϕ (ng) = ψ(n)Wϕ (g). Hence we have I(s; ϕ, ϕ′ ) = Z

Nm (A)\ GLm (A)

=

= =

Z Z

Z

Wϕ Nm (k)\ Nm (A)

Wϕ Nm (A)\ GLm (A)

Wϕ

Nm (A)\ GLm (A) Ψ(s; Wϕ , Wϕ′ ′ )

n 0 0 In−m

h 0 0 In−m

·

· ϕ′ (nh) dn | det(h)|s−(n−m)/2 dh Z 0 ψ(n)ϕ′ (nh) dn

h 0 In−m

Nm (k)\ Nm (A)

· | det(h)|s−(n−m)/2 dh 0 Wϕ′ ′ (h)| det(h)|s−(n−m)/2 dh

h 0 In−m

where Wϕ′ ′ (h) is the ψ −1 -Whittaker function on GLm (A) associated to ϕ′ , i.e., Z ϕ′ (nh)ψ(n) dn,

Wϕ′ ′ (h) =

Nm (k)\ Nm (A)

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99

and we retain absolute convergence for Re(s) >> 0. From this point, the fact that the integrals are Eulerian is a consequence of the uniqueness of the Whittaker model for GLn . Take ϕ a smooth cusp form in a cuspidal representation π of GLn (A). Assume in addition that ϕ is factorizable, i.e., in the decomposition π = ⊗′ πv of π into a restricted tensor product of local representations, ϕ = ⊗ϕv is a pure tensor. Then as we have seen there is a choice of local Whittaker models so that Q Wϕ (g) = Wϕv (gv ). Similarly for decomposable ϕ′ we have the factorizaQ tion Wϕ′ ′ (h) = Wϕ′ ′v (hv ). If we substitute these factorizations into our integral expression, then Q since the domain of integration factors Nm (A)\ GLm (A) = Nm (kv )\ GLm (kv ) we see that our integral factors into a product of local integrals YZ hv 0 ′ W ϕv Wϕ′ ′v (hv ) · Ψ(s; Wϕ , Wϕ′ ) = 0 I n−m Nm (kv )\ GLm (kv ) v

·| det(hv )|vs−(n−m)/2 dhv . If we denote the local integrals by Z ′ Ψv (s; Wϕv , Wϕ′v ) =

W ϕv

Nm (kv )\ GLm (kv )

hv 0 Wϕ′ ′v (hv ) · 0 In−m

·| det(hv )|vs−(n−m)/2 dhv , which converges for Re(s) >> 0 by the gauge estimate of [31, Prop. 2.3.6], we see that we now have a family of Eulerian integrals. Now let us return to the question of a functional equation. As in the case of GL2 , the functional equation is essentially a consequence of the existence of the outer automorphism g 7→ ι(g) = gι = tg−1 of GLn . If we define the action of this automorphism on automorphic forms by setting ϕ(g) e = e n = ι ◦ Pn ◦ ι then our integrals naturally satisfy ϕ(gι ) = ϕ(wn gι ) and let P m m the functional equation e − s; ϕ, I(s; ϕ, ϕ′ ) = I(1 e ϕ e′ )

where e ϕ, ϕ′ ) = I(s;

Z

GLm (k)\ GLm (A)

h n e Pm ϕ

We have established the following result.

1

ϕ′ (h)| det(h)|s−1/2 dh.

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Theorem 2.1 Let ϕ ∈ Vπ be a cusp form on GLn (A) and ϕ′ ∈ Vπ′ a cusp form on GLm (A) with m < n. Then the family of integrals I(s; ϕ, ϕ′ ) define entire functions of s, bounded in vertical strips, and satisfy the functional equation e − s; ϕ, I(s; ϕ, ϕ′ ) = I(1 e ϕ e′ ).

Moreover the integrals are Eulerian and if ϕ and ϕ′ are factorizable, we have Y I(s; ϕ, ϕ′ ) = Ψv (s; Wϕv , Wϕ′ ′v ) v

with convergence absolute and uniform for Re(s) >> 0. The integrals occurring on the right-hand side of our functional equation are again Eulerian. One can unfold the definitions to find first that e − s; ϕ, e − s; ρ(wn,m )W fϕ , W f′ ′ ) I(1 e ϕ e′ ) = Ψ(1 ϕ

where the unfolded global integral is Z Z h ′ e Ψ(s; W, W ) = W x In−m−1

dx W ′ (h)| det(h)|s−(n−m)/2 dh 1

with the h integral over Nm (A)\ GLm (A) and the x integral over Mn−m−1,m (A), the space of (n − m − 1) × m matrices, ρ denoting right translation, and 1 . Im with wn−m = . . wn,m the Weyl element wn,m = wn−m 1 the standard long Weyl element in GLn−m . Also, for W ∈ W(π, ψ) we set f (g) = W (wn gι ) ∈ W(e W π , ψ −1 ). The extra unipotent integration is the reme W, W ′ ) is absolutely convergent for Re(s) >> 0. en . As before, Ψ(s; nant of P m ′ e Wϕ , W ′ ′ ) will factor For ϕ and ϕ factorizable as before, these integrals Ψ(s; ϕ as well. Hence we have Y e Wϕ , W ′ ′ ) = e v (s; Wϕv , W ′ ′ ) Ψ(s; Ψ ϕ ϕv v

where e v (s; Wv , Wv′ ) = Ψ

Z Z

hv Wv xv In−m−1

dxv Wv′ (hv )| det(hv )|s−(n−m)/2 dhv 1

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101

where now the hv integral is over Nm (kv )\ GLm (kv ) and the xv integral is over the matrix space Mn−m−1,m (kv ). Thus, coming back to our functional equation, we find that the right-hand side is Eulerian and factors as Y e v (1−s; ρ(wn,m )W fϕv , W f ′ ′ ). e e fϕ , W f′ ′ ) = Ψ I(1−s; ϕ, e ϕ e′ ) = Ψ(1−s; ρ(wn,m )W ϕv ϕ v

2.3

Eulerian Integrals for GLn × GLn

The paradigm for integral representations of L-functions for GLn × GLn is not Hecke but rather the classical papers of Rankin [52] and Selberg [54]. These were first interpreted in the framework of automorphic representations by Jacquet for GL2 × GL2 [28] and then Jacquet and Shalika in general [36]. Let (π, Vπ ) and (π ′ , Vπ′ ) be two cuspidal representations of GLn (A). Let ϕ ∈ Vπ and ϕ′ ∈ Vπ′ be two cusp forms. The analogue of the construction above would be simply Z ϕ(g)ϕ′ (g)| det(g)|s dg. GLn (k)\ GLn (A)

This integral is essentially the L2 -inner product of ϕ and ϕ′ and is not suitable for defining an L-function, although it will occur as a residue of our integral at a pole. Instead, following Rankin and Selberg, we use an integral representation that involves a third function: an Eisenstein series on GLn (A). This family of Eisenstein series is constructed using the mirabolic subgroup once again. 2.3.1

The mirabolic Eisenstein series

To construct our Eisenstein series we return to the observation that Pn \ GLn ≃ kn − {0}. If we let S(An ) denote the Schwartz–Bruhat functions on An , then each Φ ∈ S defines a smooth function on GLn (A), left invariant by Pn (A), by g 7→ Φ((0, . . . , 0, 1)g) = Φ(en g). Let η be a unitary idele class character. (For our application η will be determined by the central characters of π and π ′ .) Consider the function Z Φ(aen g)|a|ns η(a) d× a. F (g, Φ; s, η) = | det(g)|s A×

If we let P′n = Zn Pn be the parabolicof GL n associated to the partition h y (n − 1, 1) then one checks that for p′ = ∈ P′n (A) with h ∈ GLn−1 (A) 0 d

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and d ∈ A× we have, F (p′ g, Φ; s, η) = | det(h)|s |d|−(n−1)s η(d)−1 F (g, Φ; s, η) = δPs ′ (p′ )η −1 (d)F (g, Φ; s, η), n

with the integral absolutely convergent for Re(s) > 1/n, so that if we extend η to a character of P′n by η(p′ ) = η(d) in the above notation we have that F (g, Φ; s, η) is a smooth section of the normalized induced representation GLn (A) s−1/2 s−1/2 IndP′ (A) (δP′ η). Since the inducing character δP′ η of P′n (A) is invarin n n ant under P′n (k) we may form Eisenstein series from this family of sections by X E(g, Φ; s, η) = F (γg, Φ; s, η). γ∈P′n (k)\ GLn (k)

If we replace F in this sum by its definition we can rewrite this Eisenstein series as Z X s Φ(aξg)|a|ns η(a) d× a E(g, Φ; s, η) = | det(g)| k × \A× ξ∈k n −{0}

= | det(g)|s

Z

k × \A×

Θ′Φ (a, g)|a|ns η(a) d× a

and this first expression is convergent absolutely for Re(s) > 1 [36]. The second expression essentially gives the Eisenstein series as the Mellin transform of the Theta series X ΘΦ (a, g) = Φ(aξg), ξ∈k n

where in the above we have written X Θ′Φ (a, g) = Φ(aξg) = ΘΦ (a, g) − Φ(0). ξ∈k n −{0}

This allows us to obtain the analytic properties of the Eisenstein series from the Poisson summation formula for ΘΦ , namely X X ΘΦ (a, g) = Φ(aξg) = Φa,g (ξ) ξ∈k n

=

X

ξ∈k n

d Φ a,g (ξ) =

ξ∈k n −n

= |a|

X

b −1 ξ tg−1 ) |a|−n | det(g)|−1 Φ(a

ξ∈k n | det(g)|−1 ΘΦˆ (a−1 ,t g−1 )

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103

ˆ on S(An ) is defined by where the Fourier transform Φ Z ˆ Φ(y)ψ(y tx) dy. Φ(x) = A×

This allows us to write the Eisenstein series as Z s E(g, Φ, s, η) = | det(g)| Θ′Φ (a, g)|a|ns η(a) d× a |a|≥1 Z s−1 + | det(g)| Θ′Φˆ (a,t g−1 )|a|n(1−s) η −1 (a) d× a + δ(s) |a|≥1

where δ(s) =

( 0

if η is ramified

s −cΦ(0) | det(g)| s+iσ

+

s−1 ˆ | det(g)| cΦ(0) s−1+iσ

if η(a) = |a|inσ with σ ∈ R

with c a non-zero constant. From this we easily derive the basic properties of our Eisenstein series [36, Section 4]. Proposition 2.1 The Eisenstein series E(g, Φ; s, η) has a meromorphic continuation to all of C with at most simple poles at s = −iσ, 1 − iσ when η is unramified of the form η(a) = |a|inσ . As a function of g it is smooth of moderate growth and as a function of s it is bounded in vertical strips (away from the possible poles), uniformly for g in compact sets. Moreover, we have the functional equation ˆ 1 − s, η −1 ) E(g, Φ; s, η) = E(gι , Φ; where gι = tg−1 . Note that under the center the Eisenstein series transforms by the central character η −1 . 2.3.2

The global integrals

Now let us return to our Eulerian integrals. Let π and π ′ be our irreducible cuspidal representations. Let their central characters be ω and ω ′ . Set η = ωω ′ . Then for each pair of cusp forms ϕ ∈ Vπ and ϕ′ ∈ Vπ′ and each Schwartz-Bruhat function Φ ∈ S(An ) set Z ′ ϕ(g)ϕ′ (g)E(g, Φ; s, η) dg. I(s; ϕ, ϕ , Φ) = Zn (A) GLn (k)\ GLn (A)

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Since the two cusp forms are rapidly decreasing on Zn (A) GLn (k)\ GLn (A) and the Eisenstein series is only of moderate growth, we see that the integral converges absolutely for all s away from the poles of the Eisenstein series and is hence meromorphic. It will be bounded in vertical strips away from the poles and satisfies the functional equation ˆ I(s; ϕ, ϕ′ , Φ) = I(1 − s; ϕ, e ϕ e′ , Φ),

coming from the functional equation of the Eisenstein series, where we still have ϕ(g) e = ϕ(gι ) = ϕ(wn gι ) ∈ Vπe and similarly for ϕ e′ . These integrals will be entire unless we have η(a) = ω(a)ω ′ (a) = |a|inσ is unramified. In that case, the residue at s = −iσ will be Z ′ ϕ(g)ϕ′ (g)| det(g)|−iσ dg Res I(s; ϕ, ϕ , Φ) = −cΦ(0) s=−iσ

Zn (A) GLn (A)\ GLn (A)

and at s = 1 − iσ we can write the residue as Z ′ ˆ Res I(s; ϕ, ϕ , Φ) = cΦ(0) s=1−iσ

Zn (A) GLn (k)\ GLn (A)

ϕ(g) e ϕ e′ (g)| det(g)|iσ dg.

Therefore these residues define GLn (A) invariant pairings between π and π ′ ⊗ | det |−iσ or equivalently between π e and π e′ ⊗ | det |iσ . Hence a residues ′ iσ can be non-zero only if π ≃ π e ⊗ | det | and in this case we can find ϕ, ϕ′ , and Φ such that indeed the residue does not vanish. We have yet to check that our integrals are Eulerian. To this end we take the integral, replace the Eisenstein series by its definition, and unfold: Z ′ ϕ(g)ϕ′ (g)E(g, Φ; s, η) dg I(s; ϕ, ϕ , Φ) = Zn (A) GLn (k)\ GLn (A) Z ϕ(g)ϕ′ (g)F (g, Φ; s, η) dg = Zn (A) P′n (k)\ GLn (A) Z Z ϕ(g)ϕ′ (g)| det(g)|s Φ(aen g)|a|ns η(a) da dg = × Z (A) Pn (k)\ GLn (A) A Z n ϕ(g)ϕ′ (g)Φ(en g)| det(g)|s dg. = Pn (k)\ GLn (A)

We next replace ϕ by its Fourier expansion in the form X ϕ(g) = Wϕ (γg) γ∈Nn (k)\ Pn (k)

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105

and unfold to find Z ′ Wϕ (g)ϕ′ (g)Φ(en g)| det(g)|s dg I(s; ϕ, ϕ , Φ) = Nn (k)\ GLn (A) Z Z ϕ′ (ng)ψ(n) dn Φ(en g)| det(g)|s dg Wϕ (g) = Nn (k)\ Nn (A) N (A)\ GLn (A) Z n Wϕ (g)Wϕ′ ′ (g)Φ(en g)| det(g)|s dg = =

Nn (A)\ GLn (A) Ψ(s; Wϕ , Wϕ′ ′ , Φ).

This expression converges for Re(s) >> 0 by the gauge estimates as before. To continue, we assume that ϕ, ϕ′ and Φ are decomposable tensors under the isomorphisms π ≃ ⊗′ πv , π ′ ≃ ⊗′ πv′ , and S(An ) ≃ ⊗′ S(kvn ) so that we Q Q Q have Wϕ (g) = v Wϕv (gv ), Wϕ′ ′ (g) = v Wϕ′ ′v (gv ) and Φ(g) = v Φv (gv ). Then, since the domain of integration also naturally factors we can decompose this last integral into an Euler product and now write Y Ψ(s; Wϕ , Wϕ′ ′ , Φ) = Ψv (s; Wϕv , Wϕ′ ′v , Φv ), v

where Ψv (s; Wϕv , Wϕ′ ′v , Φv )

=

Z

Nn (kv )\ GLn (kv )

Wϕv (gv )Wϕ′ ′v (gv )Φv (en gv )| det(gv )|s dgv ,

still with convergence for Re(s) >> 0 by the local gauge estimates. Once again we see that the Euler factorization is a direct consequence of the uniqueness of the Whittaker models. Theorem 2.2 Let ϕ ∈ Vπ and ϕ′ ∈ Vπ′ cusp forms on GLn (A) and let Φ ∈ S(An ). Then the family of integrals I(s; ϕ, ϕ′ , Φ) define meromorphic functions of s, bounded in vertical strips away from the poles. The only possible poles are simple and occur iff π ≃ π e′ ⊗ | det |iσ with σ real and are then at s = −iσ and s = 1 − iσ with residues as above. They satisfy the functional equation fϕ , W f ′ ′ , Φ). ˆ I(s; ϕ, ϕ′ , Φ) = I(1 − s; W ϕ

Moreover, for ϕ, ϕ′ , and Φ factorizable we have that the integrals are Eulerian and we have Y I(s; ϕ, ϕ′ , Φ) = Ψv (s; Wϕv , Wϕ′ ′v , Φv ) v

with convergence absolute and uniform for Re(s) >> 0.

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We remark in passing that the right-hand side of the functional equation also unfolds as Z fϕ (g)W f ′ ′ (g)Φ(e ˆ n g)| det(g)|1−s dg ˆ = W I(1 − s; ϕ, e ϕ e′ , Φ) ϕ Nn (A)\ GLn (A)

=

Y v

fϕ , W f ′ ′ , Φ) ˆ Ψv (1 − s; W ϕ

with convergence for Re(s) > 0. To make the notation more convenient for what follows, in the case m < n for any 0 ≤ j ≤ n−m−1 let us set Z Z h dx W x Ij Ψj (s : W, W ′ ) = Nm (k)\ GLm (k) Mj,m (k) In−m−j W ′ (h)| det(h)|s−(n−m)/2 dh,

e W, W ′ ) = Ψn−m−1 (s; W, W ′ ), so that Ψ(s; W, W ′ ) = Ψ0 (s; W, W ′ ) and Ψ(s; which is still absolutely convergent for Re(s) >> 0. We need to understand what type of functions of s these local integrals are. To this end, we need to understand the local Whittaker functions. So let W ∈ W(π, ψ). Since W is smooth, there is a compact open subgroup K, of finite index in the maximal compact subgroup Kn = GLn (o), so that W (gk) = W (g) for all k ∈ K. If we let {ki } be a set of coset representatives of GLn (o)/K, using that W transforms on the left under Nn (k) via ψ and the Iwasawa decomposition on GLn (k) we see that W (g) is completely determined by the values of W (aki ) = Wi (a) for a ∈ An (k), the maximal split (diagonal) torus of GLn (k). So it suffices to understand a general Whittaker function on the torus. Let αi , i = 1, . . . , n − 1, denote the standard simple

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a1 roots of GLn , so that if a =

..

.

∈ An (k) then αi (a) = ai /ai+1 .

an By a finite function on An (k) we mean a continuous function whose translates span a finite dimensional vector space [30, 31, Section 2.2]. (For the field k× itself the finite functions are spanned by products of characters and powers of the valuation map.) The fundamental result on the asymptotics of Whittaker functions is then the following [31, Prop. 2.2]. Proposition 3.1 Let π be a generic representation of GLn (k). Then there is a finite set of finite functions X(π) = {χi } on An (k), depending only on π, so that for every W ∈ W(π, ψ) there are Schwartz–Bruhat functions φi ∈ S(kn−1 ) such that for all a ∈ An (k) with an = 1 we have X W (a) = χi (a)φi (α1 (a), . . . , αn−1 (a)). X(π)

The finite set of finite functions X(π) which occur in the asymptotics near 0 of the Whittaker functions come from analyzing the Jacquet module W(π, ψ)/hπ(n)W − W |n ∈ Nn i which is naturally an An (k)–module. Note that due to the Schwartz-Bruhat functions, the Whittaker functions vanish whenever any simple root αi (a) becomes large. The gauge estimates alluded to in Section 2 are a consequence of this expansion and the one in Proposition 3.6. Several nice consequences follow from inserting these formulas for W and W ′ into the local integrals Ψj (s; W, W ′ ) or Ψ(s; W, W ′ , Φ) [31, 33]. Proposition 3.2 The local integrals Ψj (s; W, W ′ ) or Ψ(s; W, W ′ , Φ) satisfy the following properties. 1. Each integral converges for Re(s) >> 0. For π and π ′ unitary, as we have assumed, they converge absolutely for Re(s) ≥ 1. For π and π ′ tempered, we have absolute convergence for Re(s) > 0. 2. Each integral defines a rational function in q −s and hence meromorphically extends to all of C. 3. Each such rational function can be written with a common denominator which depends only on the finite functions X(π) and X(π ′ ) and hence only on π and π ′ .

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In deriving these when m < n − 1 note that one has that h 6= 0 W x Ij In−m−j−1 implies that x lies in a compact set independent of h ∈ GLm (k) [33]. Let Ij (π, π ′ ) denote the complex linear span of the local integrals Ψj (s; W , ′ W ) if m < n and I(π, π ′ ) the complex linear span of the Ψ(s; W, W ′ , Φ) if m = n. These are then all subspaces of C(q −s ) which have “bounded denominators” in the sense of (3). In fact, these subspaces have more structure – they are modules for C[q s , q −s ] ⊂ C(q −s ). To see this, note that for any h ∈ GLm (k) we have h ′ ′ W, π (h)W = | det(h)|−s−j+(n−m)/2 Ψj (s; W, W ′ ) Ψj s; π In−m and Ψ(s; π(h)W, π ′ (h)W ′ , ρ(h)Φ) = | det(h)|−s Ψ(s; W, W ′ , Φ). So by varying h and multiplying by scalars, we see that each Ij (π, π ′ ) and I(π, π ′ ) is closed under multiplication by C[q s , q −s ]. Since we have bounded denominators, we can conclude: Proposition 3.3 Each Ij (π, π ′ ) and I(π, π ′ ) is a fractional C[q s , q −s ]–ideal of C(q −s ). Note that C[q s , q −s ] is a principal ideal domain, so that each fractional ideal Ij (π, π ′ ) has a single generator, which we call Qj,π,π′ (q −s ), as does I(π, π ′ ), which we call Qπ,π′ (q −s ). However, we can say more. In the case m < n recall that from what we have said about the Kirillov model that when we restrict Whittaker functions in W(π, ψ) to the embedded GLm (k) ⊂ Pn (k) we get all functions of compact support on GLm (k) transforming by ψ. Using this freedom for our choice of W ∈ W(π, ψ) one can show that in fact the constant function 1 lies in Ij (π, π ′ ). In the case m = n one can reduce to a sum of integrals over Pn (k) and then use the freedom one has in the Kirillov model, plus the complete freedom in the choice of Φ to show that once again 1 ∈ I(π, π ′ ). The consequence of this is that our generator can be taken to be of the form Qj,π,π′ (q −s ) = Pj,π,π′ (q s , q −s )−1 for m < n or Qπ,π′ (q −s ) = Pπ,π′ (q s , q −s )−1 for appropriate polynomials in C[q s , q −s ].

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Moreover, since q s and q −s are units in C[q s , q −s ] we can always normalize our generator to be of the form Pj,π,π′ (q −s )−1 or Pπ,π′ (q −s )−1 where the polynomial P (X) satisfies P (0) = 1. Finally, in the case m < n one can show by a rather elementary although somewhat involved manipulation of the integrals that all of the ideals Ij (π, π ′ ) are the same [33, Section 2.7]. We will write this ideal as I(π, π ′ ) and its generator as Pπ,π′ (q −s )−1 . This gives us the definition of our local L-function. Definition Let π and π ′ be as above. Then L(s, π × π ′ ) = Pπ,π′ (q −s )−1 is the normalized generator of the fractional ideal I(π, π ′ ) formed by the family of local integrals. If π ′ = 1 is the trivial representation of GL1 (k) then we write L(s, π) = L(s, π × 1). One can easily show that the ideal I(π, π ′ ) is independent of the character ψ used in defining the Whittaker models, so that L(s, π × π ′ ) is independent of the choice of ψ. So it is not included in the notation. Also, note that for π ′ = χ an automorphic representation (character) of GL1 (A) we have the identity L(s, π × χ) = L(s, π ⊗ χ) where π ⊗ χ is the representation of GLn (A) on Vπ given by π ⊗ χ(g)ξ = χ(det(g))π(g)ξ. We summarize the above in the following Theorem. Theorem 3.1 Let π and π ′ be as above. The family of local integrals form a C[q s , q −s ]–fractional ideal I(π, π ′ ) in C(q −s ) with generator the local Lfunction L(s, π × π ′ ). Another useful way of thinking of the local L-function is the following. L(s, π × π ′ ) is the minimal (in terms of degree) function of the form P (q −s )−1 , with P (X) a polynomial satisfying P (0) = 1, such that the raΨ(s; W, W ′ ) Ψ(s; W, W ′ , Φ) tios (resp. ) are entire for all W ∈ W(π, ψ) and L(s, π × π ′ ) L(s, π × π ′ ) W ′ ∈ W(π ′ , ψ −1 ), and if necessary Φ ∈ S(kn ). That is, L(s, π × π ′ ) is the standard Euler factor determined by the poles of the functions in I(π, π ′ ). One should note that since the L-factor is a generator of the ideal I(π, π ′ ), then in particular it lies in I(π, π ′ ). Since this ideal is spanned by our local integrals, we have the following useful Corollary. Corollary There is a finite collection of Wi ∈ W(π, ψ), Wi′ ∈ W(π ′ , ψ −1 ), and if necessary Φi ∈ S(kn ) such that X X L(s, π × π ′ ) = Ψ(s; Wi , Wi′ ) or L(s, π × π ′ ) = Ψ(s; Wi , Wi′ , Φi ). i

i

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For future reference, let us set e(s; W, W ′ ) = e˜(s; W, W ′ ) =

Ψ(s; W, W ′ ) , L(s, π × π ′ ) e W, W ′ ) Ψ(s; , L(s, π × π ′ )

and e(s; W, W ′ , Φ) =

ej (s; W, W ′ ) =

Ψj (s; W, W ′ ) , L(s, π × π ′ )

Ψ(s; W, W ′ , Φ) . L(s, π × π ′ )

Then all of these functions are Laurent polynomials in q ±s , i.e., elements of C[q s , q −s ]. As such they are entire and bounded in vertical strips. As above, P there are choices of Wi , Wi′ , and if necessary Φi such that e(s; Wi , Wi′ ) ≡ 1 P or e(s; Wi , Wi′ , Φi ) ≡ 1. In particular we have the following result. Corollary The functions e(s; W, W ′ ) and e(s; W, W ′ , Φ) are entire functions, bounded in vertical strips, and for each s0 ∈ C there is a choice of W , W ′ , and if necessary Φ such that e(s0 ; W, W ′ ) 6= 0 or e(s0 ; W, W ′ , Φ) 6= 0.

3.1.2

The local functional equation

Either by analogy with Tate’s thesis or from the corresponding global statement, we would expect our local integrals to satisfy a local functional equation. From the functional equations for our global integrals, we would expect e − s; ρ(wn,m )W f, W f ′ ) when these to relate the integrals Ψ(s; W, W ′ ) and Ψ(1 f, W f ′ , Φ) ˆ when m = n. This will m < n and Ψ(s; W, W ′ , Φ) and Ψ(1 − s; W indeed be the case. These functional equations will come from interpreting the local integrals as families (in s) of quasi-invariant bilinear forms on W(π, ψ) × W(π ′ , ψ −1 ) or trilinear forms on W(π, ψ) × W(π ′ , ψ −1 ) × S(kn ) depending on the case. First, consider the case when m < n. In this case we have seen that h Ψ s; π

In−m

W, π ′ (h)W ′

= | det(h)|−s+(n−m)/2 Ψ(s; W, W ′ )

f, W f ′ ) has the same quasi-invariance and one checks that Ψ(1 − s; ρ(wn,m )W as a bilinear form on W(π, ψ) × W(π ′ , ψ −1 ). In addition, if we let Yn,m

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denote the unipotent radical of the standard parabolic subgroup associated to the partition (m + 1, 1, . . . , 1) as before then we have the quasi-invariance Ψ(s; π(y)W, W ′ ) = ψ(y)Ψ(s; W, W ′ ) e − s; ρ(wn,m )W f, W f ′ ) satisfies the for all y ∈ Yn,m . One again checks that Ψ(1 same quasi-invariance as a bilinear form on W(π, ψ) × W(π ′ , ψ −1 ). For n = m we have seen that Ψ(s; π(h)W, π ′ (h)W ′ , ρ(h)Φ) = | det(h)|−s Ψ(s; W, W ′ , Φ) f, W f ′ , Φ) ˆ satisfies the same quasiand it is elementary to check that Ψ(1−s; W ′ invariance as a trilinear form on W(π, ψ) × W(π , ψ −1 ) × S(kn ). Our local functional equations will now follow from the following result [33, Propositions 2.10 and 2.11]. Proposition 3.4 (i) If m < n, then except for a finite number of exceptional values of q −s there is a unique bilinear form Bs on W(π, ψ) × W(π ′ , ψ −1 ) satisfying h ′ ′ Bs π W, π (h)W = | det(h)|−s+(n−m)/2 Bs (W, W ′ ) In−m and

Bs (π(y)W, W ′ ) = ψ(y)Bs (W, W ′ )

for all h ∈ GLm (k) and y ∈ Yn,m (k). (ii) If n = m, then except for a finite number of exceptional values of −s q there is a unique trilinear form Ts on W(π, ψ) × W(π ′ , ψ −1 ) × S(kn ) satisfying Ts (π(h)W, π ′ (h)W ′ , ρ(h)Φ) = | det(h)|−s Ts (W, W ′ , Φ) for all h ∈ GLn (k). Let us say a few words about the proof of this proposition, because it is another application of the analysis of the restriction of representations of GLn to the mirabolic subgroup Pn [33, Sections 2.10 and 2.11]. In the case where m < n the local integrals involve the restriction of the Whittaker functions in W(π, ψ) to GLm (k) ⊂ Pn , that is, the Kirillov model K(π, ψ) of π. In the case m = n one notes that S0 (kn ) = {Φ ∈ S(kn ) | Φ(0) = 0}, which has co-dimension one in S(kn ), is isomorphic to the compactly induced

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113

−1/2

n (δPn ) so that by Frobenius reciprocity a GLn (k) representation indPn (k) quasi-invariant trilinear form on W(π, ψ) × W(π ′ , ψ −1 ) × S0 (kn ) reduces to a Pn (k)-quasi-invariant bilinear form on K(π, ψ) × K(π ′ , ψ −1 ). So in both cases we are naturally working in the restriction to Pn (k). The restrictions of irreducible representations of GLn (k) to Pn (k) are no longer irreducible, but do have composition series of finite length. One of the tools for analyzing the restrictions of representations of GLn to Pn , or analyzing the irreducible representations of Pn , are the derivatives of Bernstein and Zelevinsky [2,11]. These derivatives π (n−r) are naturally representations of GLr (k) for r ≤ n. π (0) = π and since π is generic the highest derivative π (n) corresponds to the irreducible common submodule (τ, Vτ ) of all Kirillov models, and is hence the non-zero irreducible representation of GL0 (k). The poles of our local integrals can be interpreted as giving quasi-invariant pairings between derivatives of π and π ′ [11]. The s for which such pairings exist for all but the highest derivatives are the exceptional s of the proposition. There is always a unique pairing between the highest derivatives π (n) and π ′(m) , which are necessarily non-zero since they correspond to the common irreducible subspace (τ, Vτ ) of any Kirillov model, and this is the unique Bs or Ts of the proposition. As a consequence of this Proposition, we can define the local γ-factor which gives the local functional equation for our integrals.

Theorem 3.2 There is a rational function γ(s, π × π ′ , ψ) ∈ C(q −s ) such that we have e − s; ρ(wn,m )W f, W f ′ ) = ω ′ (−1)n−1 γ(s, π × π ′ , ψ)Ψ(s; W, W ′ ) Ψ(1

if m < n

or

f, W f ′ , Φ) ˆ = ω ′ (−1)n−1 γ(s, π × π ′ , ψ)Ψ(s; W, W ′ , Φ) Ψ(1 − s; W

if m = n

for all W ∈ W(π, ψ), W ′ ∈ W(π ′ , ψ −1 ), and if necessary all Φ ∈ S(kn ). Again, if π ′ = 1 is the trivial representation of GL1 (k) we write γ(s, π, ψ) = γ(s, π × 1, ψ). The fact that γ(s, π × π ′ , ψ) is rational follows from the fact that it is a ratio of local integrals. An equally important local factor, which occurs in the current formulations of the local Langlands correspondence [23, 26], is the local ε-factor.

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Definition The local factor ε(s, π × π ′ , ψ) is defined as the ratio ε(s, π × π ′ , ψ) =

γ(s, π × π ′ , ψ)L(s, π × π ′ ) . L(1 − s, π e×π e′ )

With the local ε-factor the local functional equation can be written in the form e − s; ρ(wn,m )W f, W f′) Ψ(1 Ψ(s; W, W ′ ) ′ n−1 ′ = ω (−1) ε(s, π ×π , ψ) L(1 − s, π e×π e′ ) L(s, π × π ′ )

if m < n

or

f, W f ′ , Φ) ˆ Ψ(s; W, W ′ , Φ) Ψ(1 − s; W ′ n−1 ′ = ω (−1) ε(s, π × π , ψ) L(1 − s, π e×π e′ ) L(s, π × π ′ )

if m = n .

This can also be expressed in terms of the e(s; W, W ′ ), etc.. In fact, since we know we can choose a finite set of Wi , Wi′ , and if necessary Φi so that X Ψ(s, ; Wi , W ′ )

=

X Ψ(s; Wi , W ′ , Φi )

=

i

i

or

L(s, π × π ′ )

i

i

L(s, π × π ′ )

X

e(s; Wi , Wi′ ) = 1

i

X

e(s; Wi , Wi′ , Φi ) = 1

i

we see that we can write either ε(s, π × π ′ , ψ) = ω ′ (−1)n−1

X i

or ε(s, π × π ′ , ψ) = ω ′ (−1)n−1

fi , W f′) e˜(1 − s; ρ(wn,m )W i

X i

π ′ , ψ)

C[q s , q −s ].

and hence ε(s, π × ∈ functional equation twice we get

fi , W f′ , Φ ˆ i) e(1 − s; W i

On the other hand, applying the

ε(s, π × π ′ , ψ)ε(1 − s, π e×π e′ , ψ −1 ) = 1

so that ε(s, π × π ′ , ψ) is a unit in C[q s , q −s ]. This can be restated as: Proposition 3.5 ε(s, π × π ′ , ψ) is a monomial function of the form cq −f s .

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Let us make a few remarks on the meaning of the number f occurring in the ε–factor in the case of a single representation. Assume that ψ is unramified. In this case write ε(s, π, ψ) = ε(0, π, ψ)q −f (π)s . In [34] it is shown that f (π) is a non-negative integer, f (π) = 0 iff π is unramified, that in general the space of vectors in Vπ which is fixed by the compact open subgroup f (π) ∗ K1 (p ) = g ∈ GLn (o) g ≡ 0 ···

∗ .. . ∗ 0 1

f (π) (mod p )

has dimension exactly 1, and that if t < f (π) then the dimension of the space of fixed vectors for K1 (pt ) is 0. Depending on the context, either the integer f (π) or the ideal f(π) = pf (π) is called the conductor of π. Note that the analytically defined ε-factor carries structural information about π. 3.1.3

The unramified calculation

Let us now turn to the calculation of the local L-functions. The first case to consider is that where both π and π ′ are unramified. Since they are assumed generic, they are both full induced representations from unramified n characters of the Borel subgroup [69]. So let us write π ≃ IndGL Bn (µ1 ⊗ ′ ′ ′ m · · · ⊗ µn ) and π ′ ≃ IndGL Bm (µ1 ⊗ · · · ⊗ µm ) with the µi and µj unramified characters of k× . The Satake parameterization of unramified representations associates to each of these representations the semi-simple conjugacy classes [Aπ ] ∈ GLn (C) and [Aπ′ ] ∈ GLm (C) given by

Aπ =

µ1 (̟) ..

. µn (̟)

Aπ′ =

µ′1 (̟)

..

.

. ′ µm (̟)

(Recall that ̟ is a uniformizing parameter for k, that is, a generator of p.) In the Whittaker models there will be unique normalized K = GL(o)– fixed Whittaker functions, W◦ ∈ W(π, ψ) and W◦′ ∈ W(π ′ , ψ −1 ), normalized by W◦ (e) = W◦′ (e) = 1. Let us concentrate on W◦ for the moment. Since this function is right Kn –invariant and transforms on the left by ψ under Nn we have that its values are completely determined by its values on diagonal

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matrices of the form

̟ j1

̟J =

..

. ̟ jn

for J = (j1 , . . . , jn ) ∈ Zn . There is an explicit formula for W◦ (̟ J ) in terms of the Satake parameter Aπ due to Shintani [63] for GLn and generalized to arbitrary reductive groups by Casselman and Shalika [4]. Let T + (n) be the set of n–tuples J = (j1 , . . . , jn ) ∈ Zn with j1 ≥ · · · ≥ jn . Let ρJ be the rational representation of GLn (C) with dominant weight ΛJ defined by t1 j .. ΛJ = t11 · · · tjnn . . tn

Then the formula of Shintani says that ( 0 W◦ (̟ J ) = 1/2 δBn (̟ J ) tr(ρJ (Aπ ))

if J ∈ / T + (n) if J ∈ T + (n)

under the assumption that ψ is unramified. This is proved by analyzing the recursion relations coming from the action of the unramified Hecke algebra on W◦ . We have a similar formula for W◦′ (̟ J ) for J ∈ Zm . If we use these formulas in our local integrals, we find [36, I, Prop. 2.3] J X ̟ ′ Ψ(s; W◦ , W◦ ) = W◦ W◦′ (̟ J ) I n−m + J∈T (m), jm ≥0

−1 (̟ J ) · | det(̟ J )|s−(n−m)/2 δB m X = tr(ρ(J,0) (Aπ )) tr(ρJ (Aπ′ ))q −|J|s J∈T + (m), jm ≥0

=

X

tr(ρ(J,0) (Aπ ) ⊗ ρJ (Aπ′ ))q −|J|s

J∈T + (m), jm ≥0

where we let |J| = j1 +· · ·+jm and we embed Zm ֒→ Zn by J = (j1 , · · · , jm ) 7→ (J, 0) = (j1 , · · · , jm , 0, · · · , 0). We now use the fact from invariant theory that X tr(ρ(J,0) (Aπ ) ⊗ ρJ (Aπ′ )) = tr(S r (Aπ ⊗ Aπ′ )), J∈T + (m), jm ≥0, |J|=r

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where S r (A) is the r th -symmetric power of the matrix A, and ∞ X

tr(S r (A))z r = det(I − Az)−1

r=0

for any matrix A. Then we quickly arrive at Ψ(s; W◦ , W◦′ ) = det(I − q −s Aπ ⊗ Aπ′ )−1 =

Y (1 − µi (̟)µ′j (̟)q −s )−1 i,j

a standard Euler factor of degree mn. Since the L-function cancels all poles of the local integrals, we know at least that det(I − q −s Aπ ⊗ Aπ′ ) divides L(s, π × π ′ )−1 . Either of the methods discussed below for the general calculation of local factors then shows that in fact these are equal. There is a similar calculation when n = m and Φ = Φ◦ is the characteristic function of the lattice on ⊂ kn . Also, since π unramified implies that f◦ as its normalized unramified its contragredient π e is also unramified, with W Whittaker function, then from the functional equation we can conclude that in this situation we have ε(s, π × π ′ , ψ) ≡ 1. Theorem 3.3 If π, π ′ , and ψ are all unramified, then ( Ψ(s; W◦ , W◦′ ) −1 ′ −s L(s, π × π ) = det(I − q Aπ ⊗ Aπ′ ) = Ψ(s; W◦ , W◦′ , Φ◦ )

m 0 if π and π ′ are tempered. The meromorphic continuation and “bounded denominator” statement in the case of a non-archimedean local field is now replaced by the following. Define M(π × π ′ ) to be the space of all meromorphic functions φ(s) with the property that if P (s) is a polynomial function such that P (s)L(s, π × π ′ ) is holomorphic in a vertical strip S[a, b] = {s a ≤ Re(s) ≤ b} then P (s)φ(s) is bounded in S[a, b]. Note in particular that if φ ∈ M(π × π ′ ) then the quotient φ(s)L(s, π × π ′ )−1 is entire. Theorem 3.5 The integrals Ψj (s; W, W ′ ) or Ψ(s; W, W ′ , Φ) extend to meromorphic functions of s which lie in M(π × π ′ ). In particular, the ratios ej (s; W, W ′ ) =

Ψj (s; W, W ′ ) L(s, π × π ′ )

or

e(s; W, W ′ , Φ) =

Ψ(s; W, W ′ , Φ) L(s, π × π ′ )

are entire and in fact are bounded in vertical strips. This statement has more content than just the continuation and “bounded denominator” statements in the non-archimedean case. Since it prescribes the “denominator” to be the L factor L(s, π × π ′ )−1 it is bound up with the actual computation of the poles of the local integrals. In fact, a significant part of the paper of Jacquet and Shalika [38] is taken up with the simultaneous proof of this and the local functional equations:

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Theorem 3.6 We have the local functional equations f, W f ′ ) = ω ′ (−1)n−1 γ(s, π × π ′ , ψ)Ψj (s; W, W ′ ) Ψn−m−j−1 (1 − s; ρ(wn,m )W

or

f, W f ′ , Φ) ˆ = ω ′ (−1)n−1 γ(s, π × π ′ , ψ)Ψ(s; W, W ′ , Φ). Ψ(1 − s; W

The one fact that we are missing is the statement of “minimality” of the L-factor. That is, we know that L(s, π × π ′ ) is a standard archimedean Euler factor (i.e., a product of Γ-functions of the standard type) and has the property that the poles of all the local integrals are contained in the poles of the L-factor, even with multiplicity. But we have not established that the L-factor cannot have extraneous poles. In particular, we do know that we can achieve the local L-function as a finite linear combination of local integrals. Towards this end, Jacquet and Shalika enlarge the allowable space of local integrals. Let Λ and Λ′ be the Whittaker functionals on Vπ and Vπ′ associated ˆ = Λ ⊗ Λ′ with the Whittaker models W(π, ψ) and W(π ′ , ψ −1 ). Then Λ defines a continuous linear functional on the algebraic tensor product Vπ ⊗ Vπ′ which extends continuously to the topological tensor product Vπ⊗π′ = ˆ π′ , viewed as representations of GLn (k) × GLm (k). Vπ ⊗V Before proceeding, let us make a few remarks on smooth representations. If (π, Vπ ) is the space of smooth vectors in an irreducible admissible unitary representation, then the underlying Harish-Chandra module is the space of Kn -finite vectors Vπ,K . Vπ then corresponds to the (Casselman-Wallach) canonical completion of Vπ,K [66]. The category of Harish-Chandra modules is appropriate for the algebraic theory of representations, but it is useful to work in the category of smooth admissible representations for automorphic forms. If in our context we take the underlying Harish-Chandra modules Vπ,K and Vπ′ ,K then their algebraic tensor product is an admissible HarishChandra module for GLn (k) × GLm (k). The associated smooth admissible representation is the canonical completion of this tensor product, which is in fact Vπ⊗π′ , the topological tensor product of the smooth representations π and π ′ . It is also the space of smooth vectors in the unitary tensor product of the unitary representations associated to π and π ′ . So this completion is a natural place to work in the category of smooth admissible representations. Now let ˆ W(π ⊗ π ′ , ψ) = {W (g, h) = Λ(π(g) ⊗ π ′ (h)ξ)|ξ ∈ Vπ⊗π′ }.

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Then W(π⊗π ′ , ψ) contains the algebraic tensor product W(π, ψ)⊗W(π ′ , ψ −1 ) and is again equal to the topological tensor product. Now we can extend all our local integrals to the space W(π ⊗ π ′ , ψ) by setting Z Z h , h dx | det(h)|s−(n−m)/2 dh Ψj (s; W ) = W x Ij In−m−j and Ψ(s; W, Φ) =

Z

W (g, g)Φ(en g)| det(g)|s dh

for W ∈ W(π ⊗ π ′ , ψ). Since the local integrals are continuous with respect to the topology on the topological tensor product, all of the above facts remain true, in particular the convergence statements, the local functional equations, and the fact that these integrals extend to functions in M(π ×π ′ ). At this point, let Ij (π, π ′ ) = {Ψj (s; W )|W ∈ W(π ⊗ π ′ )} and let I(π, π ′ ) be the span of the local integrals {Ψ(s; W, Φ)|W ∈ W(π ⊗π ′ , ψ), φ ∈ S(kn )}. Once again, in the case m < n we have that the space Ij (π, π ′ ) is independent of j and we denote it also by I(π, π ′ ). These are still independent of ψ. So we know from above that I(π, π ′ ) ⊂ M(π × π ′ ). The remainder of [38] is then devoted to showing the following. Theorem 3.7 I(π, π ′ ) = M(π × π ′ ). As a consequence of this, we draw the following useful Corollary. Corollary There is a Whittaker function W in W(π ⊗ π ′ , ψ) such that L(s, π × π ′ ) = Ψ(s; W ) if m < n or finite collectionof functions Wi ∈ W(π ⊗ P π ′ , ψ) and Φi ∈ S(kn ) such that L(s, π × π ′ ) = i Ψ(s; Wi , Φi ) if m = n.

In the cases of m = n − 1 or m = n, Jacquet and Shalika can indeed get the local L-function as a finite linear combination of integrals involving only K-finite functions in W(π, ψ) and W(π ′ , ψ −1 ), that is, without going to the completion of W(π, ψ) ⊗ W(π ′ , ψ −1 ), but this has not been published. As a final result, let us note that in [12] it is established that the linear functionals e(s; W ) = Ψ(s; W )L(s, π × π ′ )−1 and e(s; W, Φ) = Ψ(s; W, Φ) L(s, π × π ′ )−1 are continuous on W(π ⊗ π ′ , ψ), uniformly for s in compact sets. Since there is a choice of W ∈ W(π ⊗ π ′ , ψ) such that e(s; W ) ≡ 1 P or Wi ∈ W(π ⊗ π ′ , ψ) and Φi ∈ S(kn ) such that e(s; Wi , Φi ) ≡ 1, as a result of this continuity and the fact that the algebraic tensor product W(π, ψ) ⊗ W(π ′ , ψ −1 ) is dense in W(π ⊗ π ′ , ψ) we have the following result.

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Proposition 3.8 For any s0 ∈ C there are choices of W ∈ W(π, ψ), W ′ ∈ W(π ′ , ψ −1 ) and if necessary Φ such that e(s0 ; W, W ′ ) 6= 0 or e(s0 ; W, W ′ , Φ) 6= 0.

4

Global L-functions

Once again, we let k be a global field, A its ring of adeles, and fix a non-trivial continuous additive character ψ = ⊗ψv of A trivial on k. Let (π, Vπ ) be a cuspidal representation of GLn (A) (see Section 1 for all the implied assumptions in this terminology) and (π ′ , Vπ′ ) a cuspidal representation of GLm (A). Since they are irreducible we have restricted tensor product decompositions π ≃ ⊗′ πv and π ′ ≃ ⊗′ πv′ with (πv , Vπv ) and (πv′ , Vπv′ ) irreducible admissible smooth generic unitary representations of GLn (kv ) and GLm (kv ) [14,18]. Let ω = ⊗′ ωv and ω ′ = ⊗′ ωv′ be their central characters. These are both continuous characters of k× \A× . Let S be the finite set of places of k, containing the archimedean places S∞ , such that for all v ∈ / S we have that πv , πv′ , and ψv are unramified. For each place v of k we have defined the local factors L(s, πv × πv′ ) and ε(s, πv × πv′ , ψv ). Then we can at least formally define Y Y L(s, π × π ′ ) = L(s, πv × πv′ ) and ε(s, π × π ′ ) = ε(s, πv × πv′ , ψv ). v

v

We need to discuss convergence of these products. Let us first consider the convergence of L(s, π × π ′ ). For those v ∈ / S, so πv , πv′ , and ψv are ′ unramified, we know that L(s, πv × πv ) = det(I − qv−s Aπv ⊗ Aπv′ )−1 and that 1/2

the eigenvalues of Aπv and Aπv′ are all of absolute value less than qv . Thus the partial (or incomplete) L-function Y Y LS (s, π × π ′ ) = L(s, πv × πv′ ) = det(I − q −s Aπv ⊗ Aπv′ )−1 v∈S /

v∈S /

is absolutely convergent for Re(s) >> 0. Thus the same is true for L(s, π × π ′ ). For the ε–factor, we have seen that ε(s, πv × πv′ , ψv ) ≡ 1 for v ∈ / S so that the product is in fact a finite product and there is no problem with convergence. The fact that ε(s, π × π ′ ) is independent of ψ can either be checked by analyzing how the local ε–factors vary as you vary ψ, as is done in [7, Lemma 2.1], or it will follow from the global functional equation presented below.

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The Basic Analytic Properties

Our first goal is to show that these L-functions have nice analytic properties. Theorem 4.1 The global L–functions L(s, π × π ′ ) are nice in the sense that 1. L(s, π × π ′ ) has a meromorphic continuation to all of C, 2. the extended function is bounded in vertical strips (away from its poles), 3. they satisfy the functional equation L(s, π × π ′ ) = ε(s, π × π ′ )L(1 − s, π e×π e′ ).

To do so, we relate the L-functions to the global integrals. Let us begin with continuation. In the case m < n for every ϕ ∈ Vπ and ′ ϕ ∈ Vπ′ we know the integral I(s; ϕ, ϕ′ ) converges absolutely for all s. From the unfolding in Section 2 and the local calculation of Section 3 we know that for Re(s) >> 0 and for appropriate choices of ϕ and ϕ′ we have Y I(s; ϕ, ϕ′ ) = Ψv (s; Wϕv , Wϕ′v ) v

=

Y

!

Ψv (s; Wϕv , Wϕ′v ) LS (s, π × π ′ )

v∈S

= =

! Y Ψv (s; Wϕv , Wϕ′ ) v L(s, π × π ′ ) L(s, πv × πv′ ) v∈S ! Y ev (s; Wϕv , Wϕ′v ) L(s, π × π ′ ) v∈S

We know that each ev (s; Wv , Wv′ ) is entire. Hence L(s, π × π ′ ) has a meromorphic continuation. If m = n then for appropriate ϕ ∈ Vπ , ϕ′ ∈ Vπ′ , and Φ ∈ S(An ) we again have ! Y ′ ′ ev (s; Wϕv , Wϕ′v , Φv ) L(s, π × π ′ ). I(s; ϕ, ϕ , Φ) = v∈S

Once again, since each ev (s; Wv , Wv′ , Φv ) is entire, L(s, π × π ′ ) has a meromorphic continuation.

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Let us next turn to the functional equation. This will follow from the functional equation for the global integrals and the local functional equations. We will consider only the case where m < n since the other case is entirely analogous. The functional equation for the global integrals is simply ˜ − s; ϕ, I(s; ϕ, ϕ′ ) = I(1 e ϕ e′ ).

Once again we have for appropriate ϕ and ϕ′ ′

I(s; ϕ, ϕ ) =

Y

!

ev (s; Wϕv , Wϕ′ ′v )

L(s, π × π ′ )

v∈S

while on the other side ˜ − s; ϕ, I(1 e ϕ e′ ) =

Y

!

v∈S

fϕv , W f ′ ′ ) L(1 − s, π e˜v (1 − s; ρ(wn,m )W e×π e′ ). ϕv

However, by the local functional equations, for each v ∈ S we have e f f′ fv , W fv′ ) = Ψ(1 − s; ρ(wn,m )Wv , Wv ) e˜v (1 − s; ρ(wn,m )W L(1 − s, π e×π e′ )

Ψ(s; Wv , Wv′ ) L(s, π × π ′ ) ′ n−1 ′ = ωv (−1) ε(s, πv × πv , ψv )ev (s, Wv , Wv′ )

= ωv′ (−1)n−1 ε(s, πv × πv′ , ψv )

Combining these, we have ′

L(s, π × π ) =

Y

ωv′ (−1)n−1 ε(s, πv

×

!

πv′ , ψv )

v∈S

L(1 − s, π e×π e′ ).

Now, for v ∈ / S we know that πv′ is unramified, so ωv′ (−1) = 1, and also that ′ ε(s, πv × πv , ψv ) ≡ 1. Therefore Y Y ωv′ (−1)n−1 ε(s, πv × πv′ , ψv ) = ωv′ (−1)n−1 ε(s, πv × πv′ , ψv ) v ′

v∈S

= ω (−1)n−1 ε(s, π × π ′ ) = ε(s, π × π ′ ) and we indeed have L(s, π × π ′ ) = ε(s, π × π ′ )L(1 − s, π e×π e′ ).

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Note that this implies that ε(s, π × π ′ ) is independent of ψ as well. Let us now turn to the boundedness in vertical strips. For the global integrals I(s; ϕ, ϕ′ ) or I(s; ϕ, ϕ, Φ) this simply follows from the absolute convergence. For the L-function itself, the paradigm is the following. For every ′ , and Φ finite place v ∈ S we know that there is a choice of Wv,i , Wv,i v,i if necessary such that L(s, πv × πv′ ) = L(s, πv × πv′ ) =

X

X

Ψ(s; Wv,i , Wv′ ′ i )

or

Ψ(s; Wv,i , Wv′ ′ i , Φv,i ).

If m = n − 1 or m = n then by the unpublished work of Jacquet and Shalika mentioned toward the end of Section 3 we know that we have similar statements for v ∈ S∞ . Hence if m = n − 1 or m = n there are global choices ϕi , ϕ′i , and if necessary Φi such that L(s, π × π ′ ) =

X

I(s; ϕi , ϕ′i )

or

L(s, π × π ′ ) =

X

I(s; ϕi , ϕ′i , Φi ).

Then the boundedness in vertical strips for the L-functions follows from that of the global integrals. However, if m < n − 1 then all we know at those v ∈ S∞ is that there ′ −1 ˆ is a function Wv ∈ W(πv ⊗ πv′ , ψv ) = W(πv , ψv )⊗W(π v , ψv ) or a finite collection of such functions Wv,i and of Φv,i such that L(s, πv × πv′ ) = I(s; Wv )

or

L(s, πv × πv′ ) =

X

I(s; Wv,i , Φv,i ).

To make the above paradigm work for m < n − 1 we would need to rework ˆ π′ . This is the theory of global Eulerian integrals for cusp forms in Vπ ⊗V naturally the space of smooth vectors in an irreducible unitary cuspidal representation of GLn (A) × GLm (A). So we would need extend the global theory of integrals parallel to Jacquet and Shalika’s extension of the local integrals in the archimedean theory. There seems to be no obstruction to carrying this out, and then we obtain boundedness in vertical strips for L(s, π × π ′ ) in general. We should point out that if one approaches these L-functions by the method of constant terms and Fourier coefficients of Eisenstein series, then Gelbart and Shahidi have shown a wide class of automorphic L-functions, including ours, to be bounded in vertical strips [17].

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4.2

Poles of L-functions

Let us determine where the global L-functions can have poles. The poles of the L-functions will be related to the poles of the global integrals. Recall from Section 2 that in the case of m < n we have that the global integrals I(s; ϕ, ϕ′ ) are entire and that when m = n then I(s; ϕ, ϕ′ , Φ) can have at most simple poles and they occur at s = −iσ and s = 1 − iσ for σ real when π≃π e′ ⊗ | det |iσ . As we have noted above, the global integrals and global L-functions are related, for appropriate ϕ, ϕ′ , and Φ, by ! Y ev (s; Wϕv , Wϕ′ ′v ) L(s, π × π ′ ) I(s; ϕ, ϕ′ ) = v∈S

or I(s; ϕ, ϕ′ , Φ) =

Y

!

ev (s; Wϕv , Wϕ′ ′v , Φv ) L(s, π × π ′ ).

v∈S

On the other hand, we have seen that for any s0 ∈ C and any v there is a choice of local Wv , Wv′ , and Φv such that the local factors ev (s0 ; Wv , Wv′ ) 6= 0 or ev (s0 ; Wv , Wv′ , Φv ) 6= 0. So as we vary ϕ, ϕ′ and Φ at the places v ∈ S we see that division by these factors can introduce no extraneous poles in L(s, π × π ′ ), that is, in keeping with the local characterization of the Lfactor in terms of poles of local integrals, globally the poles of L(s, π × π ′ ) are precisely the poles of the family of global integrals {I(s; ϕ, ϕ′ )} or {I(s; ϕ, ϕ′ , Φ)}. Hence from Theorems 2.1 and 2.2 we have. Theorem 4.2 If m < n then L(s, π × π ′ ) is entire. If m = n, then L(s, π × π ′ ) has at most simple poles and they occur iff π ≃ π e′ ⊗ | det |iσ with σ real and are then at s = −iσ and s = 1 − iσ. If we apply this with π ′ = π e we obtain the following useful corollary.

Corollary L(s, π × π e) has simple poles at s = 0 and s = 1.

4.3

Strong Multiplicity One

Let us return to the Strong Multiplicity One Theorem for cuspidal representations. First, recall the statement: Theorem (Strong Multiplicity One) Let (π, Vπ ) and (π ′ , Vπ′ ) be two cuspidal representations of GLn (A). Suppose there is a finite set of places S such that for all v ∈ / S we have πv ≃ πv′ . Then π = π ′ .

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We will now present Jacquet and Shalika’s proof of this statement via Lfunctions [36]. First note the following observation, which follows from our analysis of the location of the poles of the L-functions. Observation For π and π ′ cuspidal representations of GLn (A), L(s, π×e π′ ) has a pole at s = 1 iff π ≃ π ′ . Thus the L-function gives us an analytic method of testing when two cuspidal representations are isomorphic, and so by the Multiplicity One Theorem, the same. Proof: If we take π and π ′ as in the statement of Strong Multiplicity One, we have that πv ≃ πv′ for v ∈ / S and hence Y Y e′ ) LS (s, π × π e) = L(s, πv × π ev ) = L(s, πv × π ev′ ) = LS (s, π × π v∈S /

v∈S /

Since the local L-factors never vanish and for unitary representations they have no poles in Re(s) ≥ 1 (since the local integrals have no poles in this region) we see that for s = 1 that L(s, π × π e′ ) has a pole at s = 1 iff S ′ L (s, π × π e ) does. Hence we have that since L(s, π × π e) has a pole at S S S s = 1, so does L (s, π × π e). But L (s, π × π e) = L (s, π × π e′ ), so that S ′ ′ both L (s, π × π e ) and then L(s, π × π e ) have poles at s = 1. But then the L-function criterion above gives that π ≃ π ′ . Now apply Multiplicity One. 2 In fact, Jacquet and Shalika push this method much further. If π is an irreducible automorphic representation of GLn (A), but not necessarily cuspidal, then it is a theorem of Langlands [44] that there are cuspidal representations, say τ1 , . . . , τr of GLn1 , . . . , GLnr with n = n1 + · · ·+ nr , such that π is a constituent of Ind(τ1 ⊗ · · · ⊗ τr ). Similarly, π ′ is a constituent of Ind(τ1′ ⊗ · · · ⊗ τr′ ′ ). Then the generalized version of the Strong Multiplicity One theorem that Jacquet and Shalika establish in [36] is the following. Theorem (Generalized Strong Multiplicity One) Given π and π ′ irreducible automorphic representations of GLn (A) as above, suppose that / S. Then there is a finite set of places S such that πv ≃ πv′ for all v ∈ r = r ′ and there is a permutation σ of the set {1, . . . , r} such that ni = n′σ(i) ′ . and τi = τσ(i) Note, the cuspidal representations τi and τi′ need not be unitary in this statement.

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Non-vanishing Results

Of interest for questions from analytic number theory, for example questions of equidistribution, are results on the non-vanishing of L-functions. The fundamental non-vanishing result for GLn is the following theorem of Jacquet and Shalika [35] and Shahidi [56, 57]. Theorem 4.3 Let π and π ′ be cuspidal representations of GLn (A) and GLm (A). Then the L-function L(s, π × π ′ ) is non-vanishing for Re(s) ≥ 1. The proof of non-vanishing for Re(s) > 1 is in keeping with the spirit of these notes [36, I, Theorem 5.3]. Since the local L-functions are never zero, to establish the non-vanishing of the Euler product for Re(s) > 1 it suffices to show that the Euler product is absolutely convergent for Re(s) > 1, and for this it is sufficient to work with the incomplete L-function LS (s, π × π ′ ) where S is as at the beginning of this Section. Then we can write Y Y LS (s, π × π ′ ) = L(s, πv × πv′ ) = det(I − qv−s Aπv ⊗ Aπv′ )−1 v∈S /

v∈S /

with absolute convergence for Re(s) >> 0. Q Recall that an infinite product (1 + an ) is absolutely convergent iff the P associated series log(1 + an ) is absolutely convergent. Let us write ′ µv,1 µv,1 .. .. and Aπv′ = Aπv = . . . µ′v,m

µv,n

1/2

1/2

and |µ′v,j | < qv . Then X log L(s, πv × πv′ ) = − log(1 − µv,i µ′v,j qv−s )

We have seen that |µv,i | < qv

i,j

=

∞ XX (µv,i µ′v,j )d i,j d=1

dqvds

=

∞ X tr(Adπv ) tr(Adπ′ ) v

d=1

dqvds

with the sum absolutely convergent for Re(s) >> 0. Then, still for Re(s) >> 0, ∞ tr(Ad ) tr(Ad ) XX πv πv′ . log(LS (s, π × π ′ )) = ds dqv v∈S / d=1

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131

e we find If we apply this to π ′ = π = π log(LS (s, π × π)) =

∞ XX | tr(Adπv )|2 dqvds v∈S / d=1

which is a Dirichlet series with non-negative coefficients. By Landau’s Lemma this will be absolutely convergent up to the its first pole, which we know is at s = 1. Hence this series, and the associated Euler product L(s, π × π e), is absolutely convergent for Re(s) > 1. An application of the Cauchy–Schwatrz inequality then implies that the series ∞ XX tr(Adπv ) tr(Adπv′ ) log(LS (s, π × π ′ )) = dqvds v∈S / d=1

is also absolutely convergent for Re(s) > 1. Thus L(s, π × π ′ ) is absolutely convergent and hence non-vanishing for Re(s) > 1. To obtain the non-vanishing on the line Re(s) = 1 requires the technique of analyzing L-functions via their occurrence in the constant terms and Fourier coefficients of Eisenstein series, which we have not discussed. They can be found in the references [35] and [56, 57] mentioned above.

4.5

The Generalized Ramanujan Conjecture (GRC)

The current version of the GRC is a conjecture about the structure of cuspidal representations. Conjecture (GRC) Let π be a (unitary) cuspidal representation of GLn (A) with decomposition π ≃ ⊗′ πv . Then the local components πv are tempered representations. However, it has an interesting interpretation in terms of L-functions which is more in keeping with the origins of the conjecture. If π is cuspidal, then at every finite place v where π v is unramified we have associated µv,1 .. a semisimple conjugacy class, say Aπv = so that . µv,n

L(s, πv ) = det(I −

qv−s Aπv )−1

n Y (1 − µv,i qv−s )−1 . = i=1

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J.W. Cogdell

If v is an archimedean place where πv is unramified, then we can similarly write n Y Γv (s + µv,i ) L(s, π) = i=1

where

( π −s/2 Γ( 2s ) Γv (s) = 2(2π)−s Γ(s)

if kv ≃ R . if kv ≃ C

Then the statement of the GRC in these terms is Conjecture (GRC for L-functions) If π is a cuspidal representation of GLn (A) and if v is a place where πv is unramified, then |µv,i | = 1 for v non-archimedean and Re(µv,i ) = 0 for v archimedean. Note that by including the archimedean places, this conjecture encompasses not only the classical Ramanujan conjectures but also the various versions of the Selberg eigenvalue conjecture [27]. −1/2 Recall that by the Corollary to Theorem 3.3 we have the bounds qv < 1/2 |µv,i | < qv for v non-archimedean, and a similar local analysis for v archimedean would give | Re(µv,i )| < 21 . The best bound for general GLn is due to Luo, Rudnick, and Sarnak [46]. They are the uniform bounds −( 21 −

qv

1 ) n2 +1

1 − 1 2 n2 +1

≤ |µv,i | ≤ qv

if v is non-archimedean

and

1 1 − for v archimedean. 2 n2 + 1 Their techniques are global and rely on the theory of Rankin–Selberg Lfunctions as presented here, a technique of persistence of zeros in families of L-functions, and a positivity argument. | Re(µv,i )| ≤

For GL2 there has been much recent progress. The best general estimates I am aware of at present are due to Kim and Shahidi [41], who use the holomorphy of the symmetric ninth power L-function for Re(s) > 1 to obtain − 91

qv

1

< |µv,i | < qv9

for i = 1, 2, and v non-archimedean,

and Kim and Sarnak, who obtain the analogous estimate for v archimedean (with possible equality) in the appendix to [39]. For some applications, the notion of weakly Ramanujan [8] can replace knowing the full GRC.

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Definition A cuspidal representation π of GLn (A) is called weakly Ramanujan if for every ǫ > 0 there is a constant cǫ > 0 and an infinite sequence of places {vm } with the property that each πvm is unramified and the Satake parameters µvm ,i satisfy −ǫ ǫ c−1 ǫ qvm < |µvm ,i | < cǫ qvm .

For example, if we knew that all cuspidal representations on GLn (A) were weakly Ramanujan, then we would know that under Langlands liftings between general linear groups, the property of occurrence in the spectral decomposition is preserved [8]. For n = 2, 3 our techniques let us show the following. Proposition 4.1 For n = 2 or n = 3 all cuspidal representations are weakly Ramanujan.

Proof: First, let π be a cuspidal representation or GLn (A). Recall that from the absolute convergence of the Euler product for L(s, π × π) we know X X | tr(Adπ )|2 v is absolutely convergent for Re(s) > 1, so that the series dqvds d v∈S / X | tr(Aπ )|2 v that in particular we have that is absolutely convergent for qvs v∈S /

Re(s) > 1. Thus, for a set of places of positive density, we have the estimate | tr(Aπv )|2 < qvǫ for each ǫ. Since Aπv = A−1 πv for components of cuspidal representations, we have the same estimate for | tr(A−1 πv )|. In the case of n = 2 and n = 3, these estimates and the fact that | det Aπv | = |ωv (̟v )| = 1 give us estimates on the coefficients of the characteristic polynomial for Aπv . For example, if n = 3 and the characteristic ǫ/2 polynomial of Aπv is X 3 + aX 2 + bX + c then we know |a| = | tr(Aπv )| < qv , ǫ/2 |b| = | tr(A−1 πv ) det(Aπv )| < qv , and |c| = | det(Aπv )| = 1. Then an application of Rouche’s theorem gives that the roots of this polynomial all lie in the circle of radius qvǫ as long as qv > 3. Applying this to both Aπv and A−1 πv we find that for our set primes of positive density above we have the estimate qv−ǫ < |µvm ,i | < qvǫ . Thus we find that for n = 2, 3 cuspidal representations of GLn are weakly Ramanujan. 2

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The Generalized Riemann Hypothesis (GRH)

This is one of the most important conjectures in the analytic theory of Lfunctions. Simply stated, it is Conjecture (GRH) For any cuspidal representation π, all the zeros of the L-function L(s, π) lie on the line Re(s) = 12 . Even in the simplest case of n = 1 and π = 1 the trivial representation this reduces to the Riemann hypothesis for the Riemann zeta function! For an interesting survey on these and other conjectures on L-functions and their relation to number theoretic problems, we refer the reader to the survey of Iwaniec and Sarnak [27].

5

Converse Theorems

Let us return first to Hecke. Recall that to a modular form f (τ ) =

∞ X

an e2πinτ

n−1

for, say, SL2 (Z) Hecke attached an L function L(s, f ) and they were related via the Mellin transform Z ∞ f (iy)y s d× y Λ(s, f ) = (2π)−s Γ(s)L(s, f ) = 0

and derived the functional equation for L(s, f ) from the modular transformation law for f (τ ) under the modular transformation law for the transformation τ 7→ −1/τ . In his fundamental paper [24] he inverted this process by taking a Dirichlet series D(s) =

∞ X an n=1

ns

and assuming that it converged in a half plane, had an entire continuation to a function of finite order, and satisfied the same functional equation as the L-function of a modular form of weight k, then he could actually reconstruct a modular form from D(s) by Mellin inversion Z 2+i∞ X 1 −2πny f (iy) = an e = (2π)−s Γ(s)D(s)y s ds 2πi 2−i∞ i

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and obtain the modular transformation law for f (τ ) under τ 7→ −1/τ from the functional equation for D(s) under s 7→ k − s. This is Hecke’s Converse Theorem. In this Section we will present some analogues of Hecke’s theorem in the context of L-functions for GLn . Surprisingly, the technique is exactly the same as Hecke’s, i.e., inverting the integral representation. This was first done in the context of automorphic representation for GL2 by Jacquet and Langlands [30] and then extended and significantly strengthened for GL3 by Jacquet, Piatetski-Shapiro, and Shalika [31]. For a more extensive bibliography and history, see [10]. This section is taken mainly from our survey [10]. Further details can be found in [7, 9].

5.1

The Results

Once again, let k be a global field, A its adele ring, and ψ a fixed non-trivial continuous additive character of A which is trivial on k. We will take n ≥ 3 to be an integer. To state these Converse Theorems, we begin with an irreducible admissible representation Π of GLn (A). In keeping with the conventions of these notes, we will assume that Π is unitary and generic, but this is not necessary. It has a decomposition Π = ⊗′ Πv , where Πv is an irreducible admissible generic representation of GLn (kv ). By the local theory of Section 3, to each Πv is associated a local L-function L(s, Πv ) and a local ε-factor ε(s, Πv , ψv ). Hence formally we can form Y Y L(s, Π) = L(s, Πv ) and ε(s, Π, ψ) = ε(s, Πv , ψv ). v

v

We will always assume the following two things about Π: 1. L(s, Π) converges in some half plane Re(s) >> 0, 2. the central character ωΠ of Π is automorphic, that is, invariant under k× . Under these assumptions, ε(s, Π, ψ) = ε(s, Π) is independent of our choice of ψ [7]. Our Converse Theorems will involve twists by cuspidal automorphic representations of GLm (A) for certain m. For convenience, let us set A(m)

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to be the set of automorphic representations of GLm (A), A0 (m) the set of m a cuspidal representations of GLm (A), and T (m) = A0 (d). d=1

Let π ′ = ⊗′ π ′ v be a cuspidal representation of GLm (A) with m < n. Then again we can formally define Y Y L(s, Π×π ′ ) = L(s, Πv ×π ′ v ) and ε(s, Π×π ′ ) = ε(s, Πv ×π ′ v , ψv ) v

v

since again the local factors make sense whether Π is automorphic or not. A consequence of (1) and (2) above and the cuspidality of π ′ is that both e × πe′ ) converge absolutely for Re(s) >> 0, where L(s, Π × π ′ ) and L(s, Π e and πe′ are the contragredient representations, and that ε(s, Π × π ′ ) is Π independent of the choice of ψ. We say that L(s, Π × π ′ ) is nice if it satisfies the same analytic properties it would if Π were cuspidal, i.e., e × πe′ ) have analytic continuations to entire 1. L(s, Π × π ′ ) and L(s, Π functions of s, 2. these entire continuations are bounded in vertical strips of finite width, 3. they satisfy the standard functional equation e × πe′ ). L(s, Π × π ′ ) = ε(s, Π × π ′ )L(1 − s, Π

The basic Converse Theorem for GLn is the following.

Theorem 5.1 Let Π be an irreducible admissible representation of GLn (A) as above. Suppose that L(s, Π × π ′ ) is nice for all π ′ ∈ T (n − 1). Then Π is a cuspidal automorphic representation. In this theorem we twist by the maximal amount and obtain the strongest possible conclusion about Π. The proof of this theorem essentially follows that of Hecke [24] and Weil [67] and Jacquet–Langlands [30]. It is of course valid for n = 2 as well. For applications, it is desirable to twist by as little as possible. There are essentially two ways to restrict the twisting. One is to restrict the rank of the groups that the twisting representations live on. The other is to restrict ramification. When we restrict the rank of our twists, we can obtain the following result.

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Theorem 5.2 Let Π be an irreducible admissible representation of GLn (A) as above. Suppose that L(s, Π × π ′ ) is nice for all π ′ ∈ T (n − 2). Then Π is a cuspidal automorphic representation. This result is stronger than Theorem 5.1, but its proof is a bit more delicate. The theorem along these lines that is most useful for applications is one in which we also restrict the ramification at a finite number of places. Let us fix a finite set of S of finite places and let T S (m) denote the subset of T (m) consisting of representations that are unramified at all places v ∈ S. Theorem 5.3 Let Π be an irreducible admissible representation of GLn (A) as above. Let S be a finite set of finite places. Suppose that L(s, Π × π ′ ) is nice for all π ′ ∈ T S (n − 2). Then Π is quasi-automorphic in the sense that there is an automorphic representation Π′ such that Πv ≃ Π′v for all v ∈ / S. Note that as soon as we restrict the ramification of our twisting representations we lose information about Π at those places. In applications we usually choose S to contain the set of finite places v where Πv is ramified. The second way to restrict our twists is to restrict the ramification at all but a finite number of places. Now fix a non-empty finite set of places S which in the case of a number field contains the set S∞ of all archimedean places. Let TS (m) denote the subset consisting of all representations π ′ in T (m) which are unramified for all v ∈ / S. Note that we are placing a grave restriction on the ramification of these representations. Theorem 5.4 Let Π be an irreducible admissible representation of GLn (A) as above. Let S be a non-empty finite set of places, containing S∞ , such that the class number of the ring oS of S-integers is one. Suppose that L(s, Π×π ′ ) is nice for all π ′ ∈ TS (n − 1). Then Π is quasi-automorphic in the sense that there is an automorphic representation Π′ such that Πv ≃ Π′v for all v ∈ S and all v ∈ / S such that both Πv and Π′v are unramified. There are several things to note here. First, there is a class number restriction. However, if k = Q then we may take S = S∞ and we have a Converse Theorem with “level 1” twists. As a practical consideration, if we let SΠ be the set of finite places v where Πv is ramified, then for applications we usually take S and SΠ to be disjoint. Once again, we are losing all information at those places v ∈ / S where we have restricted the ramification unless Πv was already unramified there.

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The proof of Theorem 5.1 essentially follows the lead of Hecke, Weil, and Jacquet–Langlands. It is based on the integral representations of Lfunctions, Fourier expansions, Mellin inversion, and finally a use of the weak form of Langlands spectral theory. For Theorems 5.2, 5.3, and 5.4, where we have restricted our twists, we must impose certain local conditions to compensate for our limited twists. For Theorem 5.2 and 5.3 there are a finite number of local conditions and for Theorem 5.4 an infinite number of local conditions. We must then work around these by using results on generation of congruence subgroups and either weak or strong approximation.

5.2

Inverting the Integral Representation

Let Π be as above and let ξ ∈ VΠ be a decomposable vector in the space VΠ of Π. Since Π is generic, then fixing local Whittaker models W(Πv , ψv ) at all places, compatibly normalized at the unramified places, we can associate Q to ξ a non-zero function Wξ (g) = Wξv (gv ) on GLn (A) which transforms by the global character ψ under left translation by Nn (A), i.e., Wξ (ng) = ψ(n)Wξ (g). Since ψ is trivial on rational points, we see that Wξ (g) is left invariant under Nn (k). We would like to use Wξ to construct an embedding of VΠ into the space of (smooth) automorphic forms on GLn (A). The simplest idea is to average Wξ over Nn (k)\ GLn (k), but this will not be convergent. However, if we average over the rational points of the mirabolic P = Pn then the sum X Uξ (g) = Wξ (pg) Nn (k)\ P(k)

is absolutely convergent. For the relevant growth properties of Uξ see [7]. Since Π is assumed to have automorphic central character, we see that Uξ (g) is left invariant under both P(k) and the center Zn (k). Suppose now that we know that L(s, Π × π ′ ) is nice for all π ′ ∈ T (m). Then we will hope to obtain the remaining invariance of Uξ from the GLn × GLm functional equation by inverting the integral representation for L(s, Π× π ′ ). With this in mind, let Q = Qm be the mirabolic subgroup of GLn which stabilizes the standard unit vector tem+1 , that is, the column vector all of whose entries are 0 except the (m + 1)th , which is 1. Note that if m = n − 1 then Q is nothing more than the opposite mirabolic P =t P−1 to P. If we let

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αm be the permutation matrix in GLn (k) given by 1 αm = Im In−m−1 −1 then Qm = α−1 m αn−1 Pαn−1 αm is a conjugate of P and for any m we have that P(k) and Q(k) generate all of GLn (k). So now set X Vξ (g) = Wξ (αm qg) N′ (k)\ Q(k)

where N′ = α−1 m Nn αm ⊂ Q. This sum is again absolutely convergent and is invariant on the left by Q(k) and Z(k). Thus, to embed Π into the space of automorphic forms it suffices to show Uξ = Vξ , for the we get invariance of Uξ under all of GLn (k). It is this that we will attempt to do using the integral representations. Now let (π ′ , Vπ′ ) be an irreducible subrepresentation of the space of automorphic forms on GLm (A) and assume ϕ′ ∈ Vπ′ is also factorizable. Let Z h n ′ Pm Uξ ϕ′ (h)| det(h)|s−1/2 dh. I(s; Uξ , ϕ ) = 1 GLm (k)\ GLm (A) This integral is always absolutely convergent for Re(s) >> 0, and for all s if π ′ is cuspidal. As with the usual integral representation we have that this unfolds into the Euler product Z h 0 ′ Wξ Wϕ′ ′ (h)| det(h)|s−(n−m)/2 dh I(s; Uξ , ϕ ) = 0 I n−m Nm (A)\ GLm (A) YZ hv 0 Wξv = Wϕ′ ′ v (hv )| det(hv )|vs−(n−m)/2 dhv 0 I n−m Nm (kv )\ GLm (kv ) v Y = Ψv (s; Wξv , Wϕ′ ′ v ). v

Note that unless π ′ is generic, this integral vanishes. Assume first that π ′ is cuspidal. Then from the local theory of Lfunctions from Section 3, for almost all finite places we have Ψv (s; Wξv , Wϕ′ ′ ) v = L(s, Πv ×π ′ v ) and for the other places Ψv (s; Wξv , Wϕ′ ′ v ) = ev (s; Wξv , Wϕ′ ′ v ) L(s, Πv ×π ′ v ) with the ev (s; Wξv , Wϕ′ ′ v ) entire and bounded in vertical strips. So in this case we have I(s; Uξ , ϕ′ ) = e(s)L(s, Π × π ′ ) with e(s) entire and

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bounded in vertical strips. Since L(s; Π × π ′ ) is assumed to be nice we may conclude that I(s; Uξ , ϕ′ ) has an analytic continuation to an entire function which is bounded in vertical strips. When π ′ is not cuspidal, it is a subrepresentation of a representation that is induced from (possibly non-unitary) P cuspidal representations σi of GLri (A) for ri < m with ri = m and is in fact, if our integral doesn’t vanish, the unique generic constituent of this induced representation. Then we can make a similar argument using this induced representation and the fact that the L(s, Π × σi ) are nice to again Q conclude that for all π ′ , I(s; Uξ , ϕ′ ) = e(s)L(s, Π × π ′ ) = e′ (s) L(s, Π × σi ) is entire and bounded in vertical strips. (See [7] for more details on this point.) Similarly, consider I(s; Vξ , ϕ′ ) for ϕ′ ∈ Vπ′ with π ′ an irreducible subrepresentation of the space of automorphic forms on GLm (A), still with Z h n ′ Pm Vξ ϕ′ (h)| det(h)|s−1/2 dh. I(s; Vξ , ϕ ) = 1 GLm (k)\ GLm (A) Now this integral converges for Re(s) 1, then either m1 ≤ n − 2 or mr ≤ n − 2 (or both). For simplicity assume mr ≤ n − 2. Let S be a finite set of places containing all archimedean places, v0 , v1 , SΠ , and Sσi for each i. Taking

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π′ = σ er ∈ T (n − 2), we have the equality of partial L-functions

LS (s, Π × π ′ ) = LS (s, Π0 × π ′ ) = LS (s, Π1 × π ′ ) Y Y = LS (s, σi × π ′ ) = LS (s + ti − tr , σi◦ × σ er◦ ). i

i

Now LS (s, σr × σ er ) has a pole at s = 1 and all other terms are non-vanishing at s = 1. Hence L(s, Π × π ′ ) has a pole at s = 1 contradicting the fact that L(s, Π × π ′ ) is nice. If m1 ≤ 2, then we can make a similar argument using e × σ1 ). So in fact we must have r = 1 and Π0 = Π1 = Ξ is cuspidal. L(s, Π Since Π0 agrees with Π at v1 and Π1 agrees with Π at v0 we see that in fact Π = Π0 = Π1 and Π is indeed cuspidal automorphic. 5.3.3

Theorem 5.3

Now consider Theorem 5.3. Since we have restricted our ramification, we no longer know that L(s, Π × π ′ ) is nice for all π ′ ∈ T (n − 2) and so Proposition 5.1 above is not immediately applicable. In this case, for each place v ∈ S we fix a vector ξv′ ∈ VΠv as in the above Lemma. (So we must assume we Q have chosen ψ so it is unramified at the places in S.) Let ξS′ = v∈S ξv′ ∈ ΠS . Consider now only vectors ξ of the form ξ S ⊗ ξS′ with ξ S arbitrary h ′ n and in VΠS and ξS fixed. For these vectors, the functions Pn−2 Uξ 1 h Pnn−2 Vξ are unramified at the places v ∈ S, so that the integrals 1 I(s; Uξ , ϕ′ ) and I(s; Vξ , ϕ′ ) vanish unless ϕ′ (h) is also unramified at those places in S. In particular, if π ′ ∈ T (n − 2) but π ′ ∈ / T S (n − 2) these integrals will vanish for all ϕ′ ∈ Vπ′ . So now, for this fixed class of ξ we actually have I(s; Uξ , ϕ′ ) = I(s; Vξ , ϕ′ ) for all ϕ′ ∈ Vπ′ for all π ′ ∈ T (n − 2). Hence, as before, Pnn−2 Uξ (In−1 ) = Pnn−2 Vξ (In−1 ) for all such ξ. Now we proceed as before. Our Fourier expansion argument is a bit more subtle since we have to work around our local conditions, which now have been imposed before this step, but we do obtain that Uξ (g) = Vξ (g) for all Q g ∈ G′ = ( v∈S K00,v (pnv v )) GS . The generation of congruence subgroups goes as before. We then use weak approximation as above, but then take for Π′ any constituent of the extension of ΠS to an automorphic representation of GLn (A). There is no use of strong multiplicity one nor any further use of the L-function in this case. More details can be found in [9].

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Let us now sketch the proof of Theorem 5.4. We fix a non-empty finite set of places S, containing all archimedean places, such that the ring oS of Sinteger has class number one. Recall that we are now twisting by all cuspidal representations π ′ ∈ TS (n − 1), that is, π ′ which are unramified at all places v∈ / S. Since we have not twisted by all of T (n − 1) we are not in a position to apply Proposition 5.1. To be able to apply that, we will have to place local conditions at all v ∈ / S. We begin by recalling the definition of the conductor of a representation Πv of GLn (kv ) and the conductor (or level) of Π itself. Let Kv = GLn (ov ) be the standard maximal compact subgroup of GLn (kv ). Let pv ⊂ ov be the unique prime ideal of ov and for each integer mv ≥ 0 set ∗ .. mv mv . ∗ K0,v (pv ) = g ∈ GLn (ov ) g ≡ (mod p ) ∗ 0 ··· 0 ∗

mv mv v and K1,v (pm v ) = {g ∈ K0,v (pv ) | gn,n ≡ 1 (mod pv ))}. Note that for mv = 0 we have K1,v (p0v ) = K0,v (p0v ) = Kv . Then for each local component v Πv of Π there is a unique integer mv ≥ 0 such that the space of K1,v (pm v )– fixed vectors in Πv is exactly one. For almost all v, mv = 0. We take the ideal Q mv v pm v = f(Πv ) as the conductor of Πv . Then the ideal n = f(Π) = v pv ⊂ o is called the conductor of Π. For each place v we fix a non-zero vector ξv◦ ∈ Πv v which is fixed by K1,v (pm v ), which at the unramified places is taken to be the vector with respect to which the restricted tensor product Π = ⊗′ Πv is ◦ ◦ v taken. Note that for g ∈ K0,v (pm v ) we have Πv (g)ξv = ωΠv (gn,n )ξv . Now fix a non-empty finite set of places S, containing the archimedean Q places if there are any. As is standard, we will let GS = v∈S GLn (kv ), Q (k ), Π = ⊗ Π , ΠS = ⊗′v∈S GS = v∈S / GL / Πv , etc. Then the compact Qn v mv S S v∈S v S ⊂ k or the ideal it determines nS = k ∩ kS nS ⊂ oS subring n = v∈S p v / Q mv is called the S–conductor of Π. Let KS1 (n) = v∈S / K1,v (pv ) and similarly S ◦ ◦ S for K0 (n). Let ξ = ⊗v∈S by KS1 (n) and / ξv ∈ Π . Then this vector is fixed Q transforms by a character under KS0 (n). In particular, since v∈S / GLn−1 (ov ) h embeds in KS1 (n) via h 7→ we see that when we restrict ΠS to GLn−1 1 the vector ξ ◦ is unramified. Now let us return to the proof of Theorem 5.4 and in particular the version of Proposition 5.1 we can salvage. For every vector ξS ∈ ΠS consider

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the functions UξS ⊗ξ ◦ and VξS ⊗ξ ◦ . When we restrict these functions to GLn−1 they become unramified for all places v ∈ / S. Hence we see that the integrals I(s; UξS ⊗ξ ◦ , ϕ′ ) and I(s; VξS ⊗ξ ◦ , ϕ′ ) vanish identically if the function ϕ′ ∈ Vπ′ is not unramified for v ∈ / S, and in particular if ϕ′ ∈ Vπ′ for π ′ ∈ T (n−1) but ′ π ∈ / TS (n − 1). Hence, for vectors of the form ξ = ξS ⊗ ξ ◦ we do indeed have that I(s; UξS ⊗ξ ◦ , ϕ′ ) = I(s; VξS ⊗ξ ◦ , ϕ′ ) for all ϕ′ ∈ Vπ′ and all π ′ ∈ T (n − 1). Hence, as in Proposition 5.1 we may conclude that UξS ⊗ξ ◦ (In ) = VξS ⊗ξ ◦ (In ) for all ξS ∈ VΠS . Moreover, since ξS was arbitrary in VΠS and the fixed vector ξ ◦ transforms by a character of KS0 (n) we may conclude that UξS ⊗ξ ◦ (g) = VξS ⊗ξ ◦ (g) for all ξS ∈ VΠS and all g ∈ GS KS0 (n). What invariance properties of the function UξS ⊗ξ ◦ have we gained from our equality with VξS ⊗ξ ◦ . Let us let Γi (nS ) = GLn (k) ∩ GS KSi (n) which we may view naturally as congruence subgroups of GLn (oS ). Now, as a function on GS KS0 (n), UξS ⊗ξ ◦ (g) is naturally left invariant under Γ0,P (nS ) = Z(k) P(k) ∩ GS KS0 (n) while VξS ⊗ξ ◦ (g) is naturally left invariant under Γ0,Q (nS ) = Z(k) Q(k) ∩ GS KS0 (n) where Q = Qn−1 . Similarly we set Γ1,P (nS ) = Z(k) P(k) ∩ GS KS1 (n) and Γ1,Q (nS ) = Z(k) Q(k) ∩ GS KS1 (n). The crucial observation for this Theorem is the following result. Proposition The congruence subgroup Γi (nS ) is generated by Γi,P (nS ) and Γi,Q (nS ) for i = 0, 1. This proposition is a consequence of results in the stable algebra of GLn due to Bass which were crucial to the solution of the congruence subgroup problem for SLn by Bass, Milnor, and Serre. This is the reason for the restriction to n ≥ 3 in the statement of Theorem 5.4. From this we get not an embedding of Π into a space of automorphic forms on GLn (A), but rather an embedding of ΠS into a space of classical automorphic forms on GS . To this end, for each ξS ∈ VΠS let us set ΦξS (gS ) = UξS ⊗ξ ◦ ((gS , 1S )) = VξS ⊗ξ ◦ ((gS , 1S )) for gS ∈ GS . Then ΦξS will be left invariant under Γ1 (nS ) and transform by a Nebentypus character χS under Γ0 (nS ) determined by the central character ωΠS of ΠS . Furthermore, it will transform by a character ωS = ωΠS under the center Z(kS ) of GS . The requisite growth properties are satisfied and hence the map ξS 7→ ΦξS defines an embedding of ΠS into the space A(Γ0 (nS )\ GS ; ωS , χS ) of classical automorphic forms on GS relative to the congruence subgroup Γ0 (nS ) with Nebentypus χS and central character ωS .

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We now need to lift our classical automorphic representation back to an adelic one and hopefully recover the rest of Π. By strong approximation for GLn and our class number assumption we have the isomorphism between the space of classical automorphic forms A(Γ0 (nS )\ GS ; ωS , χS ) and the KS1 (n) invariants in A(GLn (k)\ GLn (A); ω) where ω is the central character of Π. Hence ΠS will generate an automorphic subrepresentation of the space of automorphic forms A(GLn (k)\ GLn (A); ω). To compare this to our original Π, we must check that, in the space of classical forms, the ΦξS ⊗ξ ◦ are Hecke eigenforms for a classical Hecke algebra and that their Hecke eigenvalues agree with those from Π. We do this only for those v ∈ / S which are unramified, where it is a rather standard calculation. As we have not talked about Hecke algebras, we refer the reader to [7] for details. Now if we let Π′ be any irreducible subrepresentation of the representation generated by the image of ΠS in A(GLn (k)\ GLn (A); ω), then Π′ is automorphic and we have Π′v ≃ Πv for all v ∈ S by construction and Π′v ≃ Πv for all v ∈ / S ′ by the Hecke algebra calculation. Thus we have proven Theorem 5.4.

5.4

Converse Theorems and Liftings

In this section we would like to make some general remarks on how to apply these Converse Theorems to the problem of functorial liftings [3]. In order to apply these theorems, you must be able to control the global properties of the L-function. However, for the most part, the way we have of controlling global L-functions is to associate them to automorphic forms or representations. A minute’s thought will then lead one to the conclusion that the primary application of these results will be to the lifting of automorphic representations from some group H to GLn . Suppose that H is a split classical group, π an automorphic representation of H, and ρ a representation of the L-group of H. Then we should be able to associate an L-function L(s, π, ρ) to this situation [3]. Let us assume that ρ :L H → GLn (C) so that to π should be associated an automorphic representation Π of GLn (A). What should Π be and why should it be automorphic. We can see what Πv should be at almost all places. Since we have the (arithmetic) Langlands (or Langlands-Satake) parameterization of representations for all archimedean places and those finite places where the representations are unramified [3], we can use these to associate to πv and the map

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ρv :L Hv → GLn (C) a representation Πv of GLn (kv ). If H happens to be GLm then we in principle know how to associate the representation Πv at all places now that the local Langlands conjecture has been solved for GLm [23, 26], but in practice this is still not feasible. For other situations, we do not know what Πv should be at the ramified places. We will return to this difficulty momentarily. But for now, let’s assume we can finesse this local problem and arrive at a representation Π = ⊗′ Πv such that L(s, π, ρ) = L(s, Π). Π should then be the Langlands lifting of π to GLn associated to ρ. For simplicity of exposition, let us now assume that ρ is simply the standard embedding of L H into GLn (C) and write L(s, π, ρ) = L(s, π) = L(s, Π). We have our candidate Π for the lift of π to GLn , but how to tell whether Π is automorphic. This is what the Converse Theorem lets us do. But to apply them we must first be able to define and control the twisted L-functions L(s, π × π ′ ) for π ′ ∈ T with an appropriate twisting set T from one of our Converse Theorems. This is one reason why it is always crucial to define not only the standard L-functions for H, but also the twisted versions. If we know, from the theory of L-functions of H twisted by GLm for appropriate π ′ , that L(s, π × π ′ ) is nice and L(s, π × π ′ ) = L(s, Π × π ′ ) for twists, then we can use Theorem 5.1 or 5.2 to conclude that Π is cuspidal automorphic or Theorem 5.3 or 5.4 to conclude that Π is quasi-automorphic and at least obtain a weak automorphic lifting Π′ which is verifiably the correct representation at almost all places. At this point this relies on the state of our knowledge of the theory of twisted L-functions for H. Let us return now to the (local) problem of not knowing the appropriate local lifting πv 7→ Πv at the ramified places. We can circumvent this by a combination of global and local means. The global tool is simply the following observation. Observation Let Π be as in Theorem 5.3 or 5.4. Suppose that η is a fixed (highly ramified) character of k× \A× . Suppose that L(s, Π × π ′ ) is nice for all π ′ ∈ T ⊗ η, where T is either of the twisting sets of Theorem 5.3 or 5.4. Then Π is quasi-automorphic as in those theorems. The only thing to observe, say by looking at the local or global integrals, is that if π ′ ∈ T then L(s, Π×(π ′ ⊗η)) = L(s, (Π⊗η)×π ′ ) so that applying the Converse Theorem for Π with twisting set T ⊗η is equivalent to applying the Converse Theorem for Π ⊗ η with the twisting set T . So, by either Theorem 5.3 or 5.4, whichever is appropriate, Π ⊗ η is quasi-automorphic and hence Π is as well.

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Now, if we begin with π automorphic on H(A), we will take T to be the set of finite places where πv is ramified. For applying Theorem 5.3 we want S = T and for Theorem 5.4 we want S ∩ T = ∅. We will now take η to be highly ramified at all places v ∈ T . So at v ∈ T our twisting representations are all locally of the form (unramified principal series)⊗(highly ramified character). We now need to know the following two local facts about the local theory of L-functions for H. 1. Multiplicativity of γ-factors: If π ′ v = Ind(π ′ 1,v ⊗ π ′ 2,v ), with π ′ i,v and irreducible admissible representation of GLri (kv ), then γ(s, πv × π ′ v , ψv ) = γ(s, πv × π ′ 1,v , ψv )γ(s, πv × π ′ 2,v , ψv ) and L(s, πv × π ′ v )−1 should divide [L(s, πv × π ′ 1,v )L(s, π × π ′ 2,v )]−1 . If πv = Ind(σv ⊗ πv′ ) with σv an irreducible admissible representation of GLr (kv ) and πv′ an irreducible admissible representation of H′ (kv ) with H′ ⊂ H such that GLr × H′ is the Levi of a parabolic subgroup of H, then ev × π ′ v , ψv ). γ(s, πv × π ′ v , ψv ) = γ(s, σv × π ′ v , ψv )γ(s, πv′ × π ′ v , ψv )γ(s, σ

2. Stability of γ-factors: If π1,v and π2,v are two irreducible admissible representations of H(kv ), then for every sufficiently highly ramified character ηv of GL1 (kv ) we have γ(s, π1,v × ηv , ψv ) = γ(s, π2,v × ηv , ψv ) and L(s, π1,v × ηv ) = L(s, π2,v × ηv ) ≡ 1. Once again, for these applications it is crucial that the local theory of L-functions is sufficiently developed to establish these results on the local γ-factors. As we have seen in Section 3, both of these facts are known for GLn . To utilize these local results, what one now does is the following. At the places where πv is ramified, choose Πv to be arbitrary, except that it should have the same central character as πv . This is both to guarantee that the central character of Π is the same as that of π and hence automorphic and

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to guarantee that the stable forms of the γ–factors for πv and Πv agree. Now form Π = ⊗′ Πv . We choose our character η so that at the places v ∈ T we have that the L– and γ–factors for both πv ⊗ ηv and Πv ⊗ ηv are in their stable form and agree. We then twist by T ⊗ η for this fixed character η. If π ′ ∈ T ⊗ η, then for v ∈ T , π ′ v is of the form π ′ v = Ind(µv,1 ⊗ · · · ⊗ µv,m ) ⊗ ηv with each µv,i an unramified character of kv× . So at the places v ∈ T we have γ(s, πv × π ′ v ) = γ(s, πv × (Ind(µv,1 ⊗ · · · ⊗ µv,m ) ⊗ ηv )) Y = γ(s, πv ⊗ (µv,i ηv )) (by multiplicativity) Y = γ(s, Πv ⊗ (µv,i ηv )) (by stability)

= γ(s, Πv × (Ind(µv,1 ⊗ · · · ⊗ µv,m ) ⊗ ηv )) (by multiplicativity)

= γ(s, Πv × π ′ v ) and similarly for the L-factors. From this it follows that globally we will have L(s, π×π ′ ) = L(s, Π×π ′ ) for all π ′ ∈ T ⊗η and the global functional equation for L(s, π × π ′ ) will yield the global functional equation for L(s, Π × π ′ ). So L(s, Π × π ′ ) is nice and we may proceed as before. We have, in essence, twisted away all information about π and Π at those v ∈ T . The price we pay is that we also lose this information in our conclusion since we only know that Π is quasi-automorphic. In essence, the Converse Theorem fills in a correct set of data at those places in T to make the resulting global representation automorphic.

5.5

Some Liftings

To conclude, let us make a list of some of the liftings that have been accomplished using these Converse Theorems. Some have used the above trick of multiplicativity and stability of γ–factors to handle the ramified places. Others, principally those that involve GL2 , have adopted a technique of Ramakrishnan [51] involving a sequence of base changes and descents to get a more complete handle on the ramified places. 1. The symmetric square lifting from GL2 to GL3 by Gelbart and Jacquet [15]. 2. Non-normal cubic base change for GL2 by Jacquet, Piatetski-Shapiro, and Shalika [32].

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3. The tensor product lifting from GL2 × GL2 to GL4 by Ramakrishnan [51]. 4. The lifting of generic cusp forms from SO2n+1 to GL2n , with Kim, Piatetski-Shapiro, and Shahidi [6]. 5. The tensor product lifting from GL2 × GL3 to GL6 and the symmetric cube lifting from GL2 to GL4 by Kim and Shahidi [40]. 6. The exterior square lifting from GL4 to GL6 and the symmetric fourth power lift from GL2 to GL5 by Kim [39]. For the most part, it was Theorem 5.3 that was used in each case, with the exception of (4), where a simpler variant was used requiring twists by T S (n − 1). For the non-normal cubic base change both Theorem 5.3 with n = 3 and Theorem 5.1 with n = 2 were used.

Acknowledgments Most of what I know about L-functions for GLn I have learned through my years of work with Piatetski-Shapiro. I owe him a great debt of gratitude for all that he has taught me. For several years Piatetski-Shapiro and I have envisioned writing a book on L-functions for GLn [13]. The contents of these notes essentially follows our outline for that book. In particular, the exposition in Sections 1, 2, and parts of 3 and 4 is drawn from drafts for this project. The exposition in Section 5 is drawn from the survey of our work on Converse Theorems [10]. I would also like to thank Piatetski-Shapiro for graciously allowing me to present part of our joint efforts in these notes. I would also like to thank Jacquet for enlightening conversations over the years on his work on L-functions for GLn .

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References [1] J. Bernstein and A. Zelevinsky, Representations of GL(n, F ) where F is a non-archimedean local field, Russian Math. Surveys, 31 (1976), 1–68. [2] J. Bernstein and A. Zelevinsky, Induced representations of reductive ´ Norm. Sup., 4e s´erie, 10 (1977), 441– p–adic groups, I, Ann. scient. Ec. 472. [3] A. Borel, Automorphic L-functions, Proc. Sympos. Pure Math., 33, part 2, (1979), 27–61. [4] W. Casselman and J. Shalika, The unramified principal series of p-adic groups. II. The Whittaker function. Compositio Math., 41 (1980), 207– 231. [5] L. Clozel, Motifs et formes automorphes: applications du principle de functorialit´e, in Automorphic Forms, Shimura Varieties, and Lfunctions, I, edited by L. Clozel and J. Milne, Academic Press, Boston, 1990, 77–159. [6] J. Cogdell, H. Kim, I. Piatetski-Shapiro, and F. Shahidi, On lifting from classical groups to GLN , Publ. Math. IHES, to appear. [7] J. Cogdell and I.I. Piatetski-Shapiro, Converse theorems for GLn , Publ. Math. IHES 79 (1994), 157–214. [8] J. Cogdell and I.I. Piatetski-Shapiro, Unitarity and functoriality, Geom. and Funct. Anal., 5 (1995), 164–173. [9] J. Cogdell and I.I. Piatetski-Shapiro, Converse theorems for GLn , II J. reine angew. Math., 507 (1999), 165–188. [10] J. Cogdell and I.I. Piatetski-Shapiro, Converse theorems for GLn and their applications to liftings, to appear in the proceedings of the International Conference on Cohomology of Arithmetic Groups, Automorphic Forms, and L-functions, Tata Institute of Fundamental Research, December 1998 – January 1999. [11] J. Cogdell and I.I. Piatetski-Shapiro, Derivatives and L-functions for GLn , to appear in a volume dedicated to B. Moishezon.

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[12] J. Cogdell and I.I. Piatetski-Shapiro, On archimedean Rankin–Selberg convolutions, manuscript (1995), 8 pages. [13] J. Cogdell and I.I. Piatetski-Shapiro, L-functions for GLn , in progress. [14] D. Flath, Decomposition of representations into tensor products, Proc. Sympos. Pure Math., 33, part 1, (1979), 179–183. [15] S. Gelbart and H. Jacquet, A relation between automorphic represen´ tations of GL(2) and GL(3), Ann. Sci. Ecole Norm. Sup. (4) 11 (1978), 471–542. [16] S. Gelbart and F. Shahidi, Analytic Properties of Automorphic Lfunctions, Academic Press, San Diego, 1988. [17] S. Gelbart and F. Shahidi, Boundedness of automorphic L-functions in vertical strips, J. Amer. Math. Soc., 14 (2001), 79–107. [18] I.M. Gelfand, M.I. Graev, and I.I. Piatetski-Shapiro, Representation Theory and Automorphic Functions, Academic Press, San Diego, 1990. [19] I.M. Gelfand and D.A. Kazhdan, Representations of GL(n, K) where K is a local field, in Lie Groups and Their Representations, edited by I.M. Gelfand. John Wiley & Sons, New York–Toronto, 1971, 95–118. [20] S.I. Gelfand, Representations of the general linear group over a finite field, in Lie Groups and Their Representations, edited by I.M. Gelfand. John Wiley & Sons, New York–Toronto, 1971, 119–132. [21] R. Godement, Notes on Jacquet-Langlands’ Theory, The Institute for Advanced Study, 1970. [22] R. Godement and H. Jacquet, Zeta Functions of Simple Algebras, Springer Lecture Notes in Mathematics, No.260, Springer-Verlag, Berlin, 1972. [23] M. Harris and R. Taylor, On the geometry and cohomology of some simple Shimura varieties, preprint (1999). ¨ [24] E. Hecke, Uber die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung, Math. Ann., 112 (1936), 664–699.

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[39] H. Kim, Functoriality for the exterior square of GL4 and the symmetric fourth of GL2 , submitted (2000). [40] H. Kim and F. Shahidi, Functorial products for GL2 × GL3 and functorial symmetric cube for GL2 , submitted (2000). [41] H. Kim and F. Shahidi, Cuspidality of symmetric powers of GL(2) with applications, submitted (2000). [42] S. Kudla, The local Langlands correspondence: the non-Archimedean case, Proc. Sympos. Pure Math., 55, part 2, (1994), 365–391. [43] R.P. Langlands, Euler Products, Yale Univ. Press, New Haven, 1971. [44] R.P. Langlands, On the notion of an automorphic representation, Proc. Sympos. Pure Math., 33, part 1, (1979), 203–207. [45] R.P. Langlands, On the classification of irreducible representations of real algebraic groups, in Representation Theory and Harmonic Analysis on Semisimple Lie Groups, AMS Mathematical Surveys and Monographs, No.31, 1989, 101–170. [46] W. Luo, Z. Rudnik, and P. Sarnak, On the generalized Ramanujan conjecture for GL(n), Proc. Symp. Pure Math., 66, part 2, (1999), 301– 310. [47] I.I. Piatetski-Shapiro, Euler Subgroups, in Lie Groups and Their Representations, edited by I.M. Gelfand. John Wiley & Sons, New York– Toronto, 1971, 597–620. [48] I.I. Piatetski-Shapiro, Multiplicity one theorems, Proc. Sympos. Pure Math., 33, Part 1 (1979), 209–212. [49] I.I. Piatetski-Shapiro, Complex Representations of GL(2, K) for Finite Fields K, Contemporary Math. Vol.16, AMS, Providence, 1983. [50] D. Ramakrishnan, Pure motives and automorphic forms, Proc. Sympos. Pure Math., 55, part 2, (1991), 411–446. [51] D. Ramakrishnan, Modularity of the Rankin-Selberg L-series, and multiplicity one for SL(2), Annals of Math., 152 (2000), 45–111.

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[65] J. Tate, Number theoretic background, Proc. Symp. Pure Math., 33, part 2, 3–26. [66] N. Wallach, Real Reductive Groups, I & II, Academic Press, Boston, 1988 & 1992. ¨ [67] A. Weil, Uber die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann., 168 (1967), 149–156. [68] A. Weil, Basic Number Theory, Springer–Verlag, Berlin, 1974. [69] A. Zelevinsky, Induced representations of reductive p–adic groups, II. ´ Norm. Sup., 4e Irreducible representations of GL(n), Ann. scient. Ec. s´erie, 13 (1980), 165–210.

Representation Theory of GL(n) over Non-Archimedean Local Fields Dipendra Prasad1∗ and A. Raghuram2† 1

2

Harish-Chandra Research Institute, Jhusi, Allahabad, India School of Mathematics, Tata Institute of Fundamental Research, Colaba, Mumbai, India

Lectures given at the School on Automorphic Forms on GL(n) Trieste, 31 July - 18 August 2000

LNS0821004

∗ †

[email protected] [email protected]

Contents 1 Introduction

163

2 Generalities on representations

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3 Preliminaries on GLn (F )

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4 Parabolic induction

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5 Jacquet functors

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6 Supercuspidal representations

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7 Discrete series representations

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8 Langlands classification

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9 Certain classes of representations 9.1 Generic representations . . . . . . . . . 9.2 Tempered representations . . . . . . . . 9.3 Unramified or spherical representations . 9.4 Iwahori spherical representations . . . .

191 191 193 194 196

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10 Representations of local Galois Groups

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11 The local Langlands conjecture for GLn

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References

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Representation Theory of GL(n) over Non-Archimedean Local Fields 163

1

Introduction

The aim of these notes is to give an elementary introduction to representation theory of p-adic groups via the basic example of GL(n). We have emphasised those topics which are relevant to the theory of automorphic forms. To keep these notes to a reasonable size, we have omitted many proofs which would have required a lot more preparation. We have given references throughout the notes either to original or more authoritative sources. Most of the omitted proofs can be found in the fundamental papers of BernsteinZelevinsky [2], Bushnell-Kutzko [5], or Casselman’s unpublished notes [7] which any serious student of the subject will have to refer sooner or later. Representation theory of p-adic groups is a very active area of research. One of the main driving forces in the development of the subject is the Langlands program. One aspect of this program, called the local Langlands correspondence, implies a very intimate connection between representation theory of p-adic groups with the representation theory of the Galois group of the p-adic field. This conjecture of Langlands has recently been proved by Harris and Taylor and also by Henniart for the case of GL(n). We have given an introduction to Langlands’ conjecture for GL(n) as well as some representation theory of Galois groups in the last two sections of these notes.

2

Generalities on representations

Let G denote a locally compact totally disconnected topological group. In the terminology of [2] such a group is called an l-group. For these notes the fundamental example of an l-group is the group GLn (F ) of invertible n × n matrices with entries in a non-Archimedean local field F. By a smooth representation (π, V ) of G we mean a group homomorphism of G into the group of automorphisms of a complex vector space V such that for every vector v ∈ V, the stabilizer of v in G, given by stabG (v) = {g ∈ G : π(g)v = v}, is open. The space V is called the representation space of π. By an admissible representation (π, V ) of G we mean a smooth representation (π, V ) such that for any open compact subgroup K of G, the invariants in V under K, denoted V K , is finite dimensional. Given a smooth representation (π, V ) of G, a subspace W of V is said to be stable or invariant under G if for every w ∈ W and every g ∈ G we have π(g)w ∈ W. A smooth representation (π, V ) of G is said to be irreducible if the only G stable subspaces of V are (0) and V.

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If (π, V ) and (π ′ , V ′ ) are two (smooth) representations of G then by HomG (π, π ′ ) we denote the space of G intertwining operators from V into V ′ , i.e., HomG (π, π ′ ) is the space of all linear maps f : V → V ′ such that f (π(g)v) = π ′ (g)f (v) for all v ∈ V and all g ∈ G. Let (π, V ) be a smooth representation of G. Let V ∗ denote the space of linear functionals on V, i.e., V ∗ = HomC (V, C). There is an obvious action π ∗ of G on V ∗ given by (π ∗ (g)f )(v) = f (π(g−1 )v) for g ∈ G, v ∈ V and f ∈ V ∗ . However this representation is in general not smooth. Let V ∨ denote the subspace of V ∗ which consists of those linear functionals in V ∗ whose stabilizers are open in G under the above action. It is easily seen that V ∨ is stabilized by G and this representation denoted (π ∨ , V ∨ ) is called the contragredient representation of (π, V ). The following is a basic lemma in representation theory. Lemma 2.1 (Schur’s Lemma) Let (π, V ) be a smooth irreducible admissible representation of an l-group G. Then the dimension of HomG (π, π) is one. Proof : Let A ∈ HomG (π, π). Let K be any open compact subgroup of G. Then A takes the space of K-fixed vectors V K to itself. Choose a K such that V K 6= (0). Since V K is a finite dimensional complex vector space, there exists an eigenvector for the action of A on V K , say, 0 6= v ∈ V K with Av = λv. It follows that the kernel of (A − λ1V ) is a nonzero G invariant subspace of V. Irreducibility of the representation implies that A = λ1V . 2 Corollary 2.2 Any irreducible admissible representation of an abelian lgroup is one dimensional. Corollary 2.3 On an irreducible admissible representation (π, V ) of an lgroup G the centre Z of G , operates via a character ωπ , i.e., for all z ∈ Z we have π(z) = ωπ (z)1V . (This character ωπ is called the central character of π.) Exercise 2.4 (Dixmier’s lemma) Let V be a complex vector space of countable dimension. Let Λ be a collection of endomorphisms of V acting irreducibly on V. Let T be an endomorphism of V which commutes with every element of Λ. Then prove that T acts as a scalar on V. (Hint. Think of V as a module over the field of rational functions in oneovariable C(X) with n 1 X acting on V via T and use the fact that X−λ |λ ∈ C is an uncountable

Representation Theory of GL(n) over Non-Archimedean Local Fields 165 set consisting of linearly independent elements in the C-vector space C(X).) Deduce that if G is an l-group which is a countable union of compact sets and (π, V ) is an irreducible smooth representation of G then the dimension of HomG (π, π) is one. One of the most basic ways of constructing representations is by the process of induction. Before we describe this process we need a small digression on Haar measures. Let G be an l-group. Let dl x be a left Haar measure on G. So in particular dl (ax) = dl (x) for all a, x ∈ G. For any g ∈ G the measure dl (xg) (where x is the variable) is again a left Haar measure. By uniqueness of Haar measures there is a positive real number ∆G (g) such that dl (xg) = ∆G (g)dl (x). It is easily checked that g 7→ ∆G (g) is a continuous group homomorphism and is called the modular character of G. Further dr (x) := ∆G (x)−1 dl (x) is a right Haar measure. A left Haar measure is right invariant if and only if the modular character is trivial and in this case G is said to be unimodular. Example 2.5 Let F be a non-Archimedean local field. The group G = GLn (F ) is unimodular. (In general a reductive p-adic group is unimodular.) This may be seen as follows. Note that ∆G is trivial on [G, G] as the range of ∆G is abelian. Further by the defining relation ∆G is trivial on the centre Z. (Both these remarks are true for any group.) Hence ∆G is trivial on Z[G, G]. Observe that Z[G, G] is of finite index in G and since positive reals admits no non-trivial finite subgroups we get that ∆G is trivial. Exercise 2.6 Let B be the subgroup of all upper triangular matrices in G = GLn (F ). Then show that B is not unimodular as follows. Let d∗ y be a Haar measure on F ∗ and let dx be a Haar measure on F + . The normalized absolute value on F ∗ is defined by d(ax) = |a|F dx for all a ∈ F ∗ and x ∈ F. For b ∈ B given by y1 1 1 xi,j y2 , b= . . 1 yn show that a left Haar measure on B is given by the formula : Y Y dxi,j . db = d∗ yi i

1≤i<j≤n

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Deduce that the modular character ∆B of B is given by : n−1 3−n ∆B (diag(a1 , ..., an )) = |a1 |1−n F |a2 |F ...|an |F .

Let δB = ∆−1 B . Show that δB (b) = |det(Ad(b)|Lie(U ) )|F where Ad(b)|Lie(U ) is the adjoint representation of B on the Lie algebra Lie(U ) of U which is the unipotent radical of B. (In some places δB is used as the definition of the modular character, see for e.g., [1].) For any l-group G, let ∆G denote the modular character of G. Let B be a closed subgroup of G. Let ∆ = ∆G |B · ∆−1 B . Let (σ, W ) be a smooth representation of B. To this is associated the (normalized) induced representation IndG B (σ) whose representation space is : (1) f (bg) = ∆(b)1/2 σ(b)f (g), ∀b ∈ B, ∀g ∈ G f :G→W . (2) f (hu) = f (h) for all u ∈ Uf an open set in G

The group G acts on this space by right translation, i.e., given x ∈ G and f ∈ IndG B (σ) we have (x · f )(g) = f (gx) for all g ∈ G. We remark here that throughout these notes we deal with normalized induction. By normalizing we mean the ∆1/2 factor appearing in (1) above. This is done so that unitary representations are taken to unitary representations under induction. This simplifies some formulae like that which describes the contragredient of an induced representation and complicates some other formulae like Frobenius reciprocity. This induced representation admits a subrepresentation which is also another ‘induced’ representation called that obtained by compact induction. This is denoted as indG B (σ) whose representation space is given by G indG B (σ) = {f ∈ IndB (σ) : f is compactly supported modulo B}.

The following theorem summarises the basic properties of induced representations. Theorem 2.7 Let G be an l-group and let B be a closed subgroup of G. G 1. Both IndG B and indB are exact functors from the category of smooth representations of B to the category of smooth representations of G.

Representation Theory of GL(n) over Non-Archimedean Local Fields 167 2. Both the induction functors are transitive,i.e., if C is a closed subgroup B G of B then IndG B (IndC ) = IndC . A similar relation holds for compact induction. G ∨ ∨ 3. Let σ be a smooth representation of B. Then IndG B (σ) = indB (σ ).

4. (Frobenius Reciprocity) Let π be a smooth representation of G and σ be a smooth representation of B. 1/2 σ). (a) HomG (π, IndG B (σ)) = HomB (π|B , ∆ −1/2 σ, (π ∨ | )∨ ). (b) HomG (indG B B (σ), π) = HomB (∆

Both the above identifications are functorial in π and σ. The reader may refer to paragraphs 2.25, 2.28 and 2.29 of [2] for a proof of the above theorem. We recall some basic facts now about taking invariants under open compact subgroups. Let C be an open compact subgroup of an l-group G. Let H(G, C) be the space of compactly supported bi-C-invariant functions on G. This forms an algebra under convolution which is given by : Z f (xy −1 )g(y) dy (f ∗ g)(x) = G

with the identity element being vol(C)−1 eC where eC is the characteristic function of C as a subset of G. Let (π, V ) be a smooth admissible representation of G then the space of invariants V C is a finite dimensional module for H(G, C). The following proposition is not difficult to prove. The reader is urged to try and fix a proof of this. (See paragraph 2.10 of [2].) Proposition 2.8 Let the notations be as above. 1. The functor (π, V ) 7→ (π C , V C ) is an exact additive functor from the category of smooth admissible representations of G to the category of finite dimensional modules of the algebra H(G, C). 2. Let (π, V ) be an admissible representation of G such that V C 6= (0). Then (π, V ) is irreducible if and only if (π C , V C ) is an irreducible H(G, C) module. 3. Let (π1 , V1 ) and (π2 , V2 ) be two irreducible admissible representations of G such that both V1C and V2C are non-zero. We have π1 ≃ π2 as G representations if and only if π1C ≃ π2C as H(G, C) modules.

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Preliminaries on GLn (F )

In this section we begin the study of the group G = GLn (F ) and its representations. (Unless otherwise mentioned, from now on G will denote the group GLn (F ).) We begin by describing this group and some of its subgroups which are relevant for these notes. The emphasis is on various decompositions of G with respect to these subgroups. Let F be a non-Archimedean local field. Let OF be its ring of integers whose unique maximal ideal is PF . Let ̟F be a uniformizer for F, i.e., PF = ̟F OF . Let qF be the cardinality of the residue field kF = OF /PF . The group G = GLn (F ) is the group of all invertible n × n matrices with entries in F. A more invariant description is that it is the group of all invertible linear transformations of an n dimensional F vector space V and in this case it is denoted GL(V ). If we fix a basis of V then we can identify GL(V ) with GLn (F ). We let B denote the subgroup of G consisting of all upper triangular matrices. Let T denote the subgroup of all diagonal matrices and let U denote the subgroup of all upper triangular unipotent matrices. Note that T normalizes U and B is the semi-direct product T U. This B is called the standard Borel subgroup with U being its unipotent radical and T is called the diagonal torus. We let U − to be the subgroup of all lower triangular unipotent matrices. Let W , called the Weyl group of G, denote the group NG (T )/T where NG (T ) is the normalizer in G of the torus T. It is an easy exercise to check that NG (T ) is the subgroup of all monomial matrices and so W can be identified with Sn the symmetric group on n letters. We will usually denote a diagonal matrix in T as diag(a1 , ..., an ). We now introduce the Borel subgroup and more generally the parabolic subgroups in a more invariant manner. Let V be an n dimensional F vector space. Define a flag in V to be a strictly increasing sequence of subspaces W• = {W0 ⊂ W1 ⊂ · · · Wm = V }. The subgroup of GL(V ) which stabilizes the flag W• , i.e., with the property that gWi = Wi for all i is called a parabolic subgroup of G associated to the flag W• . If {v1 , ..., vn } is a basis of V then the stabilizer of the flags of the form W• = {(v1 ) ⊂ (v1 , v2 ) ⊂ · · · (v1 , v2 , ..., vn ) = V }

Representation Theory of GL(n) over Non-Archimedean Local Fields 169 is called a Borel subgroup. It can be seen that GL(V ) operates transitively on the set of such flags, and hence the stabilizer of any two such (maximal) flags are conjugate under GL(V ). If W• = {W0 ⊂ W1 ⊂ · · · Wm = V }, then inside the associated parabolic subgroup P , there exists the normal subgroup N consisting of those elements which operate trivially on Wi+1 /Wi for 0 ≤ i ≤ m − 1. The subgroup N is called the unipotent radical of P . It can be seen that there is a semi direct product decomposition P = M N with m−1 Y GL(Wi+1 /Wi ). M= i=0

The decomposition P = M N is called a Levi decomposition of P with N the unipotent radical, and M a Levi subgroup of P. We now introduce some of the open compact subgroups of G which will be relevant to us. We let K = GLn (OF ) denote the subgroup of elements G with entries in OF and whose determinant is a unit in OF . This is an open compact subgroup of G. The following exercise contains most of the basic properties of K. Exercise 3.1 Let V be an n dimensional F vector space. Let L be a lattice in V, i.e., an OF submodule of rank n. Show that stabG (L) is an open compact subgroup of G = GL(V ). If C is any open compact subgroup then show that there is a lattice L such that C ⊂ stabG (L). Deduce that up to conjugacy K is the unique maximal open compact subgroup of GLn (F ). For every integer m ≥ 1, the map OF → OF /Pm F induces a map K → m GLn (OF /PF ). The kernel of this map, denoted Km is called the principal congruence subgroup of level m. We also define K0 to be K. For all m ≥ 1, we have Km = {g ∈ GLn (OF ) : g − 1n ∈ ̟Fm Mn (OF )} . Note that Km is an open compact subgroup of G and gives a basis of neighbourhoods at the identity. Hence G is an l-group, i.e., a locally compact totally disconnected topological group. We are now in a position to state the main decomposition theorems which will be of use to us later in the study of representations of G.

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Theorem 3.2 (Bruhat decomposition) G=

a

BwB =

w∈W

a

BwU.

w∈W

Proof: This is an elementary exercise in basic linear algebra involving row reduction (on the left) and column reduction (on the right) by elementary operations. Since the rank of an element in G is n we end up with a monomial matrix with such operations and absorbing scalars into T ⊂ B we end up with an element in W. We leave the details to the reader. The disjointness of the union requires more work and is not obvious. We refer the reader to Theorem 8.3.8 of [14]. 2 Theorem 3.3 (Cartan decomposition) Let A = {diag(̟Fm1 , ..., ̟Fmn ) : mi ∈ Z, m1 ≤ m2 ≤ · · · ≤ mn }. a G= K ·a·K a∈A

Proof : Let g ∈ G. After fixing a basis for an n dimensional F vector space V we identify GL(V ) with GLn (F ). Let L be the standard lattice in V corresponding to this basis. Let L1 be the lattice g(L). Let r be the least integer such that ̟Fr L is contained in L1 and let L2 = ̟Fr L. The proof of the Cartan decomposition falls out from applying the structure theory of finitely generated torsion modules over principal ideal domains. In our context we would apply this to the torsion OF module L1 /L2 . We urge the reader to fill in the details. 2 Theorem 3.4 (Iwasawa decomposition) G=K ·B Proof : We sketch a proof for this decomposition for GL2 (F ). The proof for GLn (F ) uses induction on n and the same matrix manipulations as in the GL2 (F ) case is used in reducing from n to n− 1. Assume that G = GL2 (F ). Let g = ac db ∈ G. If a = 0 then write g as w(w−1 g) ∈ K · B. If a 6= 0 and if a−1 c ∈ OF then 1 0 ∗ ∗ g= ∈ K · B. ca−1 1 0 ∗

Representation Theory of GL(n) over Non-Archimedean Local Fields 171 If a−1 c ∈ / OF then replace g by kg where 1 1 + ̟Fr k= ∈ K. 1 ̟Fr We may choose r large enough such that the (modified) a and c satisfy the property a−1 c ∈ OF which gets us to the previous case and we may proceed as before. 2 Theorem 3.5 (Iwahori factorization) For m ≥ 1, Km = (Km ∩ U − ) · (Km ∩ T ) · (Km ∩ U ). Proof : This is an easy exercise in row and column reduction. See paragraph 3.11 of [2]. 2 We are now in a position to begin the study of representations of G = GLn (F ). The purpose of these notes is to give a detailed account of irreducible representations of G with emphasis on those topics which are relevant to the theory of automorphic forms. The first point to notice is that most representations we deal with are infinite dimensional. We leave this as the following exercise. Exercise 3.6 1. Show that the derived group of GLn (F ) is SLn (F ), the subgroup of determinant one elements. Any character χ of F ∗ gives a character g 7→ χ(det(g)) of G = GLn (F ) and every character of G looks like this. 2. Show that a finite dimensional smooth irreducible representation of G is one dimensional and hence is of the form g 7→ χ(det(g)). Note that any representation π of G can be twisted by a character χ denoted as π ⊗ χ whose representation space is the same as π and is given by (π ⊗ χ)(g) = χ(det(g))π(g). It is trivial to see that π is irreducible if and only if π ⊗ χ is irreducible and that ωπ⊗χ = ωπ χn as characters of the centre Z ≃ F ∗ . For any complex number s we will denote π(s) to be the representation π ⊗ | · |sF . Even if one is primarily interested in irreducible representations, some natural constructions such as parabolic induction defined in the next section, force us to also consider representations which are not irreducible. However

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they still would have certain finiteness properties. We end this section with a quick view into these finiteness statements. Let (π, V ) be a smooth representation of G. We call π to be a finitely generated representation if there exist finitely many vectors v1 , ..., vm in V such that the smallest G invariant subspace of V containing these vectors is V itself. We say that π has finite length if there is a sequence of G invariant subspaces (0) = V0 ⊂ V1 ⊂ · · · ⊂ Vm = V such that each successive quotient Vi+1 /Vi for 0 ≤ i ≤ m − 1 is an irreducible representation of G. The following theorem is a non-trivial theorem with a fairly long history and various mathematician’s work has gone into proving various parts of it (especially Harish-Chandra, R. Howe, H. Jacquet and I.N. Bernstein). (See Theorem 4.1 of [2]. The introduction of this same paper has some relevant history.) Theorem 3.7 Let (π, V ) be a smooth representation of G = GLn (F ). Then the following are equivalent : 1. π has finite length. 2. π is admissible and is finitely generated. Exercise 3.8

1. Prove the above theorem for G = GL1 (F ) = F ∗ .

2. Show that the sum of all characters of F ∗ on which the uniformizer ̟F acts trivially is an admissible representation which is not of finite length. 3. Let f be the characteristic function of OF× thought of as a subset of F ∗ . Let V be the representation of F ∗ generated by f inside the regular representation of F ∗ on Cc∞ (F ∗ ). Show that V gives an example of a smooth finitely generated representation which is not of finite length.

4

Parabolic induction

One important way to construct representations of G is by the process of parabolic induction. Let P = M N be a parabolic subgroup of G = GLn (F ). Recall that P is the stabilizer of some flag and its unipotent radical is the subgroup which

Representation Theory of GL(n) over Non-Archimedean Local Fields 173 acts trivially on all successive quotients. More concretely, for a partition n = n1 + n2 + · · · + nk of n, let P = P (n1 , n2 , ..., nk ) be the standard parabolic subgroup given by block upper triangular matrices : g1 ∗ ∗ ∗ g ∗ ∗ 2 : gi ∈ GLn (F ) . P = i . . gk

The unipotent radical of P is the block upper triangular unipotent matrices given by : 1n1 ∗ ∗ ∗ 1 ∗ ∗ n 2 NP = N = . . 1nk and the Levi subgroup of P is the block diagonal subgroup : g 1 k Y g 2 GLni (F ). MP = M = : gi ∈ GLni (F ) ≃ . i=1 gk

Let (ρ, W ) be a smooth representation of M. Since M is the quotient of P by N we can inflate ρ to a representation of P (sometimes referred to as ‘extending it trivially across N’) also denoted ρ. Now we can consider IndG P (ρ). (See § 2.) We say that this representation is parabolically induced from M to G. To recall, IndG P (ρ) consists of all locally constant functions f : G → Vρ such that f (pg) = δP (p)1/2 ρ(p)f (g) where δP (p) = |det(Ad(p)|Lie(N ) )|F . (See Exercise 2.6.) The following theorem contains the basic properties of parabolic induction. Theorem 4.1 Let P = M N be a parabolic subgroup of G = GLn (F ). Let (ρ, W ) be a smooth representation of M. 1. The functor ρ 7→ IndG P (ρ) is an exact additive functor from the category of smooth representations of M to the category of smooth representations of G.

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G 2. IndG P (ρ) = indP (ρ). G ∨ ∨ 3. IndG P (ρ ) ≃ IndP (ρ) .

4. If ρ is unitary then so is IndG P (ρ). 5. If ρ is admissible then so is IndG P (ρ). 6. If ρ is finitely generated then so is IndG P (ρ). Proof : General properties of induction stated in Theorem 2.7 gives (1). Iwasawa decomposition (Theorem 3.4) implies (2). Theorem 2.7 and (2) imply (3). For a proof of (4) observe that given an M invariant unitary structure (see Definition 7.1) on W we can cook up a G invariant unitary structure on IndG P (ρ) by integrating functions against each other which is justified by (2). We urge the reader to fill in the details. We now prove (5). It suffices to prove the space of vectors fixed by Km (for any m) is finite dimensional where Km is the principal congruence subgroup of level m. Let f ∈ IndG P (ρ) which is fixed by Km . Note that the Iwasawa decomposition (Theorem 3.4) gives that P \G/Km is a finite set. Let g1 , ..., gr be a set of representatives for this double coset decomposition. We may and shall choose these elements to be in K. The function f is completely determined by its 1/2 values on the elements gi . Since we have f (mgi k) = δP (m)ρ(m)f (gi ) for all m ∈ M and k ∈ Km we get that f (gi ) is fixed by M ∩ (gi Km gi−1 ) = M ∩ Km which is simply the principal congruence subgroup of level m for M. Hence each of the f (gi ) takes values in a finite dimensional space by admissibility Km is finite dimensional. of ρ which implies that IndG P (ρ) Now for the proof of (6) we give the argument to show that the space of locally constant functions on P \G is a finitely generated representation of G. The argument for general parabolically induced representations is similar. To prove that the space of locally constant functions on P \G is finitely generated, it is sufficient to treat the case when P is the Borel subgroup B of the group of upper triangular matrices. (We do this because the essence of the argument is already seen in the case of B\G. It is however true that a subrepresentation of a finitely generated representation of G is itself finitely generated. See Theorem 4.19 of [2].) The proof of finite generation of the space of locally constant functions on B\G depends on the Iwahori factorization (Theorem 3.5). Recall that if

Representation Theory of GL(n) over Non-Archimedean Local Fields 175 Km is a principal congruence subgroup of level m then we have Km = (Km ∩ U − ) · (Km ∩ T ) · (Km ∩ U ). − = K ∩ U − to We note that there are elements in T which shrink Km m the identity. For example if we take the matrix

µ = diag(1, ̟F , ̟F2 , ..., ̟Fn−1 ) − to the identity, then the powers of µ have the property that they shrink Km −i − i i.e., limi→∞ µ Km µ = {1}. Let χX denote the characteristic function of a subset X of a certain −i . This ambient space. Look at the translates of χB·Km − by the powers µ will give us, µ−i · χBKm − = χ − i = χ − i BKm µ Bµ−i Km µ −i , we get the characteristic function of Therefore translating χBKm − by µ BC for arbitrarily small open compact subgroups C. These characteristic functions together with their G translates clearly span all the locally constant functions on B\G, completing the proof of (6). 2

We would like to emphasize that even if ρ is irreducible the representation IndG P (ρ) is in general not irreducible. However the above theorem assures us, using Theorem 3.7, that it is of finite length. In general, understanding when the induced representation is irreducible is an extremely important one and this is well understood for G. We will return to this point in the section on Langlands classification (§ 8). One instance of this is given in the next example. Example 4.2 (Principal series) Let χ1 , χ2 , ..., χn be n characters of F ∗ . Let χ be the character of B associated to these characters, i.e., a1 ∗ ∗ ∗ a2 ∗ ∗ = χ1 (a1 )χ2 (a2 )...χn (an ). χ . . an The representation π(χ) = π(χ1 , ..., χn ) of G obtained by parabolically inducing χ to G is called a principal series representation of G. It turns out that π(χ) is reducible if and only if there exist i 6= j such that χi = χj | · |F where |·|F is the normalized multiplicative absolute value on F. In particular if χ1 , ..., χn are all unitary then π(χ) is a unitary irreducible representation.

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Jacquet functors

Parabolic induction constructs representations of GLn (F ) from representations of its Levi subgroups. There is a dual procedure, more precisely, an adjoint functor, which constructs representations of Levi subgroups from representations of GLn (F ). The importance and basic properties of this construction was noted by Jacquet for the GLn (F ) case, which was generalized to all reductive groups by Harish-Chandra. Definition 5.1 Let P = M N be the Levi decomposition of a parabolic subgroup P of G = GLn (F ). For a smooth representation (ρ, V ) of P , define ρN to be the largest quotient of ρ on which N operates trivially. Let ρ(N ) = V (N ) = {n · v − v|n ∈ N, v ∈ V }. Then ρN = V /V (N ). This ρN is called the Jacquet functor of ρ. If ρ is a smooth representation of G then the Jacquet functor ρN of ρ is just that of ρ restricted to P. It is easily seen that the Jacquet functor ρN is a representation for M. This is seen by noting that M normalizes N. There is an easy and important characterization of this subspace V (N ) and this is given in the following lemma. Lemma 5.2 For a smooth representation V of N , V (N ) is exactly the space of vectors v ∈ V such that Z n · v dn = 0, KN

where KN is an open compact subgroup of N and dn is a Haar measure on N . (The integral is actually a finite sum.) Proof : The main property of N used in this lemma is that it is a union of open compact subgroups. Clearly the integral of the vectors of the form n · v − v on an open compact subgroup of N containing n is zero. On the other hand if the integral equals zero, then v ∈ V (N ) as it reduces to a similar conclusion about finite groups, which is easy to see. 2 The following theorem contains most of the basic properties of Jacquet functors. The reader is urged to compare this with Theorem 4.1.

Representation Theory of GL(n) over Non-Archimedean Local Fields 177 Theorem 5.3 Let (π, V ) be a smooth representation of G. Let P = M N be a parabolic subgroup of G. Then 1. The Jacquet functor V → VN is an exact additive functor from the category of smooth representations of G to the category of smooth representations of M. 2. (Transitivity) Let Q ⊂ P be standard parabolic subgroups of G with Levi decompositions P = MP NP and Q = MQ NQ . Hence MQ ⊂ MP , NP ⊂ NQ , and MQ (NQ ∩ MP ) is a parabolic subgroup of MP with MQ as a Levi subgroup and NQ ∩ MP as the unipotent radical, and we have (πNP )NQ ∩MP ≃ πNQ . 3. (Frobenius Reciprocity) For a smooth representation σ of M, 1/2

HomG (π, IndG P (σ)) ≃ HomM (πN , σδP ). 4. If π is finitely generated then so is πN . 5. If π is admissible then so is πN . Proof : For the proof of (1) it suffices to prove that V1 ∩ V (N ) = V1 (N ) which is clear from the previous lemma. We urge the reader to fix proofs of (2) and (3). For (2) note that NQ = NP (NQ ∩ MP ). For (3) it is an exercise in modifying the proof of the usual Frobenius reciprocity while using the definitions of parabolic induction and Jacquet functors. An easy application of Iwasawa decomposition (Theorem 3.4) gives (4). Statement (5) is originally due to Jacquet. We sketch an argument below as it is important application of Iwahori factorization. Let P = P (n1 , ..., nk ) be a standard parabolic subgroup and let P = M N be its Levi decomposition. The principal congruence subgroup Km of level m admits an Iwahori factorization which looks like + − 0 Km Km Km = Km + = K ∩ N and K 0 = K ∩ M and K − = K ∩ N − where where Km m m m m m P − = M N − is the opposite parabolic subgroup obtained by simply taking transposes of all elements in P. Let A : V → VN be the canonical projection. Under this projection we show that 0 A(V Km ) = (VN )Km .

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Clearly, using admissibility of π, this would prove admissibility of πN because 0 is the principal congruence subgroup of level m of M and these form a Km basis of neighbourhoods at the identity for M. 0 Since A is a P equivariant map, we get that A(V Km ) ⊂ (VN )Km . Let 0 v¯ ∈ (VN )Km . Choose v ∈ V such that A(v) = v¯. For any z ∈ ZM , the centre of M, given by m1 ̟F 1n1 ̟Fm2 1n2 z= . mk ̟F 1nk let t(z) = maxi,j |mi − mj |. Let v1 = z −1 v. Then A(v1 ) = A(z −1 v) = 0. z −1 A(v) = z −1 v¯. Since z is central in M , A(v1 ) is also fixed by Km − z −1 ⊂ stab (v). Hence K − will fix Choose t(z) ≫ 0 such that zKm G m −1 − and A(v ) is fixed by K 0 and v1 = z v. To summarize, v1 is fixed by Km 1 m + . (The latter because N acts trivially on V .) Km N m ∩ P ) = 1. Let v2 = R Choose a Haar measure on G such that vol(K−1 ¯. The upshot is that Km π(k)v1 dk. It is easy to check that A(v2 ) = z v 0

given v¯ ∈ (VN )Km there exists z ∈ ZM such that z −1 v¯ ∈ A(V Km ). 0 Now if v¯1 , ..., v¯r are any linearly independent vectors in (VN )Km then there exists z ∈ ZM such that z −1 v¯1 , ..., z −1 v¯ are in A(V Km ) and are also 0 linearly independent. This implies that the dimension of (VN )Km is bounded 2 above by that of A(V Km ) and hence they must be equal. We now compute the Jacquet functor of the principal series representation introduced in Example 4.2. As a notational convenience for a smooth representation π of finite length of any l-group H, we denote by πss the semisimplification of π, i.e., if π = π0 ⊃ π1 ⊃ · · · ⊃ πn = {0} with each πi /πi+1 n−1 irreducible, then πss = ⊕i=0 (πi /πi+1 ). By the Jordan-Holder theorem, πss is independent of the filtration π = π0 ⊃ π1 ⊃ · · · ⊃ πn = {0}.

Theorem 5.4 (Jacquet functor for principal series) Let π be the prinGL (F ) cipal series representation π(χ) = IndB n (χ). Then the Jacquet functor of π with respect to the Borel subgroup B = T U is given as a module for T as X 1/2 (πU )ss ≃ χw δB w∈W

χw

where denotes the character of the torus obtained by twisting χ by the element w in the Weyl group W , i.e χw (t) = χ(w(t)).

Representation Theory of GL(n) over Non-Archimedean Local Fields 179 Proof : The representation space of π can be thought of as a certain space of “functions on G/B twisted by the character χ”; more precisely, π can be thought of as the space of locally constant functions on G/B with values in a sheaf Eχ obtained from the character χ of the Borel subgroup B. If Y is a closed subspace of a topological space X “of the kind that we are considering here”, e.g. locally closed subspaces of the flag variety, then there is an exact sequence, 0 → Cc∞ (X − Y, Eχ |X−Y ) → Cc∞ (X, Eχ ) → Cc∞ (Y, Eχ |Y ) → 0. It follows that Mackey’s theory (originally for finite groups) about restriction of an induced representation to a subgroup holds good for p-adic ` groups too. Hence using the Bruhat decomposition GL(n) = w∈W BwB, and denoting B ∩ wBw−1 to be T · Uw , we have X 1/2 w (ResB IndG indB B (χ))ss = B∩wBw −1 ((χδB ) ) w∈W

=

X

1/2

w indB T ·Uw ((χδB ) )

w∈W

=

X

1/2

Cc∞ (U/Uw , (χδB )w ).

w∈W

We now note that the largest quotient of Cc∞ (U/Uw ) on which U operates trivially is one dimensional (obtained by integrating a function with respect to a Haar measure on U/Uw ) on which the action of the torus T is the sum of positive roots which are not in Uw which can be seen to be [δB · −1 −w .) This implies that stands for the w translate of δB (δB )−w ]1/2 . (Here δB 1/2 w ∞ the largest quotient of Cc (U/Uw , (χδB ) ) on which U operates trivially is 1/2 the 1 dimensional T -module on which T operates by the character χw δB . Hence, i h X GL (F ) 1/2 IndB n (χ) ≃ χw δB . U ss

w∈W

2 GL (F )

GLn (F )

Corollary 5.5 If HomGLn (F ) (IndB n (χ), IndB χ′ = χw for some w in W , the Weyl group.

(χ′ )) is nonzero then

Proof : This is a simple consequence of the Frobenius reciprocity combined with the calculation of the Jacquet functor done above. 2

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Example 5.6 Let π(χ) denote a principal series representation of GL2 (F ). Then 1/2

1/2

1. If χ 6= χw , then π(χ)U ≃ χδB ⊕ χw δB . 2. If χ = χw , then π(χ)U is a non-trivial extension of T -modules: 1/2

1/2

0 → χδB → π(χ)U → χδB → 0. Exercise 5.7 With notation as in the previous exercise, prove that a principal series representation of GL2 (F ) induced from a unitary character is irreducible. Exercise 5.8 Let G be an abelian group with characters χ1 and χ2 . Prove that if χ1 6= χ2 , then any exact sequence of G-modules, 0 → χ1 → V → χ2 → 0, splits. We end this section with a theorem about Jacquet modules for parabolically induced representations. This theorem is at the basis for considering supercuspidal representations (which are those representations for which all Jacquet modules are trivial). Theorem 5.9 Let P = M N be a parabolic subgroup of G = GLn (F ). Let σ be a smooth irreducible representation of M. Let (π, V ) be any ‘subquotient’ of IndG P (ρ). Then the Jacquet module πN of π is non-zero. Proof : Note that if π is subrepresentation of the induced representation then it easily follows from Frobenius reciprocity. Indeed, we have 1/2

(0) 6= HomG (π, IndG P (ρ)) = HomM (πN , δP ρ) which implies the πN 6= (0). In general if π is a subquotient then there is a ‘trick’ due to Harish-Chandra (see Corollary 7.2.2 in [7]) using which π can be realized as a subrepresentation of a Weyl group ‘twist’ of the induced representation, which will bring us to the above case. Since this trick is very important and makes its presence in quite a few arguments we delineate it as the following theorem. 2

Representation Theory of GL(n) over Non-Archimedean Local Fields 181 We will need a little bit of notation before we can state this theorem. Let P = M N be a parabolic subgroup of G. Let x ∈ NG (M ) - the normalizer in G of M. Then we can consider the representation xρ of M whose representation space is same as that of ρ and the action is given by xρ(m) = ρ(x−1 mx). x Now we can consider the induced representation IndG P ( ρ). Note that this representation is in general not equivalent to the original induced representation. Theorem 5.10 Let P = M N be a parabolic subgroup of G = GLn (F ). Let σ be an smooth irreducible representation of M. Let (π, V ) be any irreducible subquotient of IndG P (σ). Then there exists an element w ∈ W ∩NG (M ) (where w W is the Weyl group) such that π is a subrepresentation of IndG P ( σ).

6

Supercuspidal representations

A very important and novel feature of p-adic groups (compared to real reductive groups) is the existence of supercuspidal representations. We will see that these representations are the building blocks of all irreducible admissible representations of p-adic groups. A complete set of supercuspidals for GLn (F ) was constructed by Bushnell and Kutzko in their book [5]. The local Langlands correspondence, proved by Harris and Taylor and also by Henniart [9], interprets supercuspidal representations of GLn (F ) in terms of irreducible n dimensional representations of the Galois group of F. Before we come to the definition of a supercuspidal representation, we need to define the notion of a matrix coefficient of a representation. For a smooth representation (π, V ) of GLn (F ) recall that (π ∨ , V ∨ ) denotes the contragredient representation of (π, V ). For vectors v in π and v ∨ in π ∨ , define the matrix coefficient fv,v∨ to be the function on GLn (F ) given by fv,v∨ (g) = hv ∨ , π(g)vi. Theorem 6.1 Let (π, V ) be an irreducible admissible representation of G = GLn (F ). Then the following are equivalent : 1. One matrix coefficient of π is compactly supported modulo the centre. 2. Every matrix coefficient of π is compactly supported modulo the centre. 3. The Jacquet functors of π (for all proper parabolic subgroups) are zero. 4. The representation π does not occur as a subquotient of any representation parabolically induced from any proper parabolic subgroup.

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A representation satisfying any one of the above conditions of the theorem is called a supercuspidal representation. We refer the reader to paragraph 3.21 of [2] for a proof of this theorem. We note that every irreducible admissible representation of GL1 (F ) = F ∗ is supercuspidal. However for GLn (F ) with n ≥ 2 since parabolic induction is of no use in constructing supercuspidal representations, a totally new approach is needed. One way to construct supercuspidal representations is via induction from certain finite dimensional representations of compact open subgroups. This has been a big program in recent times which has been completed for the case of GLn by Bushnell and Kutzko in their book [5]. They prove that any supercuspidal representation of GLn (F ) is obtained by induction from a finite dimensional representation of certain open compact modulo centre subgroup of GLn (F ). Here is a sample of such a construction in the simplest possible situation. These are what are called depth zero supercuspidal representations of GLn (F ). By a cuspidal representation of the finite group GLn (Fq ) we mean an irreducible representation for which all the Jacquet functors are zero (equivalently there are no non-zero vectors fixed by N (Fq ) for unipotent radicals N of any proper parabolic subgroup). Cuspidal representations of GLn (Fq ) are completely known by the work of J.Green [8] in the 50’s. Theorem 6.2 Consider a representation of GLn (Fq ) to be a representation of GLn (OF ) via the natural surjection from GLn (OF ) to GLn (Fq ). If σ is an irreducible cuspidal representation of GLn (Fq ) thought of as a representation of GLn (OF ) and χ is a character of F ∗ whose restriction to OF× is the same as the central character of σ then χ · σ is a representation of F ∗ GLn (OF ). GLn (F ) (χ · σ). Then π Let π be the compactly induced representation indF ∗ GL n (OF ) is an irreducible admissible supercuspidal representation of GLn (F ). The proof we give is based on Proposition 1.5 in [6]. To begin with we need a general lemma which describes the restriction of an induced representation in one special context that we are interested in. Lemma 6.3 Let H be an open compact-mod-centre subgroup of a unimodular l-group G. Let (σ, W ) be a smooth finite dimensional representation of H. Let π = indG H (σ) be the compact induction of σ to a representation π of G. Then the restriction of π to H is given by : M g π|H = indG (σ) ≃ IndH H H∩g −1 Hg ( σ|H∩g −1 Hg ) H g∈H\G/H

Representation Theory of GL(n) over Non-Archimedean Local Fields 183 where gσ is the representation of g−1 Hg given by gσ(g−1 hg) = σ(h) for all h ∈ H. Proof of Lemma 6.3 : For g ∈ G, let Vg = C ∞ (HgH, σ) denote the space of smooth functions f on HgH such that f (h1 gh2 ) = σ(h1 )f (gh2 ). Since HgH is open in G and is also compact mod H we get a canonical injection Vg → indG H (σ) given by extending functions by zero outside HgH. All the inclusions Vg → π gives us a canonical map M

Vg → π.

g∈H\G/H

This map is an isomorphism of H modules. This can be seen as follows : This map is clearly injective. (Consider supports of the functions.) Surjectivity follows from the definition of π. Note also that Vg is H stable if H acts by right shifts and the map Vg → π is H equivariant. g The map sending f → f¯ from Vg to IndH H∩g −1 Hg ( σ|H∩g −1 Hg ) where f¯(h) = f (gh) is easily checked to be a well-defined H equivariant bijection. This proves the lemma. 2 Proof of Theorem 6.2 : Let H denote the subgroup F ∗ GLn (OF ) which is an open and compact-mod-centre subgroup of G = GLn (F ). For brevity let σ also denote the representation χ · σ of H. We will prove irreducibility, admissibility and supercuspidality of π separately, which will prove the theorem. Irreducibility: We begin by proving irreducibility of π. We make the following claim : Claim : For all g ∈ G − H, we have HomH∩g−1 Hg (σ, gσ) = (0). Assuming the claim for the time being we prove the theorem as follows. The claim and the previous lemma implies that σ appears in π with multiplicity one, i.e., dim(HomH (σ, π|H)) = 1. Now suppose π is not irreducible then there exists an exact sequence of non-trivial G modules : 0 −→ π1 −→ π −→ π2 −→ 0. We note that since H is compact-mod-centre, any representation (like σ, π1 , π2 and π) on which F ∗ acts by a character is necessarily semi-simple as an HG module. Since π = indG H σ it embeds in IndH (σ) as a G module and hence so

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does π1 . Using Frobenius reciprocity 2.7.4(a) we get that HomH (π1 , σ) 6= (0), i.e., σ occurs in π1 . By Frobenius reciprocity 2.7.4(b) ∨ ∨ HomG (indG H σ, π2 ) = HomH (σ, (π2 |H ) ).

Since H is an open subgroup ((π2∨ )H )∨ = π2 as H-modules. Thus we get that σ occurs in π2 also but this contradicts the fact that σ occurs with multiplicity one in π. It suffices now to prove the claim. Since g ∈ G − H, we can assume without loss of generality, using the Cartan decomposition (see Theorem 3.3), m that g = diag(̟Fm1 , ..., ̟F n−1 , 1) where m1 ≥ ... ≥ mn−1 ≥ 0. Choose k such that mk ≥ 1 and mk+1 = 0. Suppose that HomH∩g−1 Hg (σ, gσ) 6= (0). Then we also have HomNk (O)∩g−1 Nk (O)g (σ, gσ) 6= (0) where Nk is the unipotent radical of the standard parabolic subgroup corresponding to the partition n = k + (n − k). We have Nk (O) ∩ gNk (O)g−1 ⊂ Nk (P) - the notations being obvious. Since σ is inflated from GLn (Fq ), gσ is trivial on Nk (O) ∩ g−1 Nk (O)g, i.e., HomNk (O)∩g−1 Nk (O)g (σ, 1) 6= (0). This contradicts the fact that (σ, W ) is a cuspidal representation of GLn (Fq ). (We urge the reader to justify this last statement.) Admissibility: We now prove that π is an admissible representation. It suffices to show that for m ≥ 1, π Km is finite dimensional, where Km is the principal congruence subgroup of level m. Note that π Km consists of locally constant functions f : G → W such that f (hgk) = σ(h)f (g), ∀h ∈ H, ∀g ∈ G, ∀k ∈ Km . Using the Cartan decomposition (see Theorem 3.3), we may choose representatives for the double cosets H\G/Km which look like g = a · k, where a = diag(̟Fm1 , · · · , ̟Fmn ) with 0 = m1 ≤ m2 ≤ · · · ≤ mn , and k ∈ K/Km . For such an a, let t(a) = max{mi+1 − mi : 1 ≤ i ≤ n − 1}. Note that if t(a) ≥ m, and if t(a) = mj+1 − mj , then Uj (O) ⊂ gKm g−1 ∩ H where Uj is the unipotent radical of the maximal parabolic corresponding to the partition n = j + (n − j). Therefore, σ(u) · f (g) = f (ug) = f (g · g−1 ug) = f (g), i.e., f (g) is a vector in W which is fixed by Uj (O) and hence Uj (Fq ). Cuspidality of (σ, W ) implies that f (g) = 0. This shows that if 0 6= f ∈ π Km , then f can be supported only on double cosets HakKm with t(a) < m, which is a finite set, proving the finite dimensionality of π Km .

Representation Theory of GL(n) over Non-Archimedean Local Fields 185 Supercuspidality: Let P = M · N be a parabolic subgroup of G. To prove that π is supercuspidal, we need to show that the Jacquet module πN = (0). It suffices to show that (πN )∗ = (0). Note that HomN (π, C) ∼ = (π ∗ )N is the space of N -fixed vectors in the vector space dual of π. We will show that (π ∗ )N = 0. ∗ Since π = indG H (σ), the dual vector space π may be identified with ∞ ∗ C (H\G, σ ) which is the space of locally constant functions φ on G with values in W ∗ such that φ(hg) = σ ∗ (h)φ(g). (We urge the reader to justify this identification; it boils down to saying that the dual of a direct sum of vector spaces is the direct product of vector spaces.) Hence (π ∗ )N consists of locally constant functions φ : G → W ∗ such that φ(hgn) = σ ∗ (h)φ(g),

∀h ∈ H, ∀g ∈ G, ∀n ∈ N.

Using Iwasawa decomposition, we may take representatives for double cosets H\G/N to lie in M . For m ∈ M , note that N (O) = N ∩ H = H ∩ mN m−1 . Hence for all h ∈ N (O), we have σ ∗ (h)φ(m) = φ(hm) = φ(m · m−1 hm) = φ(m), i.e., φ(m) ∈ W ∗ is fixed by N (Fq ). Cuspidality of (σ, W ) implies cuspidality of (σ ∗ , W ∗ ) which gives that φ(m) = 0. Hence (π ∗ )N = (0). 2 Exercise 6.4 With notations as in Theorem 6.2 show that GL (F )

GL (F )

n n (χ · σ). (χ · σ) = IndF ∗ GL indF ∗ GL n (OF ) n (OF )

We next have the following basic theorem which justifies the assertion made in the beginning of this section that supercuspidal representations are the building blocks of all irreducible representations. This theorem will be refined quite a bit in the section on Langlands classification. Theorem 6.5 Let π be an irreducible admissible representation of G = GLn (F ). Then there exists a Levi subgroup M and a supercuspidal representation ρ of M such that π is a subrepresentation of the representation of G obtained from ρ by the process of parabolic induction.

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Proof : Let P be a parabolic subgroup of GLn (F ) which is smallest for the property that the Jacquet functor with respect to P is non-zero. (So P = G is a possibility which occurs if and only if the representation is supercuspidal.) If P = M N , it is clear from theorem 5.3 that any irreducible subquotient of πN is a supercuspidal representation of M . Let ρ be an irreducible quotient of πN as an M module. Since HomM (πN , ρ) 6= (0), it follows from the Frobenius reciprocity that GLn (F )

HomGLn (F ) (π, IndP

−1/2

(δP

ρ)) 6= (0),

which gives a realization of π inside a representation parabolically induced from a supercuspidal representation. 2 Given the previous theorem one might ask if given an irreducible π ‘how many’ induced representations can it occur in? The next theorem says that π occurs in essentially only one such induced representation. Such a uniqueness assertion then allows us to talk of the supercuspidal support of the given representation. This is the following theorem due to Bernstein and Zelevinsky. (see Theorem 2.9 of [3].) We need some notation to state the theorem. If π is a representation of finite length then let JH(π) denote the set of equivalence classes of irreducible subquotients of π. Further, let JH0 (π) denote the set of all irreducible subquotients of π counted with multiplicity. So each ω in JH(π) is contained in JH0 (π) with some multiplicity. Theorem 6.6 Let P = M N and P ′ = M ′ N ′ be standard parabolic subgroups of G = GLn (F ). Let σ (resp. σ ′ ) be an irreducible supercuspidal G ′ ′ representation of M (resp. M ′ ). Let π = IndG P (σ) and π = IndP ′ (σ ). Then the following are equivalent. 1. There exists w ∈ W such that wM w−1 = M ′ and w σ = σ ′ . 2. HomG (π, π ′ ) 6= (0). 3. JH(π) ∩ JH(π ′ ) is not empty. 4. JH0 (π) = JH0 (π ′ ).

7

Discrete series representations

Note that one part of Theorem 6.1 says that every matrix coefficient of an irreducible supercuspidal representation is compactly supported modulo the

Representation Theory of GL(n) over Non-Archimedean Local Fields 187 centre. In general analytic behaviour of matrix coefficients dictate properties of the representation. Now we consider a larger class of representations which are said to be in the discrete series for G. These have a characterization in terms of their matrix coefficients being square integrable modulo the centre. We need a few definitions now. Albeit we have mentioned the notion of a unitary representation before (see Theorem 4.1) we now give a definition. Definition 7.1 Let (π, V ) be a smooth representation of G. We say π is a unitary representation of G if V has an inner product h , i (also called an unitary structure) which is G invariant, i.e., hπ(g)v, π(g)wi = hv, wi for all g ∈ G and all v, w ∈ V. In general, V equipped with h , i need only be a pre-Hilbert space. Definition 7.2 Let (π, V ) be a smooth irreducible representation of G. We say π is essentially square integrable if there is a character χ : F ∗ → R>0 such that |fv,v∨ (g)|2 χ(det g) is a function on Z\G for every matrix coefficient fv,v∨ of π, and Z |fv,v∨ (g)|2 χ(det g) dg < ∞.

Z\G

If χ can be taken to be trivial then π is said to be a square integrable representation and in this case it is said to be in the discrete series for G. Exercise 7.3 1. Let ν be a unitary character of F ∗ . Let L2 (Z\G, ν) denote the space of measurable functions f : G → C such that f (zg) = ν(z)f (g) for all z ∈ Z and g ∈ G and Z |f (g)|2 dg < ∞. Z\G

Show that L2 (Z\G, ν) is a unitary representation of G where G acts on these functions by right shifts. 2. Let (π, V ) be a discrete series representation. Check that its central character ωπ is a unitary character. Let 0 6= l ∈ V ∨ . Show that the map v 7→ fv,l is a non-zero G equivariant map from π into L2 (Z\G, ωπ ). Hence deduce that π is a unitary representation which embeds as a subrepresentation of L2 (Z\G, ωπ ). The classification of discrete series representations is due to Bernstein and Zelevinsky. We summarize the results below. To begin with we need a

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result on when a representation parabolically induced from a supercuspidal representation is reducible. (See Theorem 4.2 of [3].) Theorem 7.4 Let P = P (n1 , ..., nk ) = M N be a standard parabolic subgroup of G. Let σ = σ1 ⊗ · · · ⊗ σk be an irreducible representation of M with every σi an irreducible supercuspidal representation of GLni (F ). The parabolically induced representation IndG P (σ) is reducible if and only if there exist 1 ≤ i, j ≤ k with i 6= j, ni = nj and σi ≃ σj (1) = σj | · |F . We now consider some very specific reducible parabolically induced representations. Let n = ab and let σ be an irreducible supercuspidal representation of GLa (F ). Let P = M N be the standard parabolic subgroup corresponding to n = a + a + ... + a the sum taken b times, so M = GLa (F ) × ... × GLa (F ) the product taken b times. Let ∆ denote the segment ∆ = (σ, σ(1), ..., σ(b − 1)) thought of as the representation σ ⊗ σ(1) ⊗ ... ⊗ σ(b − 1) of M. Let IndG P (∆) denote the corresponding parabolically induced representation of G. By Theorem 7.4 this representation is reducible. We are now in a position to state the following theorem. (This theorem, as recorded in Theorem 9.3 of Zelevinsky’s paper [15], is normally attributed to Bernstein.) Theorem 7.5 With notations as above, we have : 1. For any segment ∆ the induced representation IndG P (∆) has a unique irreducible quotient. This irreducible quotient will be denoted Q(∆). 2. For any segment ∆ the representation Q(∆) is an essentially square integrable representation. Every essentially square integrable representation of G is equivalent to some Q(∆) for a uniquely determined ∆, i.e., for a uniquely determined a, b and σ. 3. The representation Q(∆) is square integrable if and only if it is unitary and also if and only if σ((b − 1)/2) is unitary. Example 7.6 (Steinberg Representation) One important example of a discrete series representation is obtained as follows. Take a = 1 and b = n with the notations as above. Hence the parabolic subgroup P is just (1−n)/2 as a (supercuspidal) the standard Borel subgroup B. Let σ = | · |F

Representation Theory of GL(n) over Non-Archimedean Local Fields 189 representation of GLa (F ) = F ∗ . Then the representation IndG P (∆) is just the regular representation of G on smooth functions on B\G. The corresponding Q(∆) is called the Steinberg representation of G. We will denote the Steinberg representation for GLn (F ) by Stn . It is a square integrable representation with trivial central character. It can also be defined as the alternating sum X Stn = (−1)rank(P ) Cc∞ (P \G), B⊂P

where B is a fixed Borel subgroup of G, P denotes a parabolic subgroup of G containing B, and rank(P ) denotes the rank of [M, M ] where M is a Levi subgroup of P . The essentially square integrable representations Q(∆) are also called generalized Steinberg representations. Exercise 7.7 Show that the Jacquet module with respect to B of the Steinberg representation of GLn (F ) is one dimensional. What is the character of T on this one dimensional space? Exercise 7.8 Show that the trivial representation of GLn (F ) is not essentially square integrable.

8

Langlands classification

In this section we state the Langlands classification for all irreducible admissible representations of GLn (F ). The supercuspidal representations were introduced in § 6. Then we used supercuspidal representations to construct all the essentially square integrable representations (the Q(∆)’s) in § 7. Not every representation is accounted for by this construction as for instance the trivial representation is not in this set (unless n = 1!). The Langlands classification builds every irreducible representation starting from the essentially square integrable ones. The theorem stated below is due to Zelevinsky. (See Theorem 6.1 of [15].) Before we can state the theorem we need the notion of when two segments are linked. Let ∆ = (σ, ..., σ(b − 1)) and ∆′ = (σ ′ , ..., σ ′ (b′ − 1)) be two segments where σ (resp. σ ′ ) is an irreducible supercuspidal representation of GLa (F ) (resp. GLa′ (F )). We say ∆ and ∆′ are linked if neither of them is included in the other and their union is a segment (so in particular a = a′ ). We say that ∆ precedes ∆′ if they are linked and there is a positive integer r such that σ ′ = σ(r).

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Theorem 8.1 (Langlands classification) For 1 ≤ i ≤ k, let ∆i be a segment for GLni (F ). Assume that for i < j, ∆i does not precede ∆j . Let n = n1 + ... + nk . Let G = GLn (F ) and let P be the parabolic subgroup of G corresponding to this partition. Then : 1. The parabolically induced representation IndG P (Q(∆1 )⊗···⊗Q(∆k )) admits a unique irreducible quotient which will be denoted Q(∆1 , ..., ∆k ). 2. Any irreducible representation of G is equivalent to some Q(∆1 , ..., ∆k ) as above. 3. If ∆′1 , ..., ∆′k′ is another set of segments satisfying the hypothesis then we have Q(∆1 , ..., ∆k ) ≃ Q(∆′1 , ..., ∆′k′ ) if and only if k = k′ and ∆i = ∆′s(i) for some permutation s of {1, ..., k}. Exercise 8.2 Give a realization of the trivial representation of G as a representation Q(∆1 , ..., ∆k ) in the Langlands classification. Exercise 8.3 (Representations of GL2 (F )) In this exercise all the irreducible admissible representations of GL2 (F ) are classified. Let G = GL2 (F ). 1. Let χ be a character of F ∗ . Then g 7→ χ(det(g)) gives a character of G. Show that these are all the finite dimensional irreducible admissible representations of G. 2. Let χ be a character of G as above. Let St2 be the Steinberg representation of G and let St2 (χ) = St2 ⊗ χ be the twist by χ. Show that any principal series representation is either irreducible or is reducible in which case, up to semi-simplification it looks like χ ⊕ St2 (χ) for some character χ. 3. Use the Langlands classification and make a list of all irreducible admissible representations of G. Show that any irreducible admissible representation is either one dimensional or is infinite dimensional and in which case it is either supercuspidal, or of the form St2 (χ) or is an irreducible principal series representation π(χ1 , χ2 ).

Representation Theory of GL(n) over Non-Archimedean Local Fields 191

9

Certain classes of representations

In this section we examine certain important classes of representations of G = GLn (F ). These notions are especially important in the theory of automorphic forms. (For readers familiar with such terms, local components of global representations tend to have such properties.)

9.1

Generic representations

Recall our notation that U is the unipotent radical of the standard Borel subgroup B = T U where T is the diagonal torus. (So U is the subgroup of all upper triangular unipotent matrices.) Since T normalizes U we get an ˆ consisting of all characters of U. A maximal orbit action of T on the space U ˆ is called a generic orbit and any character of U in a generic orbit of T on U is called a generic character of U. It is easily seen that a character of U is generic if and only if its stabilizer in T is the centre Z. We can construct (generic) characters of U as follows. Let u = u(i, j) be any element in U. Let ψF be a non-trivial additive character of U. Let a = (a1 , ..., an−1 ) be any (n − 1) tuple of elements of F. We get a character of U by u 7→ ψa (u) = ψF (a1 u(1, 2) + a2 u(2, 3) + ... + an−1 u(n − 1, n)). The reader is urged to check that every character of U is one such ψa . Further it is easily seen that ψa is generic if and only if each ai is non-zero. We will now fix one generic character of U , denoted Ψ. Fix for once and for all a non-trivial additive character ψF of F such that the maximal fractional ideal of F on which ψF is trivial is OF . We fix the following generic character of U given by: u 7→ Ψ(u) = ψF (u(1, 2) + ... + u(n − 1, n)). Now consider the representation IndG U (Ψ) which is the induction from U to G of the character Ψ. Note that the representation space of IndG U (Ψ) consists of all smooth functions f on G such that f (ng) = Ψ(n)f (g) for all n ∈ U and g ∈ G. Definition 9.1 An irreducible admissible representation (π, V ) is said to be generic if HomG (π, IndG U (Ψ)) is non-zero. If π is generic then using Frobenius reciprocity we get that there is a non-zero linear functional ℓ : V → C such

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that ℓ(π(n)v) = Ψ(n)ℓ(v) for all v ∈ V and n ∈ U. Such a linear functional is called a Whittaker functional for π. If π is generic then the representation space V may be realized on a certain space of functions f with the property that f (ng) = Ψ(n)f (g) for all n ∈ U and g ∈ G and the action of G on V is by right shifts on this realization. Such a realization is called a Whittaker model for π. One of the main results on generic representations is the following theorem (due to Shalika [13]) which says that Whittaker models are unique when they exist. This theorem is important for the global theory, for instance, to prove multiplicity one for automorphic forms on GL(n). Also, one way to attach local Euler factors for generic representations of G is via Whittaker models. Theorem 9.2 (Multiplicity one for Whittaker models) Let (π, V ) be an irreducible admissible representation of GLn (F ). Then the dimension of the space of Whittaker functionals is at most one, i.e., dimC (HomG (π, IndG U (Ψ))) ≤ 1. Put differently, if π admits a Whittaker model then it admits a unique one. The question now arises as to which representations are actually generic. The first theorem in this direction was due to Gelfand and Kazhdan which says that every irreducible admissible supercuspidal representation is generic. The classification of all generic representations of GLn (F ) is a theorem due to Zelevinsky (see Theorem 9.7 of [15]) which states that π is generic if and only if it is irreducibly induced from essentially square integrable representations. In particular one has that discrete series representations are always generic. We would like to point out that this phenomenon is peculiar to GLn (F ) and is false for general reductive p-adic groups. Theorem 9.3 (Generic representations) Let π = Q(∆1 , ..., ∆k ) be an irreducible admissible representation of GLn (F ). Then π is generic if and only if no two of the ∆i are linked in which case IndG P (Q(∆1 ) ⊗ · · · ⊗ Q(∆k )) is an irreducible representation, and π == Q(∆1 , ..., ∆k ) = IndG P (Q(∆1 ) ⊗ · · · ⊗ Q(∆k )). Exercise 9.4 1. Use the definition of genericity to show that one dimensional representations are never generic. Now verify the above theorem for one dimensional representations.

Representation Theory of GL(n) over Non-Archimedean Local Fields 193 2. Let G = GL2 (F ). Show that an irreducible admissible representation is generic if and only if it is infinite dimensional.

9.2

Tempered representations

The notion of a tempered representation is important in the global theory of automorphic forms. The importance is evidenced by what is called the generalized Ramanujan conjecture which says that every local component of a cuspidal automorphic representation of GL(n) is tempered. In this section we give the definition of temperedness and state a theorem due to Jacquet [11] which says when a representation is tempered. (It is closely related to Zelevinsky’s theorem in the previous section.) It says that a representation with unitary central character is tempered if and only if it is irreducibly induced from a discrete series representation (as against essentially square integrable representations to get the generic ones). Definition 9.5 An irreducible admissible representation (π, V ) is tempered if ωπ the central character of π is unitary and if one (and equivalently every) matrix coefficient fv,v∨ is in L2+ǫ (Z\G) for every ǫ > 0. Every unitary supercuspidal representation of G is tempered, since matrix coefficients of supercuspidals are compactly supported modulo centre and hence in L2+ǫ (Z\G). One may drop the unitarity and rephrase this as every supercuspidal being essentially tempered with an obvious meaning given to the latter. Theorem 9.6 (Tempered representations) Let π = Q(∆1 , ..., ∆k ) be an irreducible admissible representation of GLn (F ). Then π is tempered if and only if one actually has π = IndG P (Q(∆1 ) ⊗ · · · ⊗ Q(∆k )) with every Q(∆i ) being square-integrable. From Theorem 9.3 and Theorem 9.6 we get that every tempered representation of GLn (F ) is generic. We emphasize this point in light of the global context. A theorem of Shalika [13] says that every local component of a cuspidal automorphic representation of GLn is generic. The generalized Ramanujan conjecture for GLn says that every such local component is tempered. The reader is urged to construct examples of generic representations of GLn (F ) which are not tempered.

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Unramified or spherical representations

The notion of unramified representations is again of central importance in the global theory as almost all local components of global representations are unramified. An unramified representation is a GLn analogue of an unramified character of F ∗ which is just a character trivial on the units OF× . Such a character is called unramified because every character corresponding to an unramified extension via local class field theory is unramified (the norm map being surjective on the units). Recall that K = GLn (OF ) is the (up to conjugacy) unique maximal compact subgroup of G = GLn (F ). We let HK denote the spherical Hecke algebra of G which is the space of compactly supported bi-K-invariant functions on G. This is an algebra under convolution Z f (xy −1 )g(y) dy (f ∗ g)(x) = G

where dy is a Haar measure on G normalized such that vol(K) = 1. The identity element in HK is just the characteristic function of K. Exercise 9.7 (Gelfand) Show that HK is a commutative algebra. Consider the transpose map on G to show that it induces a map on HK which is both an involution and an anti-involution. (Hint: Use Theorem 3.3.) Actually more is known about this Hecke algebra and we state the following theorem of Satake regarding the structure of HK . Theorem 9.8 (Satake Isomorphism) The spherical Hecke algebra HK is canonically isomorphic to the Weyl group invariants of the space of Laurent polynomials in n variables, i.e., −1 W HK ≃ C[t1 , t−1 1 , ..., tn , tn ] .

Definition 9.9 Let (π, V ) be an irreducible admissible representation of G. It is said to be unramified if it has a non-zero vector fixed by K. The space of K fixed vectors denoted V K is a module for the spherical Hecke algebra HK . Let (π, V ) be an irreducible unramified representation of G. We know that V K is a module for HK . We can use Proposition 2.8 and the fact that HK is commutative to get that the space of fixed vectors is actually one

Representation Theory of GL(n) over Non-Archimedean Local Fields 195 dimensional and hence this gives a character of the spherical Hecke algebra which is called the spherical character associated to π. Using this proposition again gives that the spherical character uniquely determines the unramified representation. We now construct all the unramified representations of G. Let χ1 , ..., χn be n unramified characters of F ∗ (i.e., they are all trivial on the units of F ). Let π = π(χ1 , ..., χn ) be the corresponding principal series representation of GLn (F ). It is an easy exercise using the Iwasawa decomposition to see that π admits a non-zero vector fixed by K which is unique up to scalars. Hence π admits a unique subquotient which is unramified. It turns out that if the characters χi are ordered to satisfy the ‘does not precede’ condition of Theorem 8.1 then the unique irreducible quotient Q(χ1 , ..., χn ) of π is actually unramified and so is the unique unramified subquotient of π. We now state the main theorem classifying all the unramified representations of GLn (F ). Theorem 9.10 (Spherical representations) Let χ1 , ..., χn denote n unramified characters of F ∗ which are ordered to satisfy the ‘does not precede’ condition of Theorem 8.1. Then the representation Q(χ1 , ..., χn ) is an unramified representation of GLn (F ). Further, every irreducible admissible unramified representation is equivalent to such a Q(χ1 , ..., χn ). Proof : We briefly sketch the proof of the second assertion in the theorem. We show that the Q(χ1 , ..., χn ) exhaust all the unramified representations of G. This is done by appealing to Proposition 2.8 and the Satake isomorphism. The proof boils down to showing that every character of the spherical Hecke algebra is accounted for by one of the characters coming from a Q(χ1 , ..., χn ) and so by appealing to Statement (3) of Proposition 2.8 we would be done. Now via the Satake isomorphism, a character of HK determines and is determined by a set of n non-zero complex numbers which are the values taken by the character on the variables t1 , ..., tn . Now it is easy to check that the spherical character of Q(χ1 , ..., χn ) (or π(χ1 , ..., χn ) which is more easier to work with) takes the value χi (̟F ) on the variable ti . The remark that an unramified character is completely determined by its value on a uniformizer which can be any non-zero complex number finishes the proof. 2 Example 9.11 Let π = Q(χ1 , χ2 ) be an irreducible admissible unitary unramified generic representation of GL2 (F ). Then for i = 1, 2 it can be shown

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that q −1/2 ≤ |χi (̟F )| ≤ q 1/2 . In particular, there are such representations which are not tempered. The Ramanujan conjecture for GL2 (F ) would assert that if π is a local component of a cuspidal automorphic representation of GL2 then π is tempered. Note that almost all such local components are unitary unramified and generic and the Ramanujan conjecture would boil down to showing that |χi (̟F )| = 1. See the discussion on pp.332-334 of [1] for related matters and in particular as to how the Langlands program implies the Ramanujan conjecture.

9.4

Iwahori spherical representations

This is a class of representations which may not have an immediate motivation from the global theory but is important all the same for various local reasons. (Although justifying this claim may take us outside the scope of these notes as it requires more knowledge of the structure theory of p-adic groups.) The Iwahori subgroup I of G = GLn (F ) that we will be looking at is the inverse image of the standard Borel subgroup under the canonical map from GLn (OF ) → GLn (Fq ). By definition, I is a subgroup of K. We will be looking at those representations of G which have vectors fixed by I. For this reason, as before, we need to consider the Iwahori Hecke algebra HI which is the space of compactly supported bi-I-invariant functions on G with the algebra structure being given by convolution. Definition 9.12 An irreducible admissible representation (π, V ) of G is called Iwahori spherical if it has a non-zero vector fixed by the Iwahori subgroup I. In this case the space of fixed vectors V I is a simple module denoted (π I , V I ) for the Iwahori Hecke algebra HI . To begin with, unlike the spherical case, the Iwahori Hecke algebra is not commutative and admits simple modules which are not one dimensional. The structure of HI is also a little complicated to describe and would take us outside the scope of these notes. (See § 3 of Iwahori-Matsumoto [10] for the original description of this algebra. See also § 3 of Borel [4] for a more pleasanter-to-read version of the structure of HI .) We record the following rather special lemma on representations of the Iwahori Hecke algebra. See Proposition 3.6 of [4], although Borel deals only with semi-simple groups the proof goes through mutatis mutandis to our case. We need some notation.

Representation Theory of GL(n) over Non-Archimedean Local Fields 197 For any g ∈ G, let eIgI be the characteristic function of IgI as a subset of G. Clearly eIgI ∈ HI . If we normalize the Haar measure on G such that vol(I) = 1, then eI is the identity of element of HI . Lemma 9.13 Let (σ, W ) be a finite dimensional representation of HI . Then for any g ∈ G the endomorphism σ(eIgI ) is invertible. The main fact about Iwahori spherical representations is the following theorem due to Borel and Casselman. (See Theorem 3.3.3. of [7] and a strengthening of it in Lemma 4.7 of [4].) Recall our notation that B = T U is the standard Borel subgroup of G. The Iwahori subgroup I admits an ‘Iwahori-factorization’ with respect to B as I = I −I 0I + where I − = I ∩ U − , I 0 = I ∩ T which is the maximal compact subgroup of T and I + = I ∩ U. Theorem 9.14 Let (π, V ) be an admissible representation of G. Then the canonical projection from V to the Jacquet module VU induces an isomor0 phism from V I onto (VU )I . Proof : Let A : V → VU be the canonical projection map. By the proof of 0 statement (5) of Theorem 5.3 we have that A(V I ) = (VU )I . We now have to show that A is injective on V I . Suppose v ∈ V I is such that A(v) = 0. Then v ∈ V I ∩ V (U ). Choose a compact open subgroup U1 of U such that v ∈ V (U1 ) which gives that π(e R U1 )v = 0. (For any compact subset C of G we let π(eC )v stand for C π(g)v dg.) Choose r ≫ 0 such that µ−r U1 µr ⊂ I + where µ = diag(1, ̟F , ..., ̟Fn−1 ). We then get that π(eI + )(π(µ−r )v) = 0. The Iwahori factorization gives that π(eI )π(µ−r )v = 0. What we have shown is that there exists a g ∈ G such that π(eI )π(g)v = 0. Since v ∈ V I this implies that π(eIgI )v = 0. Now appealing to Lemma 9.13 we get that v = 0. 2 Theorem 9.15 (Iwahori spherical representations) Let χ1 , ..., χn be unramified characters of F ∗ . Let π(χ) = π(χ1 , ..., χn ) be the corresponding principal series representation. Then any irreducible subquotient of π(χ) is Iwahori spherical and every Iwahori spherical representation arises in this manner.

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Proof : Let π be any irreducible subquotient of π(χ). We know from Theorem 5.9 that the Jacquet module πU is non-zero. We also know from the computation of the Jacquet module of π(χ) in Theorem 5.4 that I 0 acts trivially on π(χ)U since every χi is unramified and hence I 0 acts trivially on 0 πU , i.e., (πU )I 6= (0). Using Theorem 9.14 we get therefore that π I 6= (0), i.e., π is Iwahori spherical. Now let π be any Iwahori spherical representation of G. We know that 0 I πU 6= (0) by the above theorem and hence there is an unramified character χ of T such that HomT (πU , χ) 6= (0). Appealing to Frobenius reciprocity finishes the proof. 2 Corollary 9.16 Let (π, V ) be any irreducible admissible representation of G = GLn (F ). Then the dimension of space of fixed vectors under I is bounded above by the order of the Weyl group W, i.e., dimC (V I ) ≤ n! Equivalently, the dimension of any simple module for the Iwahori Hecke algebra HI is bounded above by n!. Proof : The corollary would follow if we show that the space of I fixed vectors of an uramified principal series π(χ) is exactly n! which is the order of the Weyl group. This follows from Theorem 9.14 and Theorem 5.4. 2

10

Representations of local Galois Groups

The Galois group GF = Gal(F¯ /F ) has distinguished normal subgroups IF , the Inertia subgroup, and PF the wild Inertia subgroup which is contained in IF . The Inertia subgroup sits in the following exact sequence, ¯ q /Fq ) → 1. 1 → IF → Gal(F¯ /F ) → Gal(F ¯ q /Fq ) is given by the natural Here the mapping from Gal(F¯ /F ) to Gal(F action of the Galois group of a local field on its residue field. The Inertia group can be thought of as the Galois group of F¯ over the 1/d maximal unramified extension F un of F . Let F t = ∪(d,q)=1 F un (̟F ). The field F t is known to be the maximal tamely ramified extension of F un (An extension is called tamely ramified if the index of ramification is coprime to the characteristic of the residue field.) We have,

Representation Theory of GL(n) over Non-Archimedean Local Fields 199

Gal(F¯ /F )/IF IF /PF

ˆ ≃ Z Y ≃ Z× ℓ . ℓ6=p

One defines the Weil group WF of F to be the subgroup of Gal(F¯ /F ) ¯ q /Fq ) is an integral power of the Frobenius autowhose image inside Gal(F q morphism x → x . Representations of the Weil group which are continuous on the inertia subgroup are exactly those representations of the Weil group for which the image of the Inertia subgroup is finite. The Weil group is a dense subgroup of the Galois group hence an irreducible representation of the Galois group defines an irreducible representation of the Weil group. It is easy to see that a representation of the Weil group can, after twisting by a character, be extended to a representation of the Galois group. Local class field theory implies that the maximal abelian quotient of the Weil group of F is naturally isomorphic to F ∗ , and hence 1 dimensional representations of WF are in bijective correspondence with characters of GL1 (F ) = F ∗ . It is this statement of abelian class field theory which is generalized by the local Langlands correspondence. However, there is a slight amount of change one needs to make, and instead of taking the Weil group, one needs to take what is called the Weil-Deligne group whose representations are the same as representations of WF on a vector space V together with a nilpotent endomorphism N such that wN w−1 = |w|N ¯ q /Fq ) is the i-th power of the where |w| = q −i if the image of w in Gal(F Frobenius. One can identify representations of the Weil-Deligne group to representations of WF ×SL2 (C) via the Jacobson-Morozov theorem. It is usually much easier to work with WF × SL2 (C) but the formulation with the nilpotent operators appears more naturally in considerations of ℓ-adic cohomology of Shimura varieties where the nilpotent operator appears as the ‘monodromy’ operator. We end with the following important proposition for which we first note that any field extension K of degree n of a local field F gives rise to an inclusion of the Weil group WK inside WF as a subgroup of index n. Since

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characters of WK are, by local class field theory, identified to characters of K ∗ , a character of K ∗ gives by induction a representation of WF of dimension n. Proposition 10.1 If (n, q) = 1, then any irreducible representation of WF of dimension n is induced from a character χ of K ∗ for a field extension K of degree n. Proof : Since irreducible representations of the Weil group and Galois groups are the same, perhaps after a twist, we will instead work with the Galois group. Let ρ be an irreducible representation of the Galois group Gal(F¯ /F ) of dimension n > 1. We will prove that there exists an extension L of F of degree greater than 1, and an irreducible representation of Gal(F¯ /L) which induces to ρ. This will complete the proof of the proposition by induction on n. The proof will be by contradiction. We will assume that ρ is not induced from any proper subgroup. We recall that there is a filtration on the Galois group PF ⊂ IF ⊂ GF = Gal(F¯ /F ). Since PF is a pro-p group, all its irreducible representations of dimension greater than 1 are powers of p. Since n is prime to p, this implies that there is a 1 dimensional representation of PF which appears in ρ restricted to PF . Since PF is a normal subgroup of the Galois group, it implies that the restriction of ρ to PF is a sum of characters. All the characters must be the same by Clifford theory. (Otherwise, the representation ρ is induced from the stabilizer of any χ-isotypical component.) So, under the hypothesis that ρ is not induced from any proper subgroup, PF acts via scalars on ρ. Since the exact sequence, 1 → PF → IF → IF /PF → 1 is a split exact sequence, let M be a subgroup in IF which goes isomorphically to IF /PF . Take a character of the abelian group M appearing in ρ. As PF operates via scalars, the corresponding 1 dimensional space is invariant under IF . It follows that ρ restricted to IF contains a character. Again IF being normal, ρ restricted to I is a sum of characters which must be all the same under the assumption that ρ is not induced from any proper subgroup. Since G/IF is pro-cyclic, this is not possible. 2

Representation Theory of GL(n) over Non-Archimedean Local Fields 201

11

The local Langlands conjecture for GLn

It is part of abelian class field theory that for a local field F , the characters of F ∗ can be identified to the characters of the Weil group WF of F . Langlands visualised a vast generalisation of this in the late 60’s to non-abelian representations of WF which is now a theorem due to M.Harris and R.Taylor, and another proof was supplied shortly thereafter by G.Henniart. The general conjecture of Langlands uses a slight variation of the Weil group, called the Deligne-Weil group, denoted WF′ and defined to be WF′ = WF × SL2 (C). Theorem 11.1 (Local Langlands Conjecture) There exists a natural bijective correspondence between irreducible admissible representations of GLn (F ) and n-dimensional representations of the Weil-Deligne group WF′ of F which are semi-simple when restricted to WF and algebraic when restricted to SL2 (C). The correspondence reduces to class field theory for n = 1, and is equivariant under twisting and taking duals. The correspondence in the conjecture is called the local Langlands correspondence, and the n-dimensional representation of WF′ associated to a representation π of GLn (F ) is called the Langlands parameter of π. The Langlands correspondence is supposed to be natural in the sense that there are L-functions and ǫ-factors attached to pairs of representations of WF′ , and also to pairs of representations of GLn (F ), characterising these representations, and the correspondence is supposed to be the unique correspondence preserving these. We will not define L-functions and ǫ-factors here, but refer the reader to the excellent survey article of Kudla [12] on the subject. The results of Bernstein-Zelevinsky reviewed in section 7 and 8 reduce the Langlands correspondence between irreducible representations of GLn (F ) and representations of WF′ to one between irreducible supercuspidal representations of GLn (F ) and irreducible representations of WF , as can be seen as follows. An n-dimensional representation σ of the Weil-Deligne group WF′ of F which is semi-simple when restricted to WF and algebraic when restricted to SL2 (C) is of the form σ=

i=r X i=1

σi ⊗ Sp(mi )

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where σi are irreducible representations of WF of dimension say ni , and Sp(mi ) is the unique mi dimensional irreducible representation of SL2 (C). Assuming the Langlands correspondence between irreducible supercuspidal representations of GLn (F ) and irreducible representations of WF , we have representations πi of GLni (F ) naturally associated to the representations σi of dimension ni of WF . To the product mi ni , and a supercuspidal representation πi of GLni (F ), Theorem 7.5 associates an essentially square integrable representation to the segment (πi (− mi2−1 ), · · · , πi ( mi2+1 )), which we denote by Stmi (πi ). Now order the representations so that for i < j, the segment m −1 m +1 (πi (− mi2−1 ), · · · , πi ( mi2+1 )), does not precede (πj (− j2 ), · · · , πj ( j2 )). By Theorem 8.1, the parobolically induced representation IndG P (Stm1 (π1 ) ⊗ · · · ⊗ Stmr (πr )), admits a unique irreducible quotient which is the representation of GLn (F ) P associated by the Langlands correspondence to σ = i=r i=1 σi ⊗ Sp(mi ).

Example 11.2 The irreducible admissible representations π of GL2 together with the associated representation σπ of the Weil-Deligne group is as follows. 1. Principal series representation, induced from a pair of characters (χ1 , χ2 ) of F ∗ . These representations are irreducible if and only if χ1 χ−1 2 6= ±1 | · | . For irreducible principal series representation, the associated Langlands parameter is χ1 ⊕ χ2 , where χ1 , χ2 are now being treated as characters of WF . 2. Twists of Steinberg. The Langlands parameter of the Steinberg is the standard 2 dimensional representation of SL2 (C). 3. Twists of the trivial representation. The Langlands parameter of the trivial representation is | · |1/2 ⊕ | · |−1/2 . 4. The rest, which is exactly the set of supercuspidal representations of GL2 (F ). In odd residue characteristic, these representations can be constructed by what is called the Weil representation, and the representation of GL2 (F ) so constructed are parametrized by characters of the invertible elements of quadratic field extensions of F . By proposition 10.1, every 2 dimensional representation of WF , when the residue characteristic is odd, is also given by induction of such a character on a quadratic extension which is the Langlands parameter.

Representation Theory of GL(n) over Non-Archimedean Local Fields 203 Acknowledgments These notes are based on the lectures given by the first author in the workshop ‘Automorphic Forms on GL(n)’ organized by Professors G. Harder and M.S. Raghunathan at ICTP, Trieste in August 2000. The authors would like to thank the organizers for inviting them for this nice conference; the European commission for financial support; Professor L. G¨ottsche for having taken care of all the details of our stay as well as the conference so well.

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References [1] D. Bump, Automorphic Forms and Representations, Cambridge studies in Advanced Mathematics 55, (1997). [2] I.N. Bernstein and A.V. Zelevinsky, Representation Theory of GL(n, F ) where F is a non-Archimedea local field, Russian Math. Survey, 31:3, 1-68, (1976). [3] I.N. Bernstein and A.V. Zelevinsky, Induced representations of reduc´ tive p-adic groups. I, Ann. Sci. Ecole Norm. Sup., (4) 10, no. 4, 441–472 (1977). [4] A. Borel, Admissible representations of a semi-simple group over a local field with vectors fixed under am Iwahori subgroup, Inv. Math., 35, 233-259 (1976). [5] C.J. Bushnell and P.C. Kutzko, The admissible dual of GL(N ) via compact open subgroups, Princeton university press, Princeton, (1993). [6] H. Carayol, Representations cuspidales du groupe lineaire, Ann. Sci. ´ Ecole. Norm. Sup., (4), Vol 17, 191-225 (1984). [7] W. Casselman, An introduction to the theory of admissible representations of reductive p-adic groups, Unpublished notes. [8] J.A. Green,The characters of the finite general linear groups, Trans. Amer. Math. Soc. 80, 402–447 (1955). [9] G. Henniart, Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique. (French) [A simple proof of the Langlands conjectures for GL(n) over a p-adic field], Invent. Math., 139, no. 2, 439–455 (2000). [10] N. Iwahori and H. Matsumoto, On some Bruhat decompositions and the structure of the Hecke algebra rings of p-adic Chevalley groups, Publ. Math. I.H.E.S., 25, 5-48 (1965). [11] H. Jacquet, Generic representations, in Non-Commutative harmonic analysis, LNM 587, Springer Verlag, 91-101 (1977).

Representation Theory of GL(n) over Non-Archimedean Local Fields 205 [12] Stephen S. Kudla, The local Langlands correspondence: the nonArchimedean case, Motives (Seattle, WA, 1991), 365–391, Proc. Sympos. Pure Math., 55, Part 2, Amer. Math. Soc., Providence, RI (1994). [13] J.A. Shalika, The multiplicity one theorem for GLn , Ann. of Math., Vol 100, 171-193 (1974). [14] T.A. Springer, Linear Algebraic Groups, 2nd Edition, Progress in Mathematics, Vol 9, Birkh¨auser, (1998). [15] A.V. Zelevinsky, Induced representations of reductive p-adic groups II, ´ Ann. Sci. Ecole Norm. Sup., (4) Vol 13, 165-210 (1980).

The Langlands Program (An overview) G. Harder∗

Mathematisches Institut der Universit¨ at Bonn, Bonn, Germany

Lectures given at the School on Automorphic Forms on GL(n) Trieste, 31 July - 18 August 2000

LNS0821005

∗

[email protected]

Contents Introduction

211

I. A Simple Example

211

II. The General Picture

223

References

234

The Langlands Program (An overview)

211

Introduction The Langlands program predicts a correspondence between two types of objects. On the one side we have automorphic representations π and on the other side we have some arithmetic objects M which may be called motives or even objects of a more general nature. Both these objects produce Lfunctions and the correspondence should be defined by the equality of these L-functions. A special case is the Weil-Taniyama conjecture which has been proved by Wiles-Taylor and others. I. A Simple Example On his home-page under www.math.ias.edu Langlands considers a couple of very explicit and simple examples of this correspondence and here I reproduce one of these examples together with some further explanation. This example is so simple that the statement of the theorem can be explained to everybody who has some basic education in mathematics. The first object is a pair of integral, positive definite, quaternary quadratic forms P (x, y, u, v) = x2 + xy + 3y 2 + u2 + uv + 3v 2 Q(x, y, u, v) = 2(x2 + y 2 + u2 + v 2 ) + 2xu + xv + yu − 2yv These forms have discriminant 112 and I mention that these two quadratic forms Q, P are the only integral, positive definite quaternary forms with discriminant 112 . This may not be so easy to verify but it is true. (Rainer Schulze-Pillot pointed out that this is actually not true; there is a third form S(x, y, u, v) = x2 + 4(y 2 + u2 + v 2 ) + xu + 4yu + 3yv + 7uv but the two forms above are sufficient for the following considerations (see [He1])). This pair will give us automorphic forms, we come to this point later. The second object is an elliptic curve E, for us this is simply a polynomial G(x, y) = y 2 + y − x3 + x2 + 10x + 20. This object is a diophantine equation, for any commutative ring R with identity we can consider the set of solutions {(a, b) ∈ R2 |G(a, b) = 0}

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In the case where R is a field k we add a point at infinity (we should consider ˜ y, z) = y 2 z +yz 2 −x3 +x2 z +10xz 2 +20z 3 ) the homogenized polynomial G(x, and define E(k) = {(a, b) ∈ k2 |G(a, b) = 0} ∪ {∞} ˜ b, c) = 0}/k∗ . = {(a, b, c) ∈ k3 \ {(0, 0, 0)} | G(a, We come back to the first object. For any integer n we can define the numbers r(P, n) = #{γ ∈ Z4 |P (γ) = n} , r(Q, n) = #{γ ∈ Z4 |Q(γ) = n} in classical terms: We consider the number of representations of n by the two forms. We can encode these numbers in generating series X X Θ(P, t) = r(P, n)tn = tP (γ) n

Θ(Q, t) =

γ∈Z4

X X r(Q, n)tn = tQ(γ) n

γ∈Z4

Of course it is not so difficult to write a few terms of these series Θ(P, t) = 1 + 4t + 4t2 + 8t3 + 20t4 + 16t5 + 32t6 + 16t7 + 36t8 + 28t9 + 40t10 + 4t11 + 64t12 + 40t13 + 64t14 + 56t15 + 68t16 + 40t17 + 100t18 + 48t19 + 104t20 + . . . Θ(Q, t) = 1 + 12t2 + 12t3 + 12t4 + 12t5 + 24t6 + 24t7 + 36t8 + 36t9 + 48t10 + 72t12 + 24t13 + 48t14 + 60t15 + 84t16 + 48t17 + 84t18 + 48t19 + 96t20 + . . . . Now we return to our second object. For any prime p we can reduce our polynomial G(x, y) mod p and we can look at the solutions of our equation G(x, y) = 0 in the field Fp with p elements. Actually this equation defines what is called a curve over Fp and if we add the point at infinity we get a projective curve. We say that this curve is smooth over Fp (or we say that we have good reduction) if for any point in the algebraic closure ¯ p ) the vector of partial derivatives (a, b) ∈ E(F (

∂G ∂G (a, b), (a, b)) 6= 0. ∂x ∂y

The Langlands Program (An overview)

213

A simple calculation shows that we get a smooth curve over Fp except for p = 11. For any p we may ask: What is the number of solutions of our equation over Fp and this means we want to know what #E(Fp ) is. To get a rough idea of what will happen we do the following: We choose an a ∈ Fp and to find a point (a, b) ∈ E(Fp ) we have to solve the quadratic equation y 2 + y = a3 − a2 − 10a − 20 in Fp . If p 6= 2 then this equation has a solution in Fp if and only if the element a3 − a2 − 10a − 20 + 1/4 is a square in Fp . Now we know that exactly half the elements in F∗p are squares and hence our chance to hit a square is roughly 1/2. But if we hit a square then we get two solutions for our equation -unless the number above should be zero- therefore we can expect that the number of solutions is roughly p. For p 6= 11 we define the number ap by #E(Fp ) = p + 1 − ap , so this number ap measures the deviation from our expectation. We have the celebrated theorem by Hasse √ For p 6= 11 we have the estimate | ap | ≤ 2 p . Again we can produce a list of values of ap for small primes 2 3 5 7 13 17 19 −2 −1 1 −2 4 −2 0 Now we can formulate a theorem which is a special case of the Langlands correspondence but which was certainly known to Eichler: Theorem For all p 6= 11 we have 1 ap = (r(P, p) − r(Q, p)) 4 This is a surprising statement which is formulated in completely elementary terms. We have two diophantine problems of rather different nature, why are they related by the theorem above? I would like to say that the theorem in the form as it stands looks like a miracle. One possible interpretation is that it provides an elementary formula for the numbers ap . But from the computational side it seems to me that the

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ap are easier to compute than the representation numbers. I come back to this further down. The theorem becomes comprehensible if we establish the connection to modular forms. The following considerations go back into the 19-th century. We consider the two generating functions for our two quadratic forms. We make a substitution t → e2πiz and we observe that the functions z 7→ Θ(P, z), z 7→ Θ(Q, z) are holomorphic functions on the upper half plane H = {z | ℑ(z) > 0}. It is a classical result that these two functions are in fact modular forms of weight 2 for the congruence subgroup a b Γ0 (11) = { | a, b, c, d ∈ Z, c ≡ 0 mod 11}. c d This means that they satisfy (? = P, Q) Θ(?,

az + b ) = (cz + d)2 Θ(?, z) cz + d

for γ=

a b ∈ Γ0 (11) c d

and in addition a certain growth condition for ℑ(z) → ∞ is satisfied. [This can be verified by a classical calculation. First of all it is easy to see that in both cases the forms are invariant under z 7→ z + 1 and then the Poisson summation formula implies the rule Θ(?, z) =

1 1 Θ(?, ). 2 11z −11z

(I skip the computation, it is based on the observation that for x ∈ R4 the function X x 7→ e2πizQ(x+ω) ω∈Z4

is periodic with period Z4 . Hence it has a Fourier expansion, writing down this expansion, putting x = 0 and another small manipulation yields the assertion). Now the modularity follows. (See also [He1])] Hence we get two modular forms of weight 2 for the group Γ0 (11) and by classical dimension formulae we know that they span the vector space

The Langlands Program (An overview)

215

of these modular forms. We know that this space of modular forms is also spanned by two other forms: One of them is the Eisenstein series E(z) =

X

γ,c≡0

mod 11

1 1 − 2 (cz + d) 11

X

γ,c6≡0

mod 11

1 (cz + d)2

(this is a difference of two divergent series and this difference makes sense (this is Hecke so we are in the 20-th century)) and the other one is a cusp form, which in this case is 2πiz

f (z) = e

∞ Y

(1 − e2πniz )2 (1 − e2π11niz )2

n=1

(also classical we have the Dedekind η-function η(z) = eπiz/12 e2πniz )) It is now also in [He1] that

Q∞

n=1 (1

−

1 f (z) = (Θ(P, z) − Θ(Q, z)). 4 A small digression: Of course we would like have information on the individual Theta series. In this context we still have another theorem by Siegel. Our two quadratic forms are in fact in the same genus, that means over any p-adic ring Zp they become equivalent (but of course they are not equivalent over Z). Then we have a very general theorem by C.L. Siegel which asserts that the sum over the Theta series over a genus where the summands are multiplied by suitable weight factors (densities) gives us an Eisenstein series. In our special situation we find ∞

X 1 1 σn e2πinz Θ(P, z) + Θ(Q, z) = E(z) = 4 6 n=0

where the coefficients σn are given rather explicitly, for instance for a prime p 6= 11 we have σp = p + 1. I will say more about the other coefficients in a minute (See A below). At this point I want to meditate a second. Here are two important points to observe.

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A) If we look at the problem to understand the representation numbers we want to know the r(P, n) for all integers n. If we go back to our elliptic curve we get only numbers for each prime p, (p 6= 11). Here the theory of automorphic forms provides another remarkable and fundamental fact. The coefficients in the two series ∞

X 1 τn e2πinz f (z) = (Θ(P, e2πz ) − Θ(Q, e2πiz )) = 4 n=1

and

∞

X 1 1 E(z) = Θ(P, e2πiz ) + Θ(Q, e2πiz ) = σn e2πinz 4 6 n=0

behave multiplicatively and I explain what this means: If we have any series P F (z) = λn e2πinz then we build formally the Mellin transform. This is a Dirichlet series, it is defined by LF (s) =

∞ X

λn n−s .

n=1

(Let us ignore convergence problems, this construction has also been discussed in J. Cogdell’s lecture). Now multiplicativity means in our case that the two Mellin transforms have an Euler product expansion Lf (s) = (

Y

p6=11

LE (s) = (

Y

p6=11

1 1 − τp

p−s

+

p1−2s

)

1 1 − 11−s

1 1 ) −s 1−2s 1 − (p + 1)p + p 1 − 11−s

and this is equivalent to some recursion formulae namely τnm = τn τm , σnm = σn σm if n, m are coprime and for p 6= 11 τpr+1 = τpr τp + pτpr−1 , σpr+1 = σpr σp + pσpr−1 if r ≥ 1 and hence we know the σ, τ if we know them for prime indices. This follows from the theory of the Hecke operators which was actually designed for proving such multiplicativity formulae (See [He2]). The two

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functions are eigenfunctions for this Hecke algebra and this is equivalent to the multiplicativity of the coefficients. This makes it also clear that our two functions are the only ones which have multiplicative coefficients. B) Now we have the formula 1 1 r(P, p) + r(Q, p) = p + 1 4 6 and together with our theorem we can say that we can express the representation numbers in terms of p and ap . Combined with the theorem by Hasse we get a consequence for the asymptotic behavior of the representation numbers and this was an application Eichler had in mind. From √ |τp | = | 41 (r(P, p) − r(Q, p))| ≤ 2 p we get the asymptotic formulae r(P, p) =

12 p + O(p1/2 ) 5

12 p + O(p1/2 ). 5 Now we return to our elliptic curve and I want to give a very sketchy outline of the proof of the theorem. We consider the Riemann surface Γ0 (11)\H. It was known to Fricke that this is a curve of genus 1 over C from which two points are removed. These two points are the cusps of the action of Γ0 (11) on H, they can be represented by 0, i∞. The curve of genus 1 can also be interpreted as Γ0 (11)\H∗ where H∗ = H ∪ Q ∪ {∞} = H ∪ P1 (Q) where this space is endowed with a suitable topology. Fricke found an equation for this curve which after some manipulation can be transformed into r(Q, p) =

y 2 + y = x3 − x2 − 10x − 20 and in modern language this means that that we have a model X0 (11)/Spec(Z) of our complex curve which has good reduction at all primes p 6= 11. The Hecke operators Tp are so-called correspondences, they can be interpreted as curves Tp ⊂ Γ0 (11)\H∗ × Γ0 (11)\H∗ which consist of the following points: If a first coordinate is represented by z ∈ H then the second coordinate is represented by one of the points {pz, z/p, (z +1)/p, . . . , (z +p−1)/p}; so in general there are p + 1 second coordinates corresponding to a first coordinate and vice versa. (Of course one has to check that replacing z by another representative gives the same set of corresponding points). These Hecke operators extend to correspondences also called Tp on the model X0 (11). To

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see that this is so one has to go to the modular interpretation of X0 (11), this means roughly that X0 (11) is the parameter space for the elliptic curves with a cyclic subgroup of order 11. Then Eichler showed that these correspondences Tp have a reduction mod p and this reduction is given by the congruence formula ([Ei]) Tp

mod p = Fp +t Fp .

¯ p ) × E(F ¯ p )} and t Fp is given Here Fp is given by the graph {(x, xp ) ∈ E(F p ¯ ¯ by {(x , x) ∈ E(Fp ) × E(Fp )}. Using the trace formula the coefficient τp can be expressed in terms of the fixed points of Tp . But mod p the fixed points of Fp and t Fp are the points in E(Fp ), and this gives a very rough indication how the theorem can be proved. The Taniyama-Shimura - Weil conjecture Wiles, Taylor and others proved the general Taniyama-Weil conjecture. I want to give some indication of the content of this general theorem. The precise statement needs some finer concepts and results from the theory of automorphic forms and the arithmetic of elliptic curves. A congruence subgroup Γ ⊂ SL2 (Z) is a subgroup of finite index which can be defined by congruence conditions on the entries. To any integer N we define the subgroup n a b o Γ(N ) = ∈ SL2 (Z) | a ≡ d ≡ 1 mod N , c d and a given subgroup Γ is called a congruence subgroup, if we can find an integer N such that Γ(N ) ⊂ Γ ⊂ SL2 (Z). Such a group operates on the upper half plane H and the quotient Γ\H carries the structure of a Riemann surface, more precisely we can compactify it to a compact Riemann surface by adding a finite number of points. These finitely many points are called the cusps. A holomorphic modular form for a given congruence subgroup Γ of weight k > 0 is a holomorphic function on the upper half plane f : H −→ C which satisfies f

az + b cz + d

= (cz + d)k f (z)

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for all γ ∈ Γ, and which satisfies a growth condition in the cusps (see 4.11 (2) in [V]). Now I need some results and concepts which I cannot explain in detail. On the space of modular forms of weight k for Γ0 (N ) we have an action of a commutative algebra T which is generated by operators Tp for the primes p 6 |N . In this space of modular forms we have the subspace of cusp forms. These are forms which vanish at infinity (see [V], .....), and this subspace is invariant under the Hecke operators. It is a classical result of Hecke that this space of cusp forms is a direct sum of spaces of common eigenforms for the Hecke operators. A modular form f for Γ0 (N ) is called a new form if i) The form f and certain transforms of it is not a modular form for a congruence subgroup Γ0 (N ′ ) where N ′ | N and N ′ < N . ii) The form f is an eigenform for all the Hecke operators Tp where p 6 |N . It requires a little bit of work to show that this is a reasonable concept. To such a new form f we can attach an L-function Y Lp (f, s), L(f, s) = p

where we have attached a local Euler factor Lp (s) to any prime p: i) For the primes p 6 |N our form is an eigenform for Tp , i.e. Tp f = ap f and we put Lp (f, s) =

ap ∈ C,

1 . 1 − ap p−s + pk−1−2s

ii) For the primes p | N and p 6= 2, 3 we have ( 1 2 1−εp p−s εp = ±1 if p 6 |N Lp (s) = 1 if p2 | N. The determination of the εp requires some knowledge of local representation theory. iii) For the primes p | N and p = 2 or 3 we also have ( 1 1−εp p−s εp = ±1 Lp (s) = 1

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but here the formulation of the conditions for the cases is even more subtle. Now it is a general theorem that the completed L function Λ(f, s) = · L(f, s) is a holomorphic function in the entire plane and satisfies the functional equation Γ(s) πs

Λ(2 − s) = N 1−s W (f )N s−1 Λ(s), where W (f ) = ±1. This is a so-called automorphic L-function. Now I explain how we can attach an L function to an elliptic curve E over Q. Let us consider an elliptic curve over Q. This is simply an equation G(x, y, z) = y 2 z + a1 xyz + a3 yz 2 − x3 − a2 x2 z − a4 xz 2 − a6 z 3 = 0 with rational coefficients a1 , a3 , a2 , a4 , a6 . (This is a traditional notation, these ai are not the ap which occur in the local L-factors.) We assume that this equation defines a non singular curve and this means that for any solution (x0 , y0 , z0 ) ∈ C3 , (x0 , y0 , z0 ) 6= 0 we have ∂G ∂G ∂G (x0 , y0 , z0 ), (x0 , y0 , z0 ), (x0 , y0 , z0 ) 6= (0, 0, 0). ∂x ∂y ∂z This is equivalent to the non vanishing of the discriminant ∆ = ∆(a1 , a2 , a3 , a4 , a6 ), this is a complicated expression in the coefficients. ([Mod], the articles of Tate and Deligne (Formulaire)). A special point is the point at infinity O = (0, 1, 0). It is the only point with z0 = 0. Now we can perform substitutions in the variables, and we get new Weierstraß equations. There is a so-called minimal Weierstraß equation y2z + a ˜1 xyz + a ˜ 3 y = x3 + a ˜2 x2 + a ˜4 x + a ˜6 , where all the a ˜i ∈ Z, and where the discriminant ∆ is minimal. (See Silverman [Si], Chap. III, § 1, VIII, § 8, [Hu], Chap. 5, § 2 and [Mod], articles by Tate and Deligne.) We have an algorithm – which is implemented in Pari – which produces this minimal equation.

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If now p is a prime, we can consider the reduction mod p. This gives us an equation over the finite field Fp , and we say that our equation has good reduction mod p, if the reduced equation defines an elliptic curve, i.e. it is smooth. If the mod p reduced curve is not smooth then we have exactly one singular point P0 = (x0 , y0 , z0 ) ∈ E(Fp ) which is different from the point O, we may assume z0 = 1. Then we can introduce variables u = x−x0 , v = y−y0 and our equation mod p becomes αu2 + βuv + γv 2 + higher order terms = 0. It is not hard to see that the quadratic leading term is not identically zero. Then we have two possibilities: i) Over Fp2 we have αu2 + βuv + γv 2 = (u − ξ1 v)(u − ξ2 v) where ξ1 6= ξ2 ii) The quadratic form is itself a square, i.e. αu2 + βuv + γv 2 = (u − ξv)2 In the first case we say that E has multiplicative reduction mod p, in the second case we say that E has potentially good reduction at p. (Potentially good is much more unpleasant than multiplicative reduction.) If we are in the case i) we define εp =

1 ifξ1 , ξ2 ∈ Fp , −1 else.

From this type of bad reduction we can produce a number np (E) > 0. If p 6= 2, 3 then np (E) =

1 multiplicative reduction, 2 potentially good reduction.

If we have p = 2 or p = 3 then the rule is more complicated, in this case we need something finer than the minimal Weierstrass-equation, we need the Neron model (see [O]) to produce np (E). We define Y N= pnp (E) , p

where p runs over the primes with bad reduction. This number N is the conductor of our curve.

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Now in [We] Weil defines an Euler factor Lp (E, s) for any prime p. If we have good reduction at p, we define as before Lp (E, s) =

1 1 − ap p−s + p1−2s

where E(Fp ) = p + 1 − ap . If we have bad reduction, then the Euler factor depends on the type of the bad reduction. The precise rule is (see [We] 2). ( 1 if we have potentially good reduction, Lp (E, s) = 1 1−εp p−s if we have multiplicative reduction. Then we can define L(E, s) =

Γ(s) Y Lp (E, s). πs p

(Here Γ(s) is the Γ-function.) Then the theorem of Wiles-Taylor asserts that to any elliptic curve E/Q with conductor N we can find a new form f on Γ0 (N ) such that Lf (s) = L(E, s). Our first example is a special case of this theorem. Converse theorems I come back to the L-function to a newform f . I introduced the Mellin transform very formally but as explained in Cogdell’s lecture we can also define it by the integral Z i∞ Γ(s) dy Λ(f, s) = f (iy)y s L(f, s) = s (2π) y 0 where f has to be suitably normalized. Now we can conclude from the theory of automorphic forms that our newform satisfies f (−

1 ) = W (f )N z 2 f (z), Nz

where W (f ) = ±1. We apply this to the integral representation: We choose a positive real number A > 0 and split the integral into an integral from A

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223

to ∞ and the integral from 0 to A. Into the second term we plug in the above transformation formula and get Z ∞ Z ∞ dy 1−s s dy + W (f )N f (iy)y 2−s f (iy)y Λ(f, s) = y y 1/(N A) A Γ(s) From this integral representation we can derive that (2π) s L(f, s) has an analytic continuation into the entire plane and that we have the functional equation Λ(2 − s) = W (f )N 1−s Λ(s).

Already Hecke observed that under certain circumstances we can go the other way round. If we have a Dirichlet series D(s) =

∞ X an n=1

ns

which defines a holomorphic function, which satisfies some boundedness conditions and satisfies a suitable functional equation then it comes from a modular form. Hecke considered the case of Sl2 (Z) and Weil generalized it in ([We]) but he had to assume that these properties remained true if the series is twisted by Dirichlet characters. (See Venkataramas’s and Cogdell’s lectures.) Such a theorem is called a converse theorem. Of course one would like to prove that the L-function of an elliptic curve has these nice analytic properties and then we could get a proof of Wiles theorem. But this is not the way it works. II. The General Picture Now I want to give some vague idea of the general Langlands program. I must confess that my own understanding is very limited. But on the other hand the entire picture is so vast and a precise formulation requires an explanation of so many subtle notions that I believe that a very rough approximation may be even more helpful than a precise presentation. The first datum is a reductive group G/Q, we may very well think that G = Gln . During this summer school we have seen that automorphic cusp forms should be understood as irreducible subrepresentations of the adele group G(A) occurring in the space of cusp forms. So this is an irreducible submodule Hπ ⊂ L20 (G(Q)\G(A))

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where the subscript π stands for the isomorphism class of our module. Several lecturers told us that such a π is in fact a restricted tensor product of local representations πv of G(Qv ) and we write π=

O′

πv .

These local representations have to satisfy some constraints. For instance for almost all finite primes πp has to be in the unramified principal series (see Prasad’s notes and below) and they must be unitary. In Raghunathan’s lecture it was explained that M L20 (G(Q)\G(A)) = m(π)Hπ and we state a fundamental problem: N′ πv given to Let us assume that there is a restricted product π = us, which fulfills the above constraints. When does π occur in the space of automorphic forms and what is m(π)? Of course this question is rather vague because we should know how π is given to us, i.e. what is the rule which produces the local data {πv }. The speculative answer to this question is, that the rule should come from some kind of arithmetic object. The classical case again We come back briefly to the special case Gl2 . In our example a modular form was a holomorphic function f on the upper half plane which satisfied f (γ(z)) = (cz + d)2 f (z) for γ in some congruence subgroup Γ ⊂ Sl2 (Z). In addition we required that it should be an eigenform for the so-called Hecke operators and I explained briefly that this was equivalent to the requirement that the Mellin transform of the Fourier expansion has an Euler product expansion. Actually the Hecke operators Tp are only defined for primes p not dividing the so-called level N of our form. In our example we had N = 11. Hence we see that f provides a collection of local data {τp }p,p6|N the eigenvalues of Tp . In our example we had in fact a rather simple rule which provided the local data, we took the difference of the representation numbers. If we want to translate from the classical language to the modern language then we have to assign a representation π(f ) of Gl2 (A) to our classical

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225

modular form: This representation should occur in the space of cusp forms L20 (Gl2 (Q)\Gl2 (A)). I do not construct it but I make a list of its properties which define it uniquely. If we write O′ π(f ) = πv then i) At the finite primes p not dividing the level the representation π(f )p is in the unramified principal series and hence a unitarily induced representation G(Q )

IndB(Qpp ) λp t1 u where λp is a quasicharacter λp ( ) = |t1 |s1 |t2 |s2 . It gives two numbers 0 t2 p 0 1 0 αp = λp ( ) and βp = λp ( ). 0 1 0 p Then these numbers are related to the p-th Fourier coefficient of f by the formula √ τp = p(αp + βp ) and αp βp = ω(p). Here ω is the so-called central character, it is the restriction of π(f ) to the center. ii) In our special situation where f is holomorphic of weight two the representation π(f )∞ of Gl2 (R) will be the first discrete series representation. If we have holomorphic modular form of weight k we get the (k − 1)-th representation of the discrete series a infinity and in the formula for the ap k−1 √ the p gets changed into p 2 . The second player in the game is our elliptic curve E/Q. This elliptic curve yields an object h1 (E), this is a motive. It is not entirely clear what this means but it creates some other objects A) A compatible system of ℓ-adic representations of the Galois group ¯ Gal(Q/Q).

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G. Harder

B) The Betti cohomology H 1 (E(C), Z) together with a so-called Hodge filtration on H 1 (E(C), C). I want to say a word about A). For any prime ℓ we can look on the ℓn division points ¯ | ℓn x = 0} E[ℓn ] = {x ∈ E(Q) ¯ is an abelian group and and at this point I assume that we know that E(Q) that E[ℓn ] is isomorphic to Z/ℓn Z × Z/ℓn Z. Of course these division points will have coordinates in larger and larger extensions of Q if n goes to infinity. ¯ This means that we have a natural action of Gal(Q/Q) on all these groups and if we form the projective limit Tℓ = lim E[ℓn ] ← n

the result is a free Zℓ -module of rank 2 together with a continuous action of the Galois group. I explained what it means that E has good reduction at a prime p. It is not so difficult to see that for a prime ℓ which is different from p the action of the Galois group is unramified at this prime p, in other words the inertia group acts trivially. Hence we can define a conjugacy class [Fp ] defined by the action of the Frobenius at p and the characteristic polynomial det(Id − Fp p−s |Tℓ (E)) ∈ Zℓ [p−s ] is a well-defined quantity. Now it follows from the Lefschetz fixed point formula that in fact det(Id − Fp p−s |Tℓ (E)) = 1 − ap p−s + p1−2s . This has important consequences 1) det(Id − Fp p−s |Tℓ (E)) ∈ Z[p−s ] 2) det(Id − Fp p−s |Tℓ (E)) does not depend on ℓ. Finally we have that 3) det(Id − Fp p−s |Tℓ (E)) is defined outside a finite set of primes S ∪ {ℓ}.

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These three properties of our different Galois modules (ℓ varies) are the defining properties for compatible systems of Galois modules. Hence we can reformulate the specific result in the first section: In our example the modular form of weight two and the elliptic curve provide a collection of local data ∞) A representation π∞ and a real Hodge structure on H 1 (E(C), Z) ⊗ C For almost all primes an unramified local representation πp of Gl2 (Qp ) and an unramified two dimensional representation ρ(πp ) of the Galois group ¯ p /Qp ) such that (in the notation used in the example) Gal(Q p) the automorphic Euler factor L(πp , s) = (1 − τp p−s + p1−2s ) is equal to the arithmetic L- factor det(Id − Fp p−s |Tℓ (E)). This means that in our example we have a second rule which produces the local components of a cusp form. This rule is provided by the elliptic curve. In this particular case it is also possible to establish the local correspondence also for the ramified primes, this has been shown by Langlands, Deligne and Carayol. It is now Langlands’ idea that such a correspondence between automorN phic representations π = ′ πv and some kind of arithmetic objects M(π) should always exist. The ideas of what nature these objects are, are also conjectural. Satake’s theorem Let us assume that we picked a prime p such that G × Qp is split. If G = Gln this can be any prime. Let Kp = G(Zp ) be the maximal compact subgroup defined by some Chevalley scheme structure G/Zp , if G = Gln this could be Gln (Zp ). To these data we attach the Hecke algebra Hp = C(Kp \G(Qp )/Kp ) : It consists of the C valued functions on G(Qp ) which are compactly supported and biinvariant under Kp and the algebra structure is given by convolution. We choose a Borel subgroup B ⊂ G and a maximal torus T ⊂ B such that T (Qp ) ∩ K = T (Zp ) is the maximal compact subgroup our torus T (Qp ). Let X∗ (T ) = Hom(Gm , T ) be the module of cocharacters, let W be the Weyl group. We introduce the module of unramified characters on the torus, this is Homunram (T (Qp ), C∗ ) = Hom(T (Qp )/T (Zp ), C∗ ) = Hom(X∗ (T ), C∗ ) = Λ(T ).

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G. Harder

We also view λ ∈ Λ(T ) as a character λ : B(Qp ) → C∗ , λ 7→ λ(b) = bλ . We will consider the group of characters Hom(T ×Q Qp , Gm ) = X ∗ (T )Qp as a subgroup of Λ(T ). An element γ ∈ X ∗ (T ) defines a homomorphism T (Qp ) → Q∗p and this gives us the following element {x 7→ |γ(x)|p } ∈ Λ(T ) which we denote by |γ|. Since we have the Iwasawa decomposition G(Qp ) = B(Qp )Kp we can attach to any λ ∈ Λ(T ) a spherical function φλ (g) = φλ (bp kp ) = (λ + |ρ|)(bp ) where ρ ∈ Λ(T ) is the half sum of positive roots. This spherical function is of course an eigenfunction for Hp under convolution, i.e. for hp ∈ Hp Z ˆ p (λ)φλ (g) φλ (gx−1 )hp (x)dx = h ˆ p (λ) is a homomorphism from Hp to C. and hp 7→ h The theorem of Satake asserts that this provides an identification ∼

Hom(Hp , C) → Λ(T )/W. To such a character we can attach an induced representation G(Q )

IndB(Qpp ) (λ) = {f : G(Qp ) → C | f (bg) = (λ + |ρ|)(b)f (g)} where in addition f |K is locally constant. These representations are called the principal series representations. We denote these irreducible modules by πp = πp (λp ) and λp is the so-called Satake parameter of πp . Let us now assume for simplicity that our group G/Q is split, for instance G = Gln /Q. In this case we may choose a split torus T /Q. We have the canonical isomorphism ∼

Hom(X∗ (T ), C∗ ) → X ∗ (T ) ⊗ C∗ and the character module X ∗ (T ) can be interpreted as the cocharacter module of the dual torus Tˆ. If we interchange the roots and the coroots then ˆ which is now a Tˆ becomes the maximal split torus of the dual group G, reductive group over C. If our group is G = Gln /Q then the dual group is Gln (C).

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A general philosophy Now we come back to our automorphic form π. If we write it as a restricted tensor product, then almost all the components are in the unramified principal series and now we can view the collection of unramified components {πp (λp )} as a collection of semi simple conjugacy classes in the dual group. Now Langlands philosophy assumes the existence of a very big group L and I cannot say exactly what properties this group should have. It certainly ¯ should somehow have the Weil group W (Q/Q) in it. This Weil group is some kind of complicated modification of the Galois group. We have also ¯ p /Qp ) and these are easier to explain: The group the local Weil groups W (Q ¯ ¯ W (Qp /Qp ) ⊂ Gal(Qp /Qp ) and consists of those elements whose image in ¯ p /Fp ) is an integral power of the Frobenius. Gal(F The arithmetic object M(π) attached to π should be a representation ˆ ρ(π) : L → G which at least fulfills the following requirement: At any prime p at which π is unramified the representation ρ(π) is also “unramified”. The structure of L should be such that for an unramified πp it provides an unramified representation ¯ p /Qp ) → G(C) ˆ ρ(πp ) : W (Q such that the image of the Frobenius Fp under ρ(πp ) is in the conjugacy class of the Satake parameter of πp . An unramified representation of the Weil group is of course a represen¯ p /Qp ) in Gal(F ¯ p /Fp ), therefore it is tation of the image Z =< Fp > of W (Q enough to know the image of the Frobenius Fp . Local Langlands correspondence Of course we can also consider ramified representations ¯ p /Qp ) → G(C) ˆ ρ : W (Q and the general Langlands programme predicts also a correspondence between these representations ρ and the admissible irreducible representations of G(Qp ). Actually the situation is more complicated than that, one has to replace the Weil group be the Weil-Deligne group. This is a difficult subject, we have seen a little bit of the difficulties when we discussed the Euler

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factors of the automorphic and the arithmetic L-function in our discussion of the Taniyama-Weil conjecture. The local Langlands conjecture is proved for the group Gln by work of Laumon-Rapoport-Stuhler ([L-R-S]) in characteristic p > 0 and by HarrisTaylor ([H-T]) in characteristic 0. This is discussed in Wedhorns lectures at this summer school ([Wed]). Representations with cohomology and motives I want to discuss a special case in which I feel a bit happier. Among the representations of G(R) there is a certain class consisting of representations π∞ which have non trivial cohomology. This means that there is a finite dimensional, irreducible rational G-module E such that H • (g, K∞ , π∞ ⊗ E) 6= {0}. Then E is determined by π∞ and for any choice of E the number of such π∞ is finite. We say that an automorphic representation π is cohomological if the component π∞ has cohomology in some module E. In this case one might speculate whether we can attach a motive or better a family of motives to it. A motive is still a conjectural object but certainly simpler in nature than L. I want to give a rough idea what a motive should be. First of all I refer to Delignes theorem that for a smooth projective scheme X/Q the ℓ adic co¯ Qℓ ) provide a compatible system of Galois modules. homology groups H i (X, A motive M is a piece in the cohomology which is defined by a projector obtained from correspondences. (In the classical case these correspondences are provided by Hecke operators). Then it is clear that M also provides a compatible system of Galois representations ¯ ¯ , Qℓ )) ρ(M ) : Gal(Q/Q) → Gl(H(M and the Euler factor at an unramified prime is defined as before by ¯ , Qℓ )) ∈ Z[p−s ]. det(Id − Fp p−s | H(M If we have an unramified principal series representation πp (λp ) and we ˆ choose in addition a finite dimensional irreducible representation r : G(C) → Gln (C), then we define the Euler factor L(πp (λp ), r, s) = det(Id − r(πp (λp ))p−s ).

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If we know all these Euler factors for all choices of r then we know the conˆ jugacy class of πp (λp ) viewed as an element in G(C). Now we can speculate To any cohomological π and which occurs in the space of (cuspidal) auˆ tomorphic forms on G and to any representation r : G(C) → Gln (C) we can find a motive M (π, r) such that for all unramified primes p we have an equality of local Euler factors L(r(πp (λp ), r, s) = det(Id − ρ(M )(Fp )p−s ) There should also be a matching between π∞ and the Hodge structure on the Betti cohomology of the motive. This system of ℓ representations (now r varies) should have the property that is compatible with the operations in linear algebra: If we decompose a tensor product r1 ⊗r2 into irreducibles then the Galois representations should decompose accordingly, at least if we pass to a subgroup of finite index in the Galois group. Already in the formulation we need the properties of compatible system. The right-hand side has a property which we a priori can not expect from the left hand side: Why should the automorphic Euler factors be in Z[p−s ]?? Can such a statement ever be true? Here the assumption that π∞ has cohomology helps. Using the rational (or even the integral structure) on the cohomology we can show that in fact that L(πp , r, s) viewed as polynomial in p−s has coefficients which are algebraic integers and which all lie in a finite extension of Q which depends on πf . We say that a cohomological form π is rational if these coefficients are in Z (this was so in our example). Otherwise we say that π is defined over F if F ⊂ C is generated by the coefficients of all our Euler factors. Then we can add to our assumption in our statement above that π should be rational. Otherwise we have to invent the notion of a motive with coefficients in F . This notion has been introduced by Deligne and then we can formulate the above assertion using this concept. Of course one can ask the question in the opposite direction: Given a motive is there somewhere an automorphic cohomological representation π such that M = M (π, r) for some r? Can we find such a representation even in the space of automorphic forms on Gln ? The theorem of Wiles is a special case where the answer to this question turns out to be yes.

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Functoriality If we believe in this kind of correspondence between automorphic forms and some sort of arithmetic objects, then we get remarkable consequences for automorphic forms. Let us just stick to the cohomological case. If we have two such motives we can form their product, which for the Galois modules amounts to take their tensor product. Going backwards we should be able to construct an automorphic form π1 × π2 on some bigger group. This is the principle of functoriality, which is suggested by the philosophy. Let me give an example. We consider holomorphic modular forms of weight 2, we can even go back to our example. We have seen that our modular form provides a compatible system of two dimensional ℓ-adic representations ¯ ¯ Qℓ )) ρ(H 1 (E)) : Gal(Q/Q) :→ Gl(H 1 (E, Now we take symmetric powers of these representations, this means that we take the k-fold tensor product of these representations first and this amounts to taking the k-fold product of h1 (E) by itself. Then we have an action of the symmetric group and we can take the symmetric part. In terms of the Galois representations this means that we get an representation on the symmetric tensors ¯ ¯ Qℓ ))) ρ(Symk (H 1 (E))) : Gal(Q/Q) :→ Gl(Symk (H 1 (E, and it is certainly a legitimate question whether this comes from an automorphic form. In this particular case we can look at our problem from a different point of view. We look at the L function (let us stick to our example) Y 1 1 LE (s) = ( ) −s 1−2s 1 − τp p + p 1 − 11−s p6=11

and we rewrite the Euler factors L(πp , s) =

1 1 − τp

p−s

+

p1−2s

)=

1 (1 − αp

p−s )(1

−α ¯ p p−s )

and we mention that it follows from Hasse’s theorem that α ¯ p is in fact the complex conjugate of αp . Now we form a new L-function, we pick a k > 1 and write a local L-factor at p 1 L(πp , r, s) = k−1 k −s (1 − αp p )(1 − αp α ¯ p p−s ) . . . (1 − α ¯ kp p−s )

The Langlands Program (An overview)

233

We can form a global L-function attached to the k-th symmetric power L(π, r, s) =

Y

p;p6|N

1 (1 −

αkp p−s )(1

−

αk−1 ¯ p p−s ) . . . (1 p α

−α ¯ kp p−s )

LN (π, r, s),

where I do not say anything about the factors at the ramified primes. In our case where N = 11 the Euler factor at 11 should not depend on k. Of course we can ask whether this is again an L-function attached to an automorphic cusp form on Glk+1 . This has been shown by Gelbart and Jacquet for k = 2. Here we are again in the situation where we could try to apply converse theorems, but we do not have methods to verify the necessary analytic properties of the L-Functions (see Cogdell’s Notes). But the cases k = 3, 4 have been treated successfully by Shahidi and Kim. We come to the concept of base change. Let us assume we have a (cuspidal) automorphic form π on some reductive group over Q. Let us assume we attached to it a representation ˆ ρ(π) : L → G(C) of our group L. Let us assume that we have a field extension K/Q, then it should be possible to restrict the group L to K and we would get a restricted representation ˆ ρ(π)K : LK → G(C). (This is another of the requirements one should put on L, if we work with motives then we would just extend the motive or restrict the Galois repre¯ sentations to Gal(Q/K)). Hence we should expect that this restriction of the representation ρ(π)K would provide an automorphic form on the group G × K which then would be the lift of π to G × K . The existence of such a lifting has indeed been proved for solvable extensions by Langlands in the case G = Gl2 /F and by Arthur and Clozel for G = Gln /F . This result plays a fundamental role in the proof of the Taniyama-Weil conjecture for elliptic curves and the local Langlands correspondence for Gln by Harris and Taylor.

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References [A-C]

Arthur, J.; Clozel, L.; Simple algebras, base change, and the advanced theory of the trace formula. Annals of Mathematics Studies, 120. Princeton University Press, Princeton, NJ, 1989. xiv+230 pp.

[Ca]

Carayol, H.; Formes automorphes et repr´esentations galoisiennes. (French) [Automorphic forms and Galois representations] Seminar on Number Theory, 1981/1982, Exp. No. 31, 20 pp., Univ. Bordeaux I, Talence, 1982.

[Co]

Cogdell, J.; Notes on L-functions for Gln , this volume.

[De1]

Deligne, P.; Formes modulaires et repr´esentations ℓ-adiques. S´eminaire Bourbaki, 1968/69, Exp. 335.

[Ei]

Eichler, M.; Quatern¨are quadratische Formen und die Riemannsche Vermutung f¨ ur die Kongruenzetafunktion, Arch. 5 (1954), 355-366.

[H-T]

Harris, M.; Taylor, R.; The geometry and cohomology of some simple Shimura varieties. With an appendix by Vladimir G. Berkovich. Annals of Mathematics Studies, 151. Princeton University Press, Princeton, NJ, 2001. viii+276 pp.

[He1]

Hecke, E.; Analytische Arithmetik der positiven quadratischen Formen, Kgl. Danske Selskab, Mathematisk-fysiske Meddelser. XIII, 12, 1940, 134 S.

[He2]

¨ Hecke, E.; Uber Modulformen und Dirichletsche Reihen mit Eulerscher Produktentwicklung I,II (Mathematische Annalen Bd. 114, 1937, S.1-28, S. 316-351).

[He3]

E. Hecke, E.; Mathematische Werke, Vandenhoeck & Ruprecht, G¨ottingen, 1959.

[Hu]

Husemoller, D.; Elliptic curves. With an appendix by Ruth Lawrence. Graduate Texts in Mathematics, 111. Springer-Verlag, New York, 1987. xvi+350 pp.

[La1]

Langlands, R.; Euler products. A James K. Whittemore Lecture in Mathematics given at Yale University, 1967. Yale Mathematical Monographs, 1. Yale University Press, New Haven, Conn.-London, 1971. v+53 pp.

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[La2]

Langlands, R.; Modular forms and ℓ-adic representations. Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), pp. 361-500. Lecture Notes in Math., Vol. 349, Springer, Berlin, 1973.

[La3]

Langlands, R.; Base change for Gl(2), Ann. of Math. Studies 96, Princeton University Press, 1980.

[La4]

Langlands, R.; Automorphic representations, Shimura varieties, and motives. Ein M¨archen. Automorphic forms, representations and Lfunctions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, pp. 205–246, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I.

[L-R-S] Laumon, G.; Rapoport, M.; Stuhler, U. D-elliptic sheaves and the Langlands correspondence. Invent. Math. 113 (1993), no. 2, 217-338. [Mod] Modular Functions of One Variable IV, ed. Proc. Int. summer school, Antwerp, ed. B.J. Birch and W. Kyuk, Springer Lecture Notes 476. [O]

Ogg, A. Elliptic curves and wild ramification, Am. Journal of Math. 89, p. 1-21.

[P-R]

Prasad, D. - A. Raghuram; Representation theory of GL(n) over non-Archimedean local fields, this volume.

[Si]

Silverman, J. H.; The arithmetic of elliptic curves. Corrected reprint of the 1986 original. Graduate Texts in Mathematics, 106. SpringerVerlag, New York, 1992. xii+400 pp.

[V]

Venkataramana, T.; Classical Modular Forms, this volume.

[Wed]

Wedhorn, T.; The local Langlands correspondence for GL(n) over p-adic fields, this volume.

[We]

¨ Weil, A.; Uber die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann., 168, 1967, S. 149-156.

[Wi]

Wiles, A.; Modular elliptic curves and Fermats Last Theorem. Ann. of Math., 142 (1995), 443 -551.

The Local Langlands Correspondence for GL(n) over p-adic Fields Torsten Wedhorn* Mathematisches Institut der Universit¨ at zu K¨ oln, K¨ oln, Germany

Lectures given at the School on Automorphic Forms on GL(n) Trieste, 31 July - 18 August, 2000 LNS0821006

* [email protected]

Abstract This work is intended as an introduction to the statement and the construction of the local Langlands correspondence for GL(n) over p-adic fields. The emphasis lies on the statement and the explanation of the correspondence.

Contents Introduction

241

Notations

246

1. The local Langlands correspondence

247

1.1 The local Langlands correspondence for GL(1) . . . . . . . . . . . . . 247 1.2 Formulation of the local Langlands correspondence. . . . . . . . . .251 2. Explanation of the GL(n)-side 254 2.1 Generalities on admissible representations. . . . . . . . . . . . . . . . . . .254 2.2 Induction and the Bernstein-Zelevinsky classification for GL(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 2.3 Square integrable and tempered representations . . . . . . . . . . . . . 266 2.4 Generic representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 2.5 Definition of L- and epsilon-factors . . . . . . . . . . . . . . . . . . . . . . . . . 271 3. Explanation of the Galois side 275 3.1 Weil-Deligne representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 3.2 Definition of L- and epsilon-factors . . . . . . . . . . . . . . . . . . . . . . . . . 278 4. Construction of the correspondence 282 4.1 The correspondence in the unramified case . . . . . . . . . . . . . . . . . . 282 4.2 Some reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 4.3 A rudimentary dictionary of the correspondence . . . . . . . . . . . . 286 4.4 The construction of the correspondence after Harris and Taylor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 5. Explanation of the correspondence 292 5.1 Jacquet-Langlands theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 5.2 Special p-divisible O-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 5.3 Deformation of p-divisible O-modules . . . . . . . . . . . . . . . . . . . . . . . 306 5.4 Vanishing cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 5.5 Vanishing cycles on the universal deformation of special p-divisible O-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 Bibliography

315

Local Langlands Correspondence for GL(n) over p-adic Fields

241

Introduction

Let K be a local field, i.e. K is either the field of real or complex numbers (in which case we call K archimedean) or it is a finite extension of Q p (in which case we call K p-adic) or it is isomorphic to IFq ((t)) for a finite field IFq (in which case we call K a local function field). The local Langlands conjecture for GLn gives a bijection of the set of equivalence classes of admissible representations of GLn (K) with the set of equivalence classes of n-dimensional Frobenius semisimple representations of the Weil-Deligne group of K. This bijection should be compatible with L- and ε-factors. For the precise definitions see chap. 2 and chap. 3. If K is archimedean, the local Langlands conjecture is known for a long time and follows from the classification of (infinitesimal) equivalence classes ˇ of admissible representations of GLn (K) (for K = C this is due to Zelobenko and Na˘ımark and for K = IR this was done by Langlands). The archimedean case is particularly simple because all representations of GLn (K) can be built up from representations of GL1 (IR), GL2 (IR) and GL1 (C). See the survey article of Knapp [Kn] for more details about the local Langlands conjecture in the archimedean case. If K is non-archimedean and n = 1, the local Langlands conjecture is equivalent to local abelian class field theory and hence is known for a long time (due originally to Hasse [Has]). Of course, class field theory predates the general Langlands conjecture. For n = 2 the local (and even the global Langlands conjecture) are also known for a couple of years (in the function field case this is due to Drinfeld [Dr1][Dr2], and in the p-adic case due to Kutzko [Kut] and Tunnel [Tu]). Later on Henniart [He1] gave also a proof for the p-adic case for n = 3. If K is a local function field, the local Langlands conjecture for arbitrary n has been proved by Laumon, Rapoport and Stuhler [LRS] generalizing Drinfeld‘s methods. They use certain moduli spaces of “D-elliptic sheaves” or “shtukas” associated to a global function field. Finally, if K is a p-adic field, the local Langlands conjecture for all n has been proved by Harris and Taylor [HT] using Shimura varieties, i.e. certain moduli spaces of abelian varieties. A few months later Henniart gave a much

242

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simpler and more elegant proof [He4]. On the other hand, the advantage of the methods of Harris and Taylor is the geometric construction of the local Langlands correspondence and that it establishes many instances of compatibility between the global and the local correspondence. Hence in all cases the local Langlands conjecture for GL(n) is now a theorem! We remark that in all cases the proof of the local Langlands conjecture for n > 1 uses global methods although it is a purely local statement. In fact, even for n = 1 (i.e. the case of local class field theory) the first proof was global in nature. This work is meant as an introduction to the local Langlands correspondence in the p-adic case. In fact, approximately half of it explains the precise statement of the local Langlands conjecture as formulated by Henniart. The other half gives the construction of the correspondence by Harris and Taylor. I did not make any attempt to explain the connections between the local theory and the global theory of automorphic forms. In particular, nothing is said about the proof that the constructed map satisfies all the conditions postulated by the local Langlands correspondence, and this is surely a severe shortcoming. Hence let me at least here briefly sketch the idea roughly: Let F be a number field which is a totally imaginary extension of a totally real field such that there exists a place w in F with Fw = K. The main idea is to look at the cohomology of a certain projective system X = (Xm )m of projective (n − 1)-dimensional F -schemes (the Xm are “Shimura varieties of PEL-type”, i.e. certain moduli spaces of abelian varieties with polarizations and a level structure depending on m). This system is associated to a reductive group G over Q such that G ⊗Q Q p is equal to Q× p × GLn (K) × anisotropic mod center factors. More precisely, these anisotropic factors are algebraic groups associated to skew fields. They affect the local structure of the Xm only in a minor way, so let us ignore them for the rest of this overview. By the general theory of Shimura varieties, to every absolutely irreducible representation ξ of G ¯ ℓ -sheaf Lξ on X where ℓ 6= p is some over Q there is associated a smooth Q ¯ ℓ -vector fixed prime. The cohomology H i (X, Lξ ) is an infinite-dimensional Q space with an action of G(Af ) × Gal(F¯ /F ) where Af denotes the ring of

Local Langlands Correspondence for GL(n) over p-adic Fields

243

finite adeles of Q. We can choose ξ in such a way that H i (X, Lξ ) = 0 for i 6= n − 1. We have a map from the set of equivalence classes of irreducible admissible representations Π of G(Af ) to the set of finite-dimensional representations of WK ⊂ Gal(F¯ /F ) where WK denotes the Weil group of K by sending Π to Rξ (Π) = HomG(Af ) (Π, H n−1 (X, Lξ )). For every such Π the decomposition G(Af ) = Q × p × GLn (K) × remaining components gives a decomposition Π = Π0 ⊗ Πw ⊗ Πw . If π is a supercuspidal representation of GLn (K) then we can find a Π as above such that Πw ∼ = πχ where χ is an unramified character of K × , such that Π0 is unramified and such that Rξ (Π) 6= 0. ˜ m of Xm ⊗F K over OK and consider the Now we can choose a model X completions Rn,m of the local rings of a certain stratum of the special fibre ˜ m . These completions carry canonical sheaves ψi (namely the sheaf of of X m vanishing cycles) and their limits ψi are endowed with a canonical action × GLn (K) × D × 1/n × WK where D 1/n is the skew field with invariant 1/n and i i center K. If ρ is any irreducible representation of D × 1/n , ψ (ρ) = Hom(ρ, ψ ) is a representation of GLn (K) × WK . Via Jacquet-Langlands theory we can associate to every supercuspidal representation π of GLn (K) an irreducible representation ρ = jl(π ∨ ) of D × 1/n . Now there exists an n-dimensional representation r(π) of WK which satisfies

[π ⊗ r(π)] =

n−1 X

(−1)n−1−i [ψi (jl(π ∨ ))]

i=0

and n · [Rξ (Π) ⊗ χ(Π0 ◦ NmK/Q p )] ∈ ZZ[r(π)] where [ ] denotes the associated class in the Grothendieck group. To show × this one gives a description of H n−1 (X, Lξ )ZZp in which the [ψi (ρ)] occur.

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This way one gets sufficient information to see that the map π 7→ r(π ∨ ⊗ | |

1−n 2

)

defines the local Langlands correspondence. I now briefly describe the contents of the various sections. The first chapter starts with an introductory section on local abelian class field theory which is reformulated to give the local Langlands correspondence for GL1 . The next section contains the formulation of the general correspondence. The following two chapters intend to explain all terms and notations used in the formulation of the local Langlands correspondence. We start with some basic definitions in the theory of representations of reductive p-adic groups and give the Langlands classification of irreducible smooth representations of GLn (K). In some cases I did not find references for the statements (although everything is certainly well known) and I included a short proof. I apologize if some of those proofs are maybe somewhat laborious. After a short interlude about generic and square-integrable representations we come to the definition of L− and ε-factors of pairs of representations. In the following chapter we explain the Galois theoretic side of the correspondence. The fourth chapter starts with the proof of the correspondence in the unramified case. Although this is not needed in the sequel, it might be an illustrating example. After that we return to the general case and give a number of sketchy arguments to reduce the statement of the existence of a unique bijection satisfying certain properties to the statement of the existence of a map satisfying these properties. The third section contains a small “dictionary” which translates certain properties of irreducible admissible representations of GLn (K) into properties of the associated Weil-Deligne representation. In the fourth section the construction of the correspondence is given. It uses Jacquet-Langlands theory, and the cohomology of the sheaf of vanishing cycles on a certain inductive system of formal schemes. These notions are explained in the last chapter. Nothing of this treatise is new. For each of the topics there is a number of excellent references and survey articles. In many instances I just copied them (up to reordering). In addition to original articles my main sources, which can (and should) be consulted for more details, were [CF], [AT], [Neu],

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[Ta2] (for the number theoretic background), [Ca], [BZ1], [Cas1], [Ro] (for the background on representation theory of p-adic groups), and [Kud] (for a survey on “non-archimedean local Langlands”). Note that this is of course a personal choice. I also benefited from the opportunity to listen to the series of lectures of M. Harris and G. Henniart on the local Langlands correspondence during the automorphic semester at the IHP in Paris in spring 2000. I am grateful to the European network in Arithmetic Geometry and to M. Harris for enabling me to participate in this semester. This work is intended as a basis for five lectures at the summer school on “Automorphic Forms on GL(n)” at the ICTP in Triest. I am grateful to M.S. Raghunathan and G. Harder for inviting me to give these lectures. Further thanks go to U. G¨ ortz, R. Hill, N. Kr¨ amer and C. M¨ uller who made many helpful remarks on preliminary versions. Finally I would like to thank the ICTP to provide a pleasant atmosphere during the summer school.

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Notations Throughout we fix the following notations and conventions • p denotes a fixed prime number. • K denotes a p-adic field, i.e. a finite extension of Q p . • vK denotes the discrete valuation of K normalized such that it sends uniformizing elements to 1. • OK denotes the ring of integers of K. Further pK is the maximal ideal of OK and πK a chosen generator of pK . • κ denotes the residue field of OK , and q the number of elements in κ. • | |K denotes the absolute value of K which takes πK to q −1 . • ψ denotes a fixed non-trivial additive character of K (i.e. a continuous homomorphism K −→ { z ∈ C | |z| = 1 }). • n denotes a positive integer. ¯ of K and denote by κ • We fix an algebraic closure K ¯ the residue field ¯ of the ring of integers of K. This is an algebraic closure of κ. ¯ It is • K nr denotes the maximal unramified extension of K in K. also equal to the union of all finite unramified extensions of K in ¯ Its residue field is equal to κ K. ¯ and the canonical homomorphism nr Gal(K /K) −→ Gal(¯ κ/κ) is an isomorphism of topological groups. • ΦK ∈ Gal(¯ κ/κ) denotes the geometric Frobenius x 7→ x1/q and σK its inverse, the arithmetic Frobenius x 7→ xq . We also denote by ΦK and σK the various maps induced by ΦK resp. σK (e.g. on Gal(K nr /K)). • If G is any Hausdorff topological group we denote by Gab its maximal abelian Hausdorff quotient, i.e. Gab is the quotient of G by the closure of its commutator subgroup. • If A is an abelian category, we denote by Groth(A) its Grothendieck group. It is the quotient of the free abelian group with basis the isomorphism classes of objects in A modulo the relation [V ′ ] + [V ′′ ] = [V ] for objects V , V ′ and V ′′ in A which sit in an exact sequence 0 −→ V ′ −→ V −→ V ′′ −→ 0. For any abelian group X and any function λ which associates to isomorphism classes of objects in A an element in X and which is additive (i.e. λ(V ) = λ(V ′ ) + λ(V ′′ ) if there exists an exact sequence 0 −→ V ′ −→ V −→ V ′′ −→ 0) we denote the induced homomorphism of abelian groups Groth(A) −→ X again by λ.

Local Langlands Correspondence for GL(n) over p-adic Fields

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1 The local Langlands correspondence 1.1

The lo al Langlands orresponden e for GL(1)

(1.1.1) In this introductory section we state the local Langlands correspondence for GL1 which amounts to one of the main theorems of abelian local class field theory. For the sake of brevity we use Galois cohomology without explanation. Galois cohomology will not be needed in the sequel. (1.1.2) For any finite extension L of K of degree m and for α ∈ K × we denote by (α, L/K) ∈ Gal(L/K)ab the norm residue symbol of local class field theory. Using Galois cohomology it can be defined as follows (see e.g. [Se1] 2, for an alternative more elementary description see [Neu] chap. IV, V): The group H 2 (Gal(L/K), L× ) is cyclic of order m and up to a sign 1 ZZ/ZZ. We use now the sign convention of [Se1]. canonically isomorphic to m 1 . Let vL/K be the generator of H 2 (Gal(L/K), L× ) corresponding to − m By a theorem of Tate (e.g. [AW] Theorem 12) we know that the map ˆ q (Gal(L/K), ZZ) −→ H ˆ q+2 (Gal(L/K), L× ) which is given by cup-product H with vL/K is an isomorphism. Now we have ˆ −2 (Gal(L/K), ZZ) = H1 (Gal(L/K), ZZ) = Gal(L/K)ab H and ˆ 0 (Gal(L/K), L× ) = K ∗ /NL/K (L× ) H where NL/K denotes the norm of the extension L of K. Hence we get an isomorphism ∼

ϕL/K : Gal(L/K)ab −→ K ∗ /NL/K (L× ). We set (α, L/K) = ϕ−1 L/K ([α]) where [α] ∈ K × /NL/K (L× ) is the class of α ∈ K × .

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(1.1.3) If L is a finite unramified extension of K of degree m we also have the following description of the norm residue symbol (cf. [Se1] 2.5): Let ΦK ∈ Gal(L/K) be the geometric Frobenius (i.e. it induces on residue fields −1 the map σK : x 7→ x−q ). Then we have for α ∈ K × v (α)

(α, L/K) = ΦKK

.

(1.1.4) In the sequel we will only need the isomorphisms ϕL/K . Nevertheless let us give the main theorem of abelian local class field theory: Theorem: The map L 7→ φ(L) := NL/K (L× ) = Ker( , L/K) defines a bijection between finite abelian extensions L of K and closed subgroups of K × of finite index. If L and L′ are finite abelian extensions of K, we have L ⊂ L′ if and only if φ(L) ⊃ φ(L′ ). In this case L is characterized as the fixed field of (φ(L), L′ /K). Proof : See e.g. [Neu] chap. V how to deduce this theorem from the isomorphism Gal(L/K) ∼ = K × /NL/K (L× ) using Lubin Tate theory. We note that this is a purely local proof. (1.1.5) If we go to the limit over all finite extensions L of K, the norm residue symbol defines an isomorphism ∼

ab ¯ −→ lim K × /NL/K (L× ). lim Gal(L/K)ab = Gal(K/K) ←− L

←− L

The canonical homomorphism K × −→ lim K × /NL/K (L× ) is injective with ←− L

dense image and hence we get an injective continuous homomorphism with dense image, called the Artin reciprocity homomorphism ab ¯ ArtK : K × −→ Gal(K/K) .

¯ of K. (1.1.6) Let OK¯ be the ring of integers of the algebraic closure K ¯ Every element of Gal(K/K) defines an automorphism of OK¯ which reduces

Local Langlands Correspondence for GL(n) over p-adic Fields

249

to an automorphism of the residue field κ ¯ of OK¯ . We get a surjective map ¯ π: Gal(K/K) −→ Gal(¯ κ/κ) whose kernel is by definition the inertia group IK of K. The group Gal(¯ κ/κ) is topologically generated by the arithmetic Frobenius automorphism σK which sends x ∈ κ ¯ to xq . It contains the free abelian group hσK i generated by σK as a subgroup. ¯ is K nr , the union of all unramified extensions The fixed field of IK in K ¯ By definition we have an isomorphism of topological groups of K in K. ∼

Gal(K nr /K) −→ Gal(¯ κ/κ).

(1.1.7) The reciprocity homomorphism is already characterized as follows ab ¯ (cf. [Se1] 2.8): Let f : K × −→ Gal(K/K) be a homomorphism such that: (a) The composition ¯ −→ Gal(¯ κ/κ) K × −→ Gal(K/K) f

v (α)

is the map α 7→ ΦKK . (b) For α ∈ K × and for any finite abelian extension L of K such that α ∈ NL/K (L× ), f (α) is trivial on L. Then f is equal to the reciprocity homomorphism ArtK . (1.1.8) We keep the notations of (1.1.6). The Weil group of K is the inverse image of hσK i under π. It is denoted by WK and it sits in an exact sequence 0 −→ IK −→ WK −→ hσK i −→ 0. We endow it with the unique topology of a locally compact group such that the projection WK −→ hσK i ∼ = ZZ is continuous if ZZ is endowed with the discrete topology and such that the induced topology on IK equals the the ¯ profinite topology induced by the topology of Gal(K/K). Note that this ¯ topology is different from the one which is induced by Gal(K/K) via the ¯ inclusion WK ⊂ Gal(K/K). But the inclusion is still continuous, and it has dense image. (1.1.9) There is the following alternative definition of the Weil group: As classes in H 2 correspond to extensions of groups, we get for every finite

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extension L of K an exact sequence 1 −→ L× −→ W (L/K) −→ Gal(L/K) −→ 1 corresponding to the class vL/K . For L ⊂ L′ we get a diagram Lx× NL′ /L

1 −→ 1 −→

L′×

−→

W (L/K)

−→

Gal(L/K) x

−→ 1

−→

W (L′ /K) −→ Gal(L′ /K) −→ 1

which can be commutatively completed by an arrow W (L′ /K) −→ W (L/K) such that we get a projective system (W (L/K))L where L runs through the ¯ Its projective limit is the Weil group of set of finite extensions of K in K. K and the projective limit of the homomorphisms W (L/K) −→ Gal(L/K) ¯ is the canonical injective homomorphism WK −→ Gal(K/K) with dense image. ab (1.1.10) Denote by WK the maximal abelian Hausdorff quotient of WK , i.e. the quotient of WK by the closure of its commutator subgroup. As the ¯ map WK −→ Gal(K/K) is injective with dense image, we get an induced injective map ab ¯ W ab ֒→ Gal(K/K) . K

It follows from (1.1.9) and from the definition of ab ¯ ArtK : K × −→ Gal(K/K) ab that the image of ArtK is WK . We get an isomorphism of topological groups ∼

ab ArtK : K × −→ WK . ab This isomorphism maps O × K onto the abelianization IK of the inertia group and a uniformizing element to a geometric Frobenius elements, i.e. if πK is a uniformizer, the image of ArtK (πK ) in Gal(¯ κ/κ) is ΦK .

(1.1.11) We can reformulate (1.1.10) as follows: Denote by A1 (K) the set of isomorphism classes of irreducible complex representations (π, V ) of K × = GL1 (K) such that the stabilizer of every vector in V is an open subgroup

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of K. It follows from the general theory of admissible representations that every (π, V ) in A1 (K) is one-dimensional (see paragraph 2.1 below). Hence A1 (K) is equal to the set of continuous homomorphisms K × −→ C × where we endow C with the discrete topology. On the other hand denote by G1 (K) the set of continuous homomorphisms WK −→ C × = GL1 (C) where we endow C × with its usual topology. Now a homomorphism WK −→ C × is continuous if and only if its restriction to the inertia group IK is continuous. But IK is compact and totally disconnected hence its image will be a compact and totally disconnected subgroup of C × hence it will be finite. It follows that a homomorphism WK −→ C × is continuous for the usual topology of C × if and only if it is continuous with respect to the discrete topology of C × . Therefore (1.1.10) is equivalent to: Theorem (Local Langlands for GL1 ): There is a natural bijection between the sets A1 (K) and G1 (K). The rest of these lectures will deal with a generalization of this theorem to GLn .

1.2

Formulation of the lo al Langlands orresponden e

(1.2.1) Denote by An (K) the set of equivalence classes of irreducible admissible representations of GLn (K). On the other hand denote by Gn (K) the set of equivalence classes of Frobenius semisimple n-dimensional complex Weil-Deligne representations of the Weil group WK (see chap. 2 and chap. 3 for a definition of these notions). (1.2.2) THEOREM (Local Langlands conjecture for GLn over p-adic fields): There is a unique collection of bijections recK,n = recn : An (K) −→ Gn (K) satisfying the following properties:

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(1) For π ∈ A1 (K) we have rec1 (π) = π ◦ Art−1 K .

(2) For π1 ∈ An1 (K) and π2 ∈ An2 (K) we have L(π1 × π2 , s) = L(recn1 (π1 ) ⊗ recn2 (π2 ), s), ε(π1 × π2 , s, ψ) = ε(recn1 (π1 ) ⊗ recn2 (π2 ), s, ψ). (3) For π ∈ An (K) and χ ∈ A1 (K) we have recn (πχ) = recn (π) ⊗ rec1 (χ).

(4) For π ∈ An (K) with central character ωπ we have det ◦ recn (π) = rec1 (ωπ ).

(5) For π ∈ An (K) we have recn (π ∨ ) = recn (π)∨ where ( )∨ denotes the contragredient. This collection does not depend on the choice of the additive character ψ. (1.2.3) As the Langlands correspondence gives a bijection between representations of GLn (K) and Weil-Deligne representations of WK certain properties of and constructions with representations on the one side correspond to properties and constructions on the other side. Much of this is still an open problem. A few “entries in this dictionary” are given by the following theorem. We will prove it in chapter 4. Theorem: Let π be an irreducible admissible representation of GLn (K) and denote by ρ = (r, N ) the n-dimensional Weil-Deligne representation associated to π via the local Langlands correspondence. (1) The representation π is supercuspidal if and only if ρ is irreducible. (2) We have equivalent statements (i) π is essentially square-integrable.

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(ii) ρ is indecomposable. (iii) The image of the Weil-Deligne group WF′ (C) under ρ is not contained in any proper Levi subgroup of GLn (C). (3) The representation π is generic if and only if L(s, Ad ◦ ρ) has no pole at s = 1 (here Ad: GLn (C) −→ GL(Mn (C)) denotes the adjoint representation). (1.2.4) We are going to explain all occuring notations in the following two chapters.

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2 Explanation of the GL(n)-side 2.1

Generalities on admissible representations

(2.1.1) Throughout this chapter let G be a connected reductive group over K and set G = G(K). Then G is a locally compact Hausdorff group such that the compact open subgroups form a basis for the neighborhoods of the identity (this is equivalent to the fact that G has compact open subgroups and they are all profinite). In particular, G is totally disconnected. To understand (1.2.2) and (1.2.3) we will only need the cases where G is either a product of GLn ’s or the reductive group associated to some central skew field D over K. Nevertheless, in the first sections we will consider the general case of reductive groups to avoid case by case considerations. In fact, almost everywhere we could even work with an arbitrary locally compact totally disconnected group (see e.g. [Vi] for an exposition). (2.1.2) In the case G = GLn and hence G = GLn (K), a fundamental system of open neighborhoods of the identity is given by the open compact m subgroups Cm = 1 + πK Mn (OK ) for m ≥ 1. They are all contained in C0 = GLn (OK ), and it is not difficult to see that C0 is a maximal open compact subgroup and that any other maximal open compact subgroup is conjugated to C0 (see e.g. [Moe] 2). (2.1.3) Definition: A representation π: G −→ GL(V ) on a vector space V over the complex numbers is called admissible if it satisfies the following two conditions: (a) (V, π) is smooth, i.e. the stabilizer of each vector v ∈ V is open in G. (b) For every open subgroup H ⊂ G the space V H of H-invariants in V is finite dimensional. We denote the set of equivalence classes of irreducible admissible representations of G by A(G). For G = GLn (K) we define An (K) = A(GLn (K)).

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Note that the notions of “smoothness” and “admissibility” are purely algebraic and would make sense if we replace C by an arbitrary field. In fact for the rest of this survey article we could replace C by an arbitrary noncountable algebraically closed field of characteristic zero. We could avoid the “non-countability assumption” if we worked consequently only with admissible representations. Further, most elements of the general theory even work over algebraically closed fields of characteristic ℓ with ℓ 6= p ([Vi]). (2.1.4) As every open subgroup of G contains a compact open subgroup, a representation (π, V ) is smooth if and only if [ V = VC C

where C runs through the set of open and compact subgroups of G, and it is admissible if in addition all the V C are finite-dimensional. (2.1.5) Example: A smooth one-dimensional representation of K × is a quasi-character of K × or by abuse of language a multiplicative quasicharacter of K, i.e. a homomorphism of abelian groups K × −→ C × which is continuous for the discrete or equivalently for the usual topology of C × (cf. (1.1.11)). (2.1.6) Let H(G) be the Hecke algebra of G. Its underlying vector space is the space of locally constant, compactly supported measures φ on G with complex coefficients. It becomes an associative C-algebra (in general without unit) by the convolution product of measures. If we choose a Haar measure dg on G we can identify H(G) with the algebra of all locally constant complex-valued functions with compact support on G where the product is given by Z f1 (hg −1 )f2 (g) dg.

(f1 ∗ f2 )(h) =

G

(2.1.7) If C is any compact open subgroup of G, we denote by H(G//C) the subalgebra of H(G) consisting of those φ ∈ H(G) which are left- and

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right-invariant under C. If we choose a Haar measure of G, we can identify H(G//C) with the set of maps C\G/C −→ C with finite support. The algebra H(G//C) has a unit, given by eC := vol(C)−1 1C where 1C denotes the characteristic function of C. If C ′ ⊂ C is an open compact subgroup, H(G//C) is a C-subalgebra of H(G//C ′ ) but with a different unit element if C 6= C ′ . We have [ H(G) = H(G//C). C

(2.1.8) If (π, V ) is a smooth representation of G, the space V becomes an H(G)-module by the formula Z π(g)dφ π(φ)v = G

for φ ∈ H(G). This makes sense as the integral is essentially a finite sum by (2.1.7). S As V = C V C where C runs through the open compact subgroups of G, every vector v ∈ V satisfies v = π(eC )v for some C. In particular, V is a non-degenerate H(G)-module, i.e. H(G) · V = V . We get a functor from the category of smooth representations of G to the category of non-degenerate H(G)-modules. This functor is an equivalence of categories [Ca] 1.4. (2.1.9) Let C be an open compact subgroup of G and let (π, V ) be a smooth representation of G. Then the space of C-invariants V C is stable under H(G//C). If V is an irreducible G-module, V C is zero or an irreducible H(G//C)-module. More precisely we have Proposition: The functor V 7→ V C is an equivalence of the category of admissible representations of finite length such that every irreducible subquotient has a non-zero vector fixed by C with the category of finitedimensional H(G//C)-modules. Proof : This follows easily from [Cas1] 2.2.2, 2.2.3 and 2.2.4.

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(2.1.10) Corollary: Let (π, V ) be an admissible representation of G and let C be an open compact subgroup of G such that for every irreducible subquotient V ′ of V we have (V ′ )C 6= 0. Then the following assertions are equivalent: (1) The G-representation π is irreducible. (2) The H(G//C)-module V C is irreducible. (3) The associated homomorphisms of C-algebras H(G//C) −→ EndC (V C ) is surjective. Proof : The equivalence of (1) and (2) is immediate from (2.1.9). The equivalence of (2) and (3) is a standard fact of finite-dimensional modules of an algebra (see e.g. [BouA] chap. VIII, §13, 4, Prop. 5). (2.1.11) Corollary: Let (π, V ) be an irreducible admissible representation of G and let C ⊂ G be an open compact subgroup such that H(G//C) is commutative. Then dimC (V C ) ≤ 1. (2.1.12) For G = GLn (K) the hypothesis that H(G//C) is commutative is fulfilled for C = GLn (OK ). In this case we have H(G//C) = C[T1±1 , . . . , Tn±1 ]Sn where the symmetric group Sn acts by permuting the variables Ti . More generally, let G be unramified, which means that there exists a reductive model of G over OK , i.e. a flat affine group scheme over OK such that its special fibre is reductive and such that its generic fibre is equal to G. This is equivalent to the condition that G is quasi-split and split over an unramified extension [Ti] 1.10. If C is a hyperspecial subgroup of G (i.e. ˜ K ) for some reductive model G), ˜ the Hecke algebra it is of the form G(O H(G//C) can be identified via the Satake isomorphism with the algebra of invariants under the rational Weyl group of G of the group algebra of the cocharacter group of a maximal split torus of G [Ca] 4.1. In particular, it is commutative. (2.1.13) Under the equivalence of the categories of smooth G-representations and non-degenerate H(G)-modules the admissible representations (π, V )

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T. Wedhorn

correspond to those non-degenerate H(G)-modules such that for any φ ∈ H(G) the operator π(φ) has finite rank. In particular, we may speak of the trace of π(φ) if π is admissible. We get a distribution φ 7→ Tr(π(φ)) which is denoted by χπ and called the distribution character of π. It is invariant under conjugation. (2.1.14) We keep the notations of (2.1.13). If {π1 , . . . , πn } is a set of pairwise non-isomorphic irreducible admissible representations of G, then the set of functionals {χπ1 , . . . , χπ2 } is linearly independent (cf. [JL] Lemma 7.1). In particular, two irreducible admissible representations with the same distribution character are isomorphic. (2.1.15) Let (π, V ) be an admissible representation of G = G(K). By a theorem of Harish-Chandra [HC] the distribution χπ is represented by a locally integrable function on G which is again denoted by χπ , i.e. for every φ ∈ H(G) we have Z χπ (g) dφ. Tr π(φ) = G

The function χπ is locally constant on the set of regular semisimple (see 5.1.3 for a definition in case G = GLn (K)) elements in G (loc. cit.), and it is invariant under conjugation. Therefore it defines a function χπ : {G}reg −→ C on the set {G}reg of conjugacy classes of regular semisimple elements in G. (2.1.16) Proposition (Lemma of Schur): Let (π, V ) be an irreducible smooth representation of G. Every G-endomorphism of V is a scalar. Proof : We only consider the case that π is admissible (we cannot use (2.1.17), because its proof uses Schur’s lemma hence we should not invoke (2.1.17) if we do not want to run into a circular argument; a direct proof of the general case can be found in [Ca] 1.4, it uses the fact that C is not

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countable). For sufficiently small open compact subgroups C of G we have V C 6= 0. Hence we have only to show that every H(G//C)-endomorphism f of a finite dimensional irreducible H(G//C)-module W over C is a scalar. As C is algebraically closed, f has an eigenvalue c, and Ker(f − c idW ) is a H(G//C)-submodule different from W . Therefore f = c idW . (2.1.17) Proposition: Let (π, V ) be an irreducible and smooth complex representation of G. (1) The representation π is admissible. (2) If G is commutative, it is one-dimensional. Proof : The first assertion is difficult and can be found for G = GLn (K) in [BZ1] 3.25. It follows from the fact that every smooth irreducible representation can be embedded in a representation which is induced from a smaller group and which is admissible (more precisely it is supercuspidal, see below). Given (1) the proof of (2) is easy: By (1) we can assume that π is admissible. For any compact open subgroup C of G the space V C is finitedimensional and a G-submodule. Hence V = V C for any C with V C 6= (0) and in particular V is finite-dimensional. But it is well known that every irreducible finite-dimensional representation of a commutative group H on a vector space over an algebraically closed field is one-dimensional (apply e.g. [BouA] chap. VIII, §13, Prop. 5 to the group algebra of H). (2.1.18) Proposition: Every irreducible smooth representation of GLn (K) is either one-dimensional or infinite-dimensional. If it is one-dimensional, it is of the form χ◦det where χ is a quasi-character of K × , i.e. a continuous homomorphism K × −→ C × . We leave the proof as an exercise (show e.g. that the kernel of a finitedimensional representation π: GLn (K) −→ GLn (C) is open, deduce that π is trivial on the subgroup of unipotent upper triangular matrices U , hence π is trivial on the subgroup of GLn (K) which is generated by all conjugates of U and this is nothing but SLn (K)). (2.1.19) Let Z be the center of G. As K is infinite, Z(K) is the center Z of G. In the case G = GLn (K) we have Z = K × . For (π, V ) ∈ A(G) we

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denote by ωπ : Z −→ C × its central character, defined by ωπ (z) idV = π(z) for z ∈ Z. It exists by the lemma of Schur. For G = GLn (K), ωπ is a quasi-character of K × . (2.1.20) Proposition Assume that G/Z is a compact group. Then every irreducible admissible representation (π, V ) of G is finite-dimensional. Proof : By hypothesis we can find a compact open subgroup G0 of G such that G0 Z has finite index in G (take for example the group G0 defined in 2.4.1 below). The restriction of an irreducible representation π of G to G0 Z decomposes into finitely many irreducible admissible representations. By the lemma of Schur, Z acts on each of these representation as a scalar, hence they are also irreducible representations of the compact group G0 and therefore they are finite-dimensional. (2.1.21) Let (π, V ) be a smooth representation of G and let χ be a quasicharacter of G. The twisted representation πχ is defined as g 7→ π(g)χ(g). The G-submodules of (π, V ) are the same as the G-submodules of (πχ, V ). In particular π is irreducible if and only if πχ is irreducible. Further, if C is a compact open subgroup of G, χ(C) ⊂ C × is finite, and therefore χ is trivial on a subgroup C ′ ⊂ C of finite index. This shows that π is admissible if and only if πχ is admissible. If G = GLn (K) every quasi-character χ is of the form χ′ ◦ det where χ′ is a multiplicative quasi-character of K (2.1.18), and we write πχ′ instead of πχ. (2.1.22) Let π: G −→ GL(V ) be a smooth representation of G. Denote by V ∗ the C-linear dual of V . It is a G-module via (gλ)(v) = λ(g −1 v) which is not smooth if dim(V ) = ∞. Define V ∨ = { λ ∈ V ∗ | StabG (λ) is open }.

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This is a G-submodule π ∨ which is smooth by definition. It is called the contragredient of the G-module V . Further we have: (1) π is admissible if and only if π ∨ is admissible and in this case the biduality homomorphism induces an isomorphism V −→ (V ∨ )∨ of G-modules. (2) π is irreducible if and only if π ∨ is irreducible. (3) In the case of G = GLn (K) we can describe the contragredient also in the following way: If π is smooth and irreducible, π ∨ is isomorphic to the representation g 7→ π(t g −1 ) for g ∈ GLn (K). Assertions (1) and (2) are easy (use that (V ∨ )C = (V C )∗ for every compact open subgroup C). The last assertion is a theorem of Gelfand and Kazhdan ([BZ1] 7.3).

2.2

Indu tion and the Bernstein-Zelevinsky lassifi ation for GL(n)

(2.2.1) Fix an ordered partition n = (n1 , n2 , . . . , nr ) of n. Denote by Gn the algebraic group GLn1 × · · · × GLnr considered as a Levi subgroup of G(n) = GLn . Denote by Pn ⊂ GLn the parabolic subgroup of matrices of the form A1 A2 ∗ ... 0 ... Ar for Ai ∈ GLni and by Un its unipotent radical. If (πi , Vi ) is an admissible representation of GLni (K), π1 ⊗ . . . ⊗ πr is an admissible reprentation of Gn (K) on W = V1 ⊗ · · · ⊗ Vr . By extending this representation to Pn and by normalized induction we get a representation π1 × · · · × πr of GLn (K) whose underlying complex vector space V is explicitly defined by V =

1/2 f : GLn (K) −→ W f smooth, f (umg) = δn (m)(π1 ⊗ · · · ⊗ πr )(m)f (g) . for u ∈ Un (K), m ∈ GLn (K) and g ∈ GLn (K)

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Here we call a map f : GLn (K) −→ W smooth if its stabilizer { g ∈ GLn (K) | f (gh) = f (h) for all h ∈ GLn (K) } is open in GLn (K) (or equivalently if f is fixed by some open compact of 1/2 GLn (K) acting by right translation), and δn denotes the positive square root of the modulus character δn (m) = | det(AdUn (m))|. The group GLn (K) acts on V by right translation.

(2.2.2) Definition: An irreducible smooth representation of GLn (K) is called supercuspidal if there exists no proper partition n such that π is a subquotient of a representation of the form π1 × · · · × πr where πi is an admissible representation of GLni (K). We denote by A0n (K) ⊂ An (K) the subset of equivalence classes of supercuspidal representations of GLn (K).

(2.2.3) Let πi be a smooth representation of GLni (K) for i = 1, . . . , r. Then π = π1 ×· · ·×πr is a smooth representation of GLn (K) with n = n1 +· · ·+nr . Further it follows from the compactness of GLn (K)/Pn that if the πi are admissible, π is also admissible ([BZ1] 2.26). Further, by [BZ2] we have the following Theorem: If the πi are of finite length (and hence admissible by (2.1.17)) for all i = 1, . . . , r (e.g. if all πi are irreducible), π1 ×· · ·×πr is also admissible and of finite length. Conversely, if π is an irreducible admissible representation of GLn (K), there exists a unique partition n = n1 + · · · + nr of n and unique (up to isomorphism and ordering) supercuspidal representations πi of GLni (K) such that π is a subquotient of π1 × · · · × πr .

(2.2.4) If π is an irreducible admissible representation of GLn (K) we denote the unique unordered tuple (π1 , · · · , πr ) of supercuspidal representations such that π is a subquotient of π1 × · · · × πr the supercuspidal support.

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(2.2.5) Definition: Let π: GLn (K) −→ GL(V ) be a smooth representation. For v ∈ V and λ ∈ V ∨ the map cπ,v,λ = cv,λ : G −→ C,

g 7→ λ(π(g)v)

is called the (v, λ)-matrix coefficient of π. (2.2.6) Let (π, V ) be an admissible representation. (1) For v ∈ V = (V ∨ )∨ and λ ∈ V ∨ we have cπ,v,λ(g) = cπ∨ ,λ,v (g −1 ). (2) If χ is a quasi-character of K × we have cπχ,v,λ(g) = χ(det(g))cπ,v,λ.

(2.2.7) Theorem: Let π be a smooth irreducible representation of GLn (K). Then the following statements are equivalent: (1) π is supercuspidal. (2) All the matrix coefficents of π have compact support modulo center. (3) π ∨ is supercuspidal. (4) For any quasi-character χ of K × , πχ is supercuspidal. Proof : The equivalence of (1) and (2) is a theorem of Harish-Chandra [BZ1] 3.21. The equivalence of (2), (3) and (4) follows then from (2.2.6). (2.2.8) For any complex number s and for any admissible representation we define π(s) as the twist of π with the character | |s , i.e. the representation g 7→ | det(g)|s π(g). If π is supercuspidal, π(s) is also supercuspidal. Define a partial order on A0n (K) by π ≤ π ′ iff there exists an integer n ≥ 0 such that π ′ = π(n). Hence every finite interval ∆ is of the form ∆(π, m) = [π, π(1), . . . , π(m − 1)]. The integer m is called the length of the interval and nm is called its degree. We write π(∆) for the representation π × · · · × π(m − 1) of GLnm (K).

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Two finite intervals ∆1 and ∆2 are said to be linked if ∆1 6⊂ ∆2 , ∆2 6⊂ ∆1 , and ∆1 ∪ ∆2 is an interval. We say that ∆1 precedes ∆2 if ∆1 and ∆2 are linked and if the minimal element of ∆1 is smaller than the minimal element of ∆2 . (2.2.9) Theorem (Bernstein-Zelevinsky classification ([Ze], cf. also [Ro])): (1) For any finite interval ∆ ⊂ A0n (K) of length m the representation π(∆) has length 2m−1 . It has a unique irreducible quotient Q(∆) and a unique irreducible subrepresentation Z(∆). (2) Let ∆1 ⊂ A0n1 (K), . . . , ∆r ⊂ A0nr (K) be finite intervals such that for i < j, ∆i does not precede ∆j (this is an empty condition if ni 6= nj ). Then the representation Q(∆1 ) × · · · × Q(∆r ) admits a unique irreducible quotient Q(∆1 , . . . , ∆r ), and the representation Z(∆1 ) × · · · × Z(∆r ) admits a unique irreducible subrepresentation Z(∆1 , . . . , ∆r ). (3) Let π be a smooth irreducible representation of GLn (K). Then it is isomorphic to a representation of the form Q(∆1 , . . . , ∆r ) (resp. Z(∆′1 , . . . , ∆′r′ )) for a unique (up to permutation) collection of intervals ∆1 , . . . , ∆r (resp. ∆′1 , . . . , ∆′r′ ) such that ∆i (resp. ∆′i ) does not precede ∆j (resp. ∆′j ) for i < j. (4) Under the hypothesis of (2), the representation Q(∆1 ) × · · · × Q(∆r ) is irreducible if and only if no two of the intervals ∆i and ∆j are linked. (2.2.10) For π ∈ A0n (K) the set of π ′ in A0n (K) which are comparable with π with respect to the order defined above is isomorpic (as an ordered set) to ZZ, in particular it is totally ordered. It follows that given a tuple of intervals ∆i = [πi , . . . , πi (mi − 1)], i = 1, . . . , r we can always permute them such that ∆i does not precede ∆j for i < j. Denote by Sn (K) the set of unordered tuples (∆1 , . . . , ∆r ) where ∆i is P an interval of degree ni such that ni = n. Then (2) and (3) of (2.2.9) are equivalent to the assertion that the maps Q: Sn (K) −→ An (K),

(∆1 , . . . , ∆r ) 7→ Q(∆1 , . . . , ∆r ),

Z: Sn (K) −→ An (K),

(∆1 , . . . , ∆r ) 7→ Z(∆1 , . . . , ∆r ),

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are bijections. The unordered tuple of supercuspidal representations πi (j) for i = 1, . . . , r and j = 0, . . . , mi − 1 is called the supercuspidal support. It is the unique unordered tuple of supercuspidal representations ρ1 , . . . , ρs such that π = Q(∆1 , . . . , ∆s ) and π ′ = Z(∆1 , . . . , ∆r ) is a subquotient of ρ1 × · · · × ρs ([Ze]). (2.2.11) If Rn (K) is the Grothendieck group of the category of admissible L representations of GLn (K) of finite length and R(K) = n≥0 Rn (K), then ([π1 ], [π2 ]) 7→ [π1 × π2 ]

defines a map R(K) × R(K) −→ R(K) which makes R(K) into a graded commutative ring ([Ze] 1.9) which is isomorphic to the ring of polynomials in the indeterminates ∆ for ∆ ∈ S(K) = S n≥1 Sn (K) (loc. cit. 7.5). The different descriptions of An (K) via the maps Q and Z define a map t: R(K) −→ R(K),

Q(∆) 7→ Z(∆).

We have: Proposition: (1) The map t is an involution of the graded ring R. (2) It sends irreducible representations to irreducible representations. (3) For ∆ = [π, π(1), . . . , π(m − 1)] we have t(Q(∆)) = Q(π, π(1), . . . , π(m − 1)) where on the right-hand side we consider π(i) as intervals of length 1. (4) We have t(Q(∆1 , . . . , ∆r )) = Z(∆1 , . . . , ∆r ), t(Z(∆1 , . . . , ∆r )) = Q(∆1 , . . . , ∆r ). Proof : Assertions (1) and (3) follow from [Ze] 9.15. The second assertion had been anounced by J.N. Bernstein but no proof has been published. It has been proved quite recently in [Pr] or [Au1] (see also [Au2]). Assertion (4) is proved by Rodier in [Ro] th´eor`eme 7 under the assumption of (2).

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(2.2.12) For each interval ∆ = [π, . . . , π(m − 1)] we set ∆∨ = [π(m − 1)∨ , . . . , π ∨ ] = [π ∨ (1 − m), . . . , π ∨ (−1), π ∨ ]. It follows from [Ze] 3.3 and 9.4 (cf. also [Tad] 1.15 and 5.6) that we have ∨ Q(∆1 , . . . , ∆r )∨ = Q(∆∨ 1 , . . . , ∆r ), ∨ Z(∆1 , . . . , ∆r )∨ = Z(∆∨ 1 , . . . , ∆r ).

In particular we see that the involution on R induced by [π] 7→ [π ∨ ] commutes with the involution t in (2.2.11).

(2.2.13) Example: Let ∆ ⊂ A1 (K) be the interval ∆ = (| |(1−n)/2 , | |(3−n)/2 , . . . , | |(n−1)/2 ). The associated representation of the diagonal torus T ⊂ GLn (K) is equal to −1/2 δB where δB (t) = | det AdU (t)|K is the modulus character of the adjoint action of T on the group of unipotent upper triangular matrices U and where B is the subgroup of upper triangular matrices in GLn (K). Hence we see that π(∆) = | |(1−n)/2 × | |(3−n)/2 × . . . × | |(n−1)/2 consists just of the space of smooth functions on B\G with the action of G induced by the natural action of G on the flag variety B\G. Hence Z(∆) is the trivial representation 1 of constant functions on G/B. The representation Q(∆) = t(Z(1)) is called the Steinberg representation and denoted by St(n). It is selfdual, i.e. St(n)∨ = St(n) (in fact, it is also unitary and even square integrable, see the next section). For n = 2 the length of π(∆) is 2, hence we have St(n) = π(∆)/1.

2.3

Square integrable and tempered representations

(2.3.1) We return to the general setting where G is an arbitrary connected reductive group over K. Every character α: G −→ G m defines on K-valued

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points a homomorphism α: G −→ K × . By composition with the absolute value | |K we obtain a homomorphism |α|K : G −→ IR>0 and we set \ G0 = Ker(|α|K ). α

If G = GLn then every α is a power of the determinant, hence we have GLn (K)0 = { g ∈ GLn (K) | | det(g)|K = 1 }. Let r be a positive real number. We call an admissible representation (π, V ) of G essentially Lr if for all v ∈ V and λ ∈ V ∨ the matrix coefficient cv,λ is Lr on G0 , i.e. the integral Z |cλ,v |r dg G0

exists (where dg denotes some Haar measure of G0 ). An admissible representation is called Lr if it is essentially Lr and if it has a central character (2.1.19) which is unitary. Let Z be the center of G. Then the composition G0 −→ G −→ G/Z has compact kernel and finite cokernel. Hence, if (π, V ) has a unitary central character ωπ , the integral Z |cv,λ |r dg

Z\G

makes sense, and (π, V ) is L if and only if this integral is finite. r

(2.3.2) Proposition: Let π be an admissible representation of G which is ′ Lr . Then it is Lr for all r ′ ≥ r. Proof : This follows from [Si3] 2.5. (2.3.3) Definition: An admissible representation of G is called essentially square integrable (resp. essentially tempered) if it is essentially L2 (resp. essentially L2+ε for all ε > 0). We have similar definitions by omitting “essentially”. By (2.3.2), any (essentially) square integrable representation is (essentially) tempered.

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(2.3.4) The notion of “tempered” is explained by the following proposition (which follows from [Si1] §4.5 and [Si3] 2.6): Proposition: Let π be an irreducible admissible representation of G such that its central character is unitary. Then the following assertions are equivalent: (1) π is tempered. (2) Each matrix coefficient defines a tempered distribution on G (with the usual notion of a tempered distribution: It extends from a linear form on the locally constant functions with compact support on G to a linear form on the Schwartz space of G (the “rapidly decreasing functions on G”), see [Si1] for the precise definition in the p-adic setting). (3) The distribution character of π is tempered. (2.3.5) Example: By (2.2.7) any supercuspidal representation is essentially Lr for all r > 0. In particular it is essentially square integrable. (2.3.6) If (π, V ) is any smooth representation of G which has a central character, then there exists a unique positive real valued quasi-character χ of G such that πχ has a unitary central character (for G = GLn (K) this is clear as every quasi-character factors through the determinant (2.1.18), for arbitrary reductive groups this is [Cas1] 5.2.5). Hence for G = GLn (K) the notion of “essential square-integrability” is equivalent to the notion of “quasi-square-integrability” in the sense of [Ze]. In particular it follows from [Ze] 9.3: Theorem: An irreducible admissible representation π of GLn (K) is essentially square-integrable if and only if it is of the form Q(∆) with the notations of (2.2.9). It is square integrable if and only if ∆ is of the form [ρ, ρ(1), . . . , ρ(m − 1)] where the central character of ρ((m − 1)/2) is unitary. (2.3.7) We also have the following characterization of tempered representations in the Bernstein-Zelevinsky classification (see [Kud] 2.2): Proposition: An irreducible admissible representation Q(∆1 , . . . , ∆r ) of GLn (K) is tempered if and only if the Q(∆i ) are square integrable.

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(2.3.8) If π = Q(∆1 , . . . , ∆r ) is a tempered representation no two of the intervals ∆i = [ρi , . . . , ρi (mi − 1)] are linked as cent(∆i ) = ρi ((mi − 1)/2) has unitary central character and all elements in ∆i different from cent(∆i ) have a non-unitary central character. Therefore we have π = Q(∆1 ) × · · · × Q(∆r ).

(2.3.9) Let π = Q(∆1 , . . . , ∆r ) be an arbitrary irreducible admissible representation. For each ∆i there exists a unique real number xi such that Q(∆i )(−xi ) is square integrable. We can order the ∆i ’s such that y1 := x1 = · · · = xm1 > y2 := xm1 +1 = · · · = xm2 > · · · > ys := xms−1 +1 = · · · = xr . In this order ∆i does not precede ∆j for i < j and all ∆i ’s which correspond to the same yj are not linked. For j = 1, . . . , s set πj = Q(∆mj−1 +1 )(−yj ) × · · · × Q(∆mj )(−yj ) with m0 = 0 and ms = r. Then all πj are irreducible tempered representation, and π is the unique irreducible quotient of π1 (y1 ) × · · · × πs (ys ). This is nothing but the Langlands classification which can be generalized to arbitrary reductive groups (see [Si1] or [BW]).

2.4

Generi representations

(2.4.1) Fix a non-trivial additive quasi-character ψ: F −→ C × and let n(ψ) −n be the exponent of ψ, i.e. the largest integer n such that ψ(πK OK ) = 1. (2.4.2) Let Un (K) ⊂ GLn (K) be the subgroup of unipotent upper triangular matrices and define a one-dimensional representation θψ of Un (K) by θψ ((uij )) = ψ(u12 + · · · + un−1,n ).

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If π is any representation of GLn (K) we can consider the space of homomorphisms of Un (K)-modules HomUn (K) (π|Un (K) , θψ ). If π is smooth and irreducible we call π generic if this space is non-zero. (2.4.3) In the next few sections we collect some facts about generic representations of GLn (K) which can be found in [BZ1], [BZ2] and [Ze]. Note that in loc. cit. the term “non-degenerate” is used instead of “generic”. First of all we have: Proposition: (1) The representation π is generic if and only if π ∨ is generic. (2) For all multiplicative quasicharacters χ: K × −→ C × , π is generic if and only if χπ is generic. (3) The property of π being generic does not depend of the choice of the non-trivial additive character ψ. (2.4.4) Via the Bernstein-Zelevinsky classification we have the following characterization of generic representations ([Ze] 9.7): Theorem: An irreducible admissible representation π = Q(∆1 , . . . , ∆r ) is generic if and only if no two segments ∆i are linked. In particular we have π∼ = Q(∆1 ) × · · · × Q(∆r ).

(2.4.5) Corollary: Every essentially tempered (and in particular every supercuspidal) representation is generic. (2.4.6) If (π, V ) is generic, it has a Whittaker model: Choose a 0 6= λ ∈ HomU (K) (π|U (K) , θψ ) and define a map V −→ { f : GLn (K) −→ C | f (ug) = θ(u)f (g) for all g ∈ GLn (K), u ∈ U (K) }, v 7→ (g 7→ λ(π(g)v))

.

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This is an injective homomorphism of GLn (K)-modules if GLn (K) acts on the right-hand side by right translation, and we call its image the Whittaker model of π with respect to ψ and denote it by W(π, ψ). (2.4.7) The concept of a generic representation plays a fundamental role in the theory of automorphic forms: If π is an irreducible admissible representation of the adele valued group GLn (AL ) for a number field L, it can be decomposed in a restricted tensor product O π= πv v

where v runs through the places of L and where πv is an admissible irreducible representation of GLn (Lv ) (see Flath [Fl] for the details). If π is cuspidal, all the πv are generic by Shalika [Sh].

2.5

Definition of L- and epsilon-fa tors

(2.5.1) Let π and π ′ be smooth irreducible representations of GLn (K) and of GLn′ (K) respectively. We are going to define L- and ε-factors of the pair (π, π ′ ). We first do this for supercuspidal (or more generally for generic) representation and then use the Bernstein-Zelevinsky classification to make the general definition. Assume now that our fixed non-trivial additive character ψ (2.4.1) is ¯ Let π and π ′ be generic representations of GLn (K) unitary, i.e. ψ−1 = ψ. and GLn′ (K) respectively. To define L- and ε-factors L(π × π ′ , s) and ε(π × π ′ , s, ψ) we follow [JPPS1]. Consider first the case n = n′ . Denote by S(K n ) the set of locally constant functions φ: K n −→ C with compact support. For elements W ∈ ¯ in the Whittaker models and for any φ ∈ S(K n ) W(π, ψ), W ′ ∈ W(π ′ , ψ) define Z ′ Z(W, W , φ, s) = W (g)W ′ (g)φ((0, . . . , 0, 1)g)| det(g)|s dg Un (K)\GLn (K)

where dg is a GLn (K)-invariant measure on Un (K)\GLn (K). This is absolutely convergent if Re(s) is sufficiently large and it is a rational function of

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q −s . The set ¯ and φ ∈ S(K n ) } { Z(W, W ′ , φ, s) | W ∈ W(π, ψ), W ′ ∈ W(π ′ , ψ) generates a fractional ideal in C[q s , q −s ] with a unique generator L(π ×π ′ , s) of the form P (q −s )−1 where P ∈ C[X] is a polynomial such that P (0) = 1. Further ε(π × π ′ , s, ψ) is defined by the equality ˆ ˜ ,W ˜ ′ , 1 − s, φ) Z(W, W ′ , s, φ) Z(W n ′ ′ (−1) ε(π × π , s, ψ) = ω . π L(π ∨ × π ′∨ , 1 − s) L(π × π ′ , s) ˜ by W ˜ (g) = W (wn t g −1 ) where wn ∈ GLn (K) is the Here we define W permutation matrix corresponding to the longest Weyl group element (i.e. to the permutation which sends i to n + 1 − i). Because of (2.1.22) this is an ¯ In the same way we define W ˜ ′ ∈ W(π ′∨ , ψ). Finally element of W(π ∨ , ψ). φˆ denotes the Fourier transform of φ with respect to ψ given by Z ˆ φ(y)ψ(t y x) dy φ(x) = Kn

for x ∈ K n . ¯ and Now consider the case n′ < n. For W ∈ W(π, ψ), W ′ ∈ W(π ′ , ψ) for j = 0, 1, . . . , n − n′ − 1 define g 0 0 Z Z )W ′ (g) W ( x Ij 0 Z(W, W ′ , j, s) = Un′ (K)\GLn′ (K) Mj×n′ (K) 0 0 In−n′ −j ′

· | det(g)|s−(n−n )/2 dx dg where dg is a GLn′ (K)-invariant measure on Un′ (K)\GLn′ (K) and dx is a Haar measure on the space of (j × n′ )-matrices over K. Again this is absolutely convergent if Re(s) is sufficiently large, it is a rational function of q −s and these functions generate a fractional ideal with a unique generator L(π × π ′ , s) of the form P (q −s )−1 where P ∈ C[X] is a polynomial such that P (0) = 1. In this case ε(π × π ′ , s, ψ) is defined by ˜ ,W ˜ ′ , n − n′ − 1 − j, 1 − s) Z(wn,n′ W Z(W, W ′ , j, s) n−1 ′ ′ (−1) = ω ε(π×π , ψ, s) π L(π ∨ × π ′∨ , 1 − s) L(π × π ′ , s) where wn,n′ is the matrix In0 ′ w 0 ′ ∈ GLn (K). n−n

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Finally for n′ > n we define L(π × π ′ , s) = L(π ′ × π, s),

ε(π × π ′ , ψ, s) = ε(π ′ × π, ψ, s).

In all cases L(π × π ′ , s) does not depend on the choice of ψ, and ε(π × π ′ , s, ψ) is of the form cq −f s for a non-zero complex number c and an integer f which depend only on π, π ′ and ψ. This finishes the definition of the L- and the ε-factor for the generic case (and in particular for the supercuspidal case). Note that if π and π ′ are supercuspidal, we have Y (2.5.1.1) L(π × π ′ , s) = L(χ, s) χ

∼π where χ runs over the unramified quasi-characters of K × such that χπ ′∨ = and where L(χ, s) is the L-function of a character as defined in Tate’s thesis (see also below). In particular we have L(π × π ′ , s) = 1 for π ∈ A0n (K) and π ′ ∈ A0n′ (K) with n 6= n′ . It seems that there is no such easy way to define ε(π × π ′ , ψ, s) for supercuspidal π and π ′ . However, Bushnell and Henniart [BH] prove that ε(π × π ∨ , ψ, 1/2) = ωπ (−1)n−1 for every irreducible admissible representation π of GLn (K). (2.5.2) From the definition of the L- and ε-factor in the supercuspidal case we deduce the definition of the L- and ε-factor for pairs of arbitrary smooth irreducible representations π and π ′ by the following inductive relations using (2.2.9)(cf. [Kud]): (1) We have L(π × π ′ , s) = L(π ′ × π, s) and ε(π × π ′ , ψ, s) = ε(π ′ × π, ψ, s). (2) If π is of the form Q(∆1 , . . . , ∆r ) (2.2.9) and if π ′ is arbitrary, then L(π × π ′ , s) =

r Y

L(Q(∆i ) × π ′ , s)

ε(π × π ′ , ψ, s) =

r Y

ε(Q(∆i ) × π ′ , ψ, s).

i=i

i=i

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(3) If π is of the form Q(∆), ∆ = [σ, σ(r − 1)] and π ′ = Q(∆′ ), ∆′ = [σ ′ , σ ′ (r ′ − 1)] with r ′ ≥ r, then L(π × π ′ , s) =

r Y

L(σ × σ ′ , s + r + r ′ − 1)

i=1 r r+r Y Y−2i ′

′

ε(π × π , ψ, s) =

i=1

×

ε(σ × σ ′ , ψ, s + i + j − 1)

j=0

′ −2i−1 r+rY

j=0

L(σ ∨ × σ ′∨ , 1 − s − i − j) . L(σ × σ ′ , s + i + j − 1)

(2.5.3) Let 1: K × −→ C × be the trivial multiplicative character. For any smooth irreducible representation π of GLn (K) we define L(π, s) = L(π × 1, s), ε(π, ψ, s) = ε(π × 1, ψ, s). For n = 1, L(π, s) and ε(π, ψ, s) are the local L- and ε-factors defined in Tate’s thesis. For n > 1 and π supercuspidal, we have L(π, s) = 1, while ε(π, ψ, s) is given by a generalized Gauss sum [Bu]. (2.5.4) Let (π, V ) be a smooth and irreducible representation of GLn (K). For any non-negative integer t define a b t t Kn (t) = { ∈ GLn (OK )|c ∈ M1×n−1 (πK OK ), d ≡ 1 (mod πK OK ) }. c d In particular, we have Kn (0) = GLn (OK ). The smallest non-negative integer t such that V Kn (t) 6= (0) is called the conductor of π and denoted by f (π). By [JPPS1] (cf. also [BHK]) it is also given by the equality ε(π, ψ, s) = ε(π, ψ, 0)q −s(f (π)+nn(ψ)) where n(ψ) denotes the exponent of ψ (2.4.1).

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3 Explanation of the Galois side 3.1

Weil-Deligne representations

(3.1.1) Let WK be the Weil group of K (1.1.8) and let ϕK : WK −→ ¯ Gal(K/K) be the canonical homomorphism. ¯ A representation of WK (resp. of Gal(K/K)) is a continuous homo¯ morphism WK −→ GL(V ) (resp. Gal(K/K) −→ GL(V )) where V is a finite-dimensional complex vector space. Denote by Rep(WK ) (resp. ¯ Rep(Gal(K/K))) the category of representations of the respective group. Note that a homomorphism of a locally profinite group (e.g. WK or ¯ Gal(K/K)) into GLn (C) is continuous for the usual topology of GLn (C) if and only if it is continuous for the discrete topology. (3.1.2) For w ∈ WK we set |w| = |w|K = |Art−1 K (w)|K . Then the map WK −→ C × , w 7→ |w|s is a one-dimensional representation (i.e. a quasi-character) of WK for every complex number s. All onedimensional representations of WK which are trivial on IK (i.e. which are unramified) are of this form ([Ta1] 2.3.1). ¯ (3.1.3) As ϕK is injective with dense image, we can identify Rep(Gal(K/K)) with a full subcategory of Rep(WK ). A representation in this subcategory is called of Galois-type. By [Ta2] 1.4.5 a representation r of WK is of Galoistype if and only if its image r(WK ) is finite. Conversely, by [De2] §4.10 and (3.1.2) every irreducible representation r of WK is of the form r = r ′ ⊗ | |s for some complex number s and for some representation r ′ of Galois type. (3.1.4) A representation of Galois-type of WK is irreducible if and only ¯ if it is irreducible as a representation of Gal(K/K). Further, if σ is any irreducible representation of WK , it is of Galois type if and only if the image of its determinant det ◦σ is a subgroup of finite order of C × .

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¯ Then we have a canonical (3.1.5) Let L be a finite extension of K in K. injective homomorphism WL −→ WK with finite cokernel. Hence restriction and induction of representations give functors resL/K :Rep(WK ) −→ Rep(WL ) indL/K :Rep(WL ) −→ Rep(WK ) satisfying the usual Frobenius reciprocity. More precisely, any representation of WK becomes a representation of WL by restriction r 7→ r|WK . This defines the map resL/K . Conversely, let r: WL −→ GL(V ) be a representation of WL . Then we define indL/K (r) as the representation of WK whose underlying vector space consists of the continuous maps f : WK −→ V such that f (xw) = r(x)f (w) for all x ∈ WL and w ∈ WK . Note that in the context of the cohomology of abstract groups this functor “induction” as defined above is often called “coinduction”.

(3.1.6) Definition: A Weil-Deligne representation of WK is a pair (r, N ) where r is a representation of WK and where N is a C-linear endomorphism of V such that (3.1.6.1)

r(γ) N = |Art−1 K (γ)|K N r(γ)

for γ ∈ WK . It is called Frobenius semisimple if r is semisimple.

(3.1.7) Remark: Let (r, N ) be a Weil-Deligne representation of WK . (1) Let γ ∈ WK be an element corresponding to a uniformizer πK via ArtK . Applying (3.1.6.1) we see that N is conjugate to qN , hence every eigenvalue of N must be zero which shows that N is automatically nilpotent. (2) The kernel of N is stable under WK , hence if (r, N ) is irreducible, N is equal to zero. Therefore the irreducible Weil-Deligne representations of WK are simply the irreducible continuous representations of WK .

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(3.1.8) Let ρ1 = (r1 , N1 ) and ρ2 = (r2 , N2 ) be two Weil-Deligne representations on complex vector spaces V1 and V2 respectively. Their tensor product ρ1 ⊗ ρ2 = (r, N ) is the Weil-Deligne representation on the space V1 ⊗ V2 given by r(w)(v1 ⊗ v2 ) = r1 (w)v1 ⊗ r2 (w)v2 ,

N (v1 ⊗ v2 ) = N1 v1 ⊗ v2 + v1 ⊗ N2 v2

for w ∈ WK and vi ∈ Vi , i = 1, 2. Further, HomC (V1 , V2 ) becomes the vector space of a Weil-Deligne representation Hom(ρ1 , ρ2 ) = (r, N ) by (r(w)ϕ)(v1) = r2 (w)(ϕ(r1 (w)−1 v1 )),

(N ϕ)(v1) = N2 (ϕ(v1 ))−ϕ(N1 (v1 ))

for ϕ ∈ HomC (V1 , V2 ), w ∈ WK and v1 ∈ V1 . In particular we get the contragredient ρ∨ of a Weil-Deligne representation as the representation Hom(ρ, 1) where 1 is the trivial one-dimensional representation. (3.1.9) Consider WK as a group scheme over Q (not of finite type) which is the limit of the constant group schemes associated to the discrete groups WK /J where J runs through the open normal subgroups of IK . Denote by ′ WK the semi-direct product ′ | G WK = WK × a

where WK acts on G a by the rule wxw−1 = |w|K x. This is a group scheme (neither affine nor of finite type) over Q whose R-valued points for some | R, and the Q-algebra R without non-trivial idempotents are given by WK × law of composition is given by (w1 , x1 )(w2 , x2 ) = (w1 w2 , |w2 |−1 K x1 + x2 ). A Weil-Deligne representation of WK is the same as a complex finite′ dimensional representation of the group scheme WK whose underlying WK representation is semisimple (to see this use the fact that representations of the additive group on a finite-dimensional vector space over a field in characteristic zero correspond to nilpotent endomorphisms). ′ | C) is called The group scheme WK (or also its C-valued points WK × the Weil-Deligne group.

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(3.1.10) It follows from the Jacobson-Morozov theorem that we can also interpret a Weil-Deligne representation as a continuous complex semisimple representation of the group WK × SL2 (C). If η is such a representation, we associate a Weil-Deligne representation (r, N ) by the formulas ! 1 0 |w| 2 ) r(w) = η(w, 1 0 |w|− 2 and exp(N ) = η(1,

1 1 ). 0 1

A theorem of Kostant assures that two representations of WF × SL2 (C) are isomorphic if and only if the corresponding Weil-Deligne representations are isomorphic (see [Ko] for these facts).

3.2

Definition of L- and epsilon-fa tors

(3.2.1) Let ρ = ((r, V ), N ) be a Frobenius semisimple Weil-Deligne representation. Denote by VN the kernel of N and by VNIK the space of invariants in VN for the action of the inertia group IK . The L-factor of ρ is given by L(ρ, s) = det(1 − q −s Φ|V IK )−1 N

where Φ ∈ WK is a geometric Frobenius. If ρ and ρ′ are irreducible WeilDeligne representations of dimension n, n′ respectively, we have Y (3.2.1.1) L(ρ ⊗ ρ′ , s) = L(χ, s) χ

ab where χ runs through the unramified quasi-characters of K × ∼ such = WK ∨ ′ ′ that χ ⊗ ρ = ρ (compare (2.5.1.1)). In particular L(ρ ⊗ ρ , s) = 1 for n 6= n′ . Fix a non-trivial additive character ψ of K and let n(ψ) be the largest −n integer n such that ψ(πK OK ) = 1. Further let dx be an additive Haar measure of K.

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To define ε(ρ, ψ, s) we first define the ε-factor of the Weil group representation (r, V ). Assume first that V is one-dimensional, i.e. r = χ is a quasi-character ab χ: WK −→ C × .

Let χ be unramified (i.e. χ(IK ) = (1) or equivalently χ = | |s for some complex number s). Then we set ε(χ, ψ, dx) = χ(w)q n(ψ) voldx (OK ) = q n(ψ)(1−s) voldx (OK ) where w ∈ WK is an element whose valuation is n(ψ). If χ is ramified, let f (χ) be the conductor of χ, i.e. the smallest integer f f such that χ(ArtK (1 + πK OK )) = 1, and let c ∈ K × be an element with valuation n(ψ) + f (χ). Then we set Z ε(χ, ψ, dx) = χ−1 (ArtK (x))ψ(x)dx. c−1 OK ×

The ε-factors attached to (r, V ) with dim(V ) > 1 are characterized by the following theorem of Langlands and Deligne [De2]: Theorem: There is a unique function ε which associates with each choice of a local field K, a non-trivial additive character ψ of K, an additive Haar measure dx on K and a representation r of WK a number ε(r, ψ, dx) ∈ C × such that (1) If r = χ is one-dimensional ε(χ, ψ, dx) is defined as above. (2) ε( , ψ, dx) is multiplicative in exact sequences of representations of WK (hence we get an induced homomorphism ε( , ψ, dx): Groth(Rep(WK )) −→ C × ). (3) For every tower of finite extensions L′ /L/K and for every choice of additive Haar measures µL on L and µL′ on L′ we have ε(indL′ /L [r ′ ], ψ ◦ TrL/K , µL ) = ε([r ′ ], ψ ◦ TrL′ /K , µL′ ) for [r ′ ] ∈ Groth(Rep(WL′ )) with dim([r ′ ]) = 0. Note that we have ε(χ, ψ, αdx) = αε(χ, ψ, dx) for α > 0 and hence via inductivity ε(r, ψ, αdx) = αdim(r) ε(χ, ψ, dx). In particular if [r] ∈

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Groth(Rep(WK )) is of dimension 0, ε([r], ψ, dx) is independent of the choice of dx. Now we can define the ε-factor of the Weil-Deligne representation ρ = (r, N ) as ε(ρ, ψ, s) = ε(| |s r, ψ, dx) det(−Φ|V IK /V IK ) N

where dx is the Haar measure on K which is self-dual with respect to the Fourier transform f 7→ fˆ defined by ψ: Z ˆ f (y) = f (x)ψ(xy) dx. In other words ([Ta1] 2.2.2) it is the Haar measure for which OK gets the volume q −d/2 where d is the valuation of the absolute different of K (if the ramification index e of K/Q p is not divided by p, we have d = e − 1, in general d can be calculated via higher ramification groups [Se2]). Note that ε(ρ, ψ, s) is not additive in exact sequences of Weil-Deligne representations as taking coinvariants is not an exact functor. (3.2.2) Let ρ be an irreducible Weil-Deligne representation of dimension n, then we can define the conductor f (ρ) of ρ by the equality ε(ρ, ψ, s) = ε(ρ, ψ, 0)q −s(f (ρ)+nn(ψ)) where n(ψ) denotes the exponent of ψ (2.4.1). This is a nonnegative integer which can be explicitly expressed in terms of higher ramification groups (e.g. [Se2] VI,§2, Ex. 2). (3.2.3) For any m ≥ 1 we define the Weil-Deligne representations Sp(m) = ((r, V ), N ) by V = Ce0 ⊕ · · · ⊕ Cem−1 with r(w)ei = |w|i ei and N ei = ei+1

(0 ≤ i < m − 1),

N em−1 = 0.

In this case we have VN = VNIK = Cem−1 and Φei = q −i ei for a geometric Frobenius Φ ∈ WK . Hence the L-factor is given by 1 L(ρ, s) = . 1−s−m 1−q

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Let ψ be an additive character such that n(ψ) = 0 and let dx be the Haar measure on K which is self-dual with respect to Fourier transform as above. Then we have ε(r, ψ, dx) = q −md/2 where d is the valuation of the absolute different of K. Hence the ε-factor is given by ε(ρ, ψ, s) = (−1)m−1q

−md−(m−2)(m−1) 2

.

(3.2.4) A Frobenius semisimple Weil-Deligne representation ρ is indecomposable if and only if it has the form ρ0 ⊗ Sp(m) for some m ≥ 1 and with ρ0 irreducible. Moreover, the isomorphism class of ρ0 and m are uniquely determined by ρ ([De1] 3.1.3(ii)). Further (as in every abelian category where all objects have finite length) every Frobenius semisimple Weil-Deligne representation is the direct sum of unique (up to order) indecomposable Frobenius semisimple Weil-Deligne representations.

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4 Construction of the correspondence 4.1

The orresponden e in the unramified ase

(4.1.1) Definition: An irreducible admissible representation (π, V ) of GLn (K) is called unramified, if the space of fixed vectors under C = GLn (OK ) is non-zero, i.e. if its conductor (2.5.4) is zero. (4.1.2) Example: A multiplicative quasi-character χ: K × −→ C × is unramified if and only if χ(O × K ) = {1}. An unramified quasi-character χ is uniquely determined by its value χ(πK ) which does not depend on the choice of the uniformizing element πK . It is of the form | |s for a unique s ∈ C/(2πi(log q)−1 )ZZ. (4.1.3) Let (χ1 , . . . , χn ) be a family of unramified quasi-characters which we can view as intervals of length zero in A01 (K). We assume that for i < j, χi does not precede χj , i.e. χ−1 i χj 6= | |K . Then Q(χ1 , . . . , χn ) is an unramified representation of GLn (K). Conversely we have [Cas2] Theorem: Every unramified representation π of GLn (K) is isomorphic to a representation of the form Q(χ1 , . . . , χn ) where the χi are unramified quasi-characters of K × . (4.1.4) An unramified representation π of GLn (K) is supercuspidal if and × only if n = 1 and π is an unramified quasi-character of K . (4.1.5) Let π be an unramified representation associated to unramified quasi-characters χ1 , . . . , χn . This tuple of unramified quasi-characters induces a homomorphism T /Tc −→ C × n where T ∼ = (K × )n denotes the diagonal torus of G and where Tc ∼ = (O × K) denotes the unique maximal compact subgroup of T of diagonal matrices with coefficients in O × K.

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Thus the set of unramified representations may be identified with the set of orbits under the Weyl group Sn of GLn in Tˆ = Hom(T /Tc , C × ) = (C × )n where the last isomorphism is given by the identification T /Tc = ZZn ,

diag(t1 , . . . , tn ) 7→ (vK (t1 ), . . . , vK (tn )).

(4.1.6) To shorten notations set C = GLn (OK ). The Hecke algebra H(GLn (K)//C) is commutative and canonically isomorphic to the Sn invariants of the group algebra ([Ca] 4.1) ±1 C[X ∗ (Tˆ)] = C[X∗ (T )] ∼ = C[t±1 1 , . . . , tn ].

If (π, V ) is an unramified representation, V C is one-dimensional (2.1.11), hence we get a canonical homomorphism λπ : H(GLn (K)//C) −→ End(V C ) = C. For every h ∈ H(GLn (K)//C) the map Tˆ/ΩGLn −→ C,

π 7→ λπ (h)

can be considered as an element in C[X ∗ (Tˆ)]Sn and this defines the isomorphism ∼ H(GLn (K)//C) −→ C[X ∗ (Tˆ)]Sn .

(4.1.7) Definition: An n-dimensional Weil-Deligne representation ρ = ((r, V ), N ) is called unramified if N = 0 and if r(IK ) = {1}. (4.1.8) Every unramified n-dimensional Weil-Deligne representation ρ = ((r, C n ), N ) is uniquely determined by the GLn (C)-conjugacy class of r(Φ) =: gρ for a geometric Frobenius Φ. By definition this element is semisimple and hence we can consider this as an Sn -orbit of the diagonal torus (C × )n of GLn (C). Hence we get a bijection recn between unramified representations of GLn (K) and unramified n-dimensional Weil-Deligne representations.

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This is normalized by the following two conditions: (i) An unramified quasi-character χ of IK× corresponds to an unramified ab quasi-character rec1 (χ) of WK via the map ArtK from local class field theory. (ii) The representation Q(χ1 , . . . , χn ) (4.1.3) corresponds to the unramified Weil-Deligne representation rec1 (χ1 ) ⊕ · · · ⊕ rec1 (χn ). By the inductive definition of L- and ε-factors it follows that the bijection recn satisfies condition (2) of (1.2.2) (see also (2.5.1.1) and (3.2.1.1)). Further condition (3) for unramified characters and condition (4) are clearly okay, and condition (5) follows from the obvious fact that if unramified elements in An (K) or in Gn (K) correspond to the Sn -orbit of diag(t1 , . . . , tn ) their contragredients correspond to the orbit of diag(t1 , . . . , tn )−1 = −1 diag(t−1 1 , . . . , tn ). (4.1.9) From the global point of view, unramified representations are the “normal” ones: If O π= πv v

is an irreducible admissible representation of the adele valued group GLn (AL ) for a number field L as in (2.4.7), all but finitely many πv are unramified.

4.2

Some redu tions

(4.2.1) In this paragraph we sketch some arguments (mostly due to Henniart) which show that it suffices to show the existence of a family of maps (recn ) satisfying all the desired properties between the set of isomorphism classes of supercuspidal representations and the set of isomorphism classes of irreducible Weil-Deligne representations. We denote by A0n (K) the subset of An (K) consisting of the supercuspidal representations of GLn (K). Further let Gn0 (K) be the set of irreducible Weil-Deligne representations in Gn (K).

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(4.2.2) Reduction to the supercuspidal case: In order to prove the local Langlands conjecture (1.2.2), it suffices to show that there exists a unique collection of bijections recn : A0n (K) −→ Gn0 (K) satisfying (1.2.2) (1) to (5). Reasoning : This follows from (2.2.9) and from (3.2.4). More precisely, for any irreducible admissible representation π ∼ = Q(∆1 , . . . , ∆r ), with ∆i = [πi , πi (mi − 1)] and πi ∈ A0ni (K) define recn1 m1 +···+nr mr (π) =

r M

recni (πi ) ⊗ Sp(mi ).

i=1

Properties (1.2.2) (1) to (5) follow then (nontrivially) from the inductive description of the L- and the ε-factors.

(4.2.3) Reduction to an existence statement: If there exists a collection of bijections (recn )n as in (4.2.2), it is unique. This follows from the fact that representations π ∈ A0n (K) are already determined inductively by their ε-factors in pairs and that by (1.2.2)(1) rec1 is given by class field theory. More precisely, we have the following theorem of Henniart [He3]: Theorem: Let n ≥ 2 be an integer and let π and π ′ be representations in A0n (K). Assume that we have an equality ε(π × τ, ψ, s) = ε(π ′ × τ, ψ, s) for all integers r = 1, . . . , n − 1 and for every τ ∈ A0r (K). Then π ∼ = π′ .

(4.2.4) Injectivity: Every collection of maps recn as in (4.2.2) is automatically injective: If χ is a quasi-character of K × , its L-function L(χ, s) is given by (1 − χ(π)q s )−1 , if χ is unramified, L(χ, s) = 1, if χ is ramified.

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In particular, it has a pole in s = 0 if and only if χ = 1. Hence by (2.5.1.1) and (3.2.1.1) we have for π, π ′ ∈ A0n (K): recn (π) = recn (π ′ ) ⇔ L(recn (π)∨ ⊗ recn (π ′ ), s) has a pole in s = 0 ⇔ L(π ∨ × π ′ , s) has a pole in s = 0 ⇔ π = π′ .

(4.2.5) Surjectivity: In order to prove the local Langlands conjecture it suffices to show that there exists a collection of maps recn : A0n (K) −→ Gn0 (K) satisfying (1.2.2) (1) to (5). Reasoning : Because of the preservation of ε-factors in pairs it follows from (3.2.2) and (2.5.4) that recn preserves conductors. But by the numerical local Langlands theorem of Henniart [He2] the sets of elements in A0n (K) and Gn0 (K) which have the same given conductor and the same central character are finite and have the same number of elements. Hence the bijectivity of recn follows from its injectivity (4.2.4).

4.3

A rudimentary di tionary of the orresponden e

(4.3.1) In this section we give some examples how certain properties of admissible representations can be detected on the corresponding Weil-Deligne representation and vice versa. Throughout (π, Vπ ) denotes an admissible irreducible representation of GLn (K), and ρ = ((r, Vr ), N ) the n-dimensional Frobenius-semisimple Weil-Deligne representation associated to it via the local Langlands correspondence (1.2.2).

(4.3.2) First of all, we have of course: Proposition: The admissible representation π is supercuspidal if and only if ρ is irreducible (or equivalently iff r is irreducible and N = 0).

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(4.3.3) Write π = Q(∆1 , . . . , ∆s ) in the Bernstein-Zelevinsky classification (2.2.9), where ∆i = [πi , . . . , πi (mi − 1)] is an interval of supercuspidal representations of GLni (K). By (4.2.2) we have ρ=

s M

(recni (πi ) ⊗ Sp(mi )).

i=1

Set ρi = ((ri , Vri ), 0) = recni (πi ). The underlying representation of the Weil group of πi ⊗ Sp(mi ) is then given by ri ⊕ ri (1) ⊕ · · · ⊕ ri (mi − 1) where r(x) denotes the representation w 7→ r(w)|w|x for any representation r of WK and any real number x. We have (ri (j), 0) = recni (πi (j)). Further, if Ni is the nilpotent endomorphism of ρi ⊗Sp(mi ), its conjugacy class (which we can consider as a non-ordered partition of ni mi by the Jordan normal form) is given by the partition

Hence we get:

ni mi = mi + · · · + mi . {z } | ni -times

Proposition: The underlying WK -representation r of ρ depends only on the supercuspidal support τ1 , . . . , τt of π (2.2.10). More precisely, we have an isomorphism of Weil-Deligne representations (r, 0) ∼ = rec(τ1 ) ⊕ . . . ⊕ rec(τt ). The conjugacy class of N is given by the degree ni of πi and the length mi of the intervals ∆i as above. In particular, we have N = 0 if and only if all intervals ∆i are of length 1. (4.3.4) Example: The Steinberg representation St(n) (2.2.13) corresponds to the Weil-Deligne representation | |(1−n)/2 Sp(n). (4.3.5) Recall from (4.1.3) that π is unramified if and only if all intervals ∆i are of length 1 and consist of an unramified quasi-character of K × . Hence

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(4.3.3) shows that π is unramified if and only if ρ is unramified. We used this already in (4.1). (4.3.6) The “arithmetic information” of WK is encoded in the inertia subgroup IK . The quotient WK /IK is the free group generated by ΦK and hence “knows” only the number q of elements in the residue field of K. Therefore Weil-Deligne representations ρ = (r, N ) with r(IK ) = 1 should be particularly simple. We call such representations IK -spherical. Then r is a semisimple representation of < ΦK >∼ = ZZ. Obviously, every finite-dimensional semisimple representation of ZZ is the direct sum of onedimensional representations. Hence r is the direct sum of quasi-characters of WK which are necessarily unramified. On the GLn (K)-side let I ⊂ GLn (K) be an Iwahori subgroup, i.e. I is an open compact subgroup of GLn (K) which is conjugated to the group of matrices (aij ) ∈ GLn (OK ) with aij ∈ πK OK for i > j. We say that π is I-spherical if the space of I-fixed vectors is non-zero. By a theorem of Casselman ([Ca] 3.8, valid for arbitrary reductive groups - with the appropriate reformulation) an irreducible admissible representation is I-spherical if and only if its supercuspidal support consists of unramified quasi-characters. Altogether we get: Proposition: We have equivalent assertions: (1) The irreducible admissible representation π is I-spherical. (2) The supercuspidal support of π consists of unramified quasicharacters. (3) The corresponding Weil-Deligne representation ρ is IK -spherical. By (2.1.9) the irreducible admissible I-spherical representations are nothing but the finite-dimensional irreducible H(GLn (K)//I)-modules. The structure of the C-algebra H(GLn (K)//I) is known in terms of generators and relations ([IM]) and depends only on the isomorphism class of GLn over some algebraically closed field (i.e. the based root datum of GLn ) and on the number q.

(4.3.7) Finally, we translate several notions which have been defined for

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admissible representations of GLn (K) into properties of Weil-Deligne representations: Proposition: Let π be an irreducible admissible representation of GLn (K) and let ρ = (r, N ) be the corresponding Weil-Deligne representation. (1) We have equivalent statements (i) π is essentially square-integrable. (ii) ρ is indecomposable. (iii) The image of the Weil-Deligne group WF′ (C) under ρ is not contained in any proper Levi subgroup of GLn (C). (2) We have equivalent statements (i) π is tempered. (ii) Let η be a representation of WK × SL2 (C) associated to ρ (unique up to isomorphism) (3.1.10). Then η(WF ) is bounded. (iii) Let η be as in (ii) and let Φ ∈ WK a geometric Frobenius. Then η(Φ) has only eigenvalues of absolute value 1. (3) The representation π is generic if and only if L(s, Ad ◦ ρ) has no pole at s = 1 (here Ad: GLn (C) −→ GL(Mn (C)) denotes the adjoint representation). Proof : (1): The equivalence of (i) and (ii) follows from (2.3.6) and (3.2.4), the equivalence of (ii) and (iii) is clear as any factorization through a Levi subgroup GLn1 (C) × GLn2 (C) ⊂ GLn (C) would induce a decomposition of ρ. (3): This is [Kud] 5.2.2. (2): The equivalence of (ii) and (iii) follows from the facts that the image of the inertia group IK under η is finite, as IK is compact and totally disconnected, and that a subgroup H of semisimple elements in GLn (C) is bounded if and only if every element of H has only eigenvalues of absolute value 1 (use the spectral norm). Now ρ is indecomposable if and only if η is indecomposable. To prove the equivalence of (i) and (iii) we can therefore assume by (2.3.7) and (4.3.3) that ρ is indecomposable, i.e. that π = Q(∆) is essentially square integrable. Let x ∈ IR be the unique real number such that Q(∆)(x) is square integrable (2.3.6). Then the description of rec(Q(∆)) in (4.3.3) shows that | det(η(w))| = | det(r(w))| = |w|nx .

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Now π is square integrable if and only if its central character is unitary. But by property (4) of the local Langlands classification the central character of π is given by det ◦r. Hence (iii) is equivalent to (i) by the following lemma whose proof we leave as an exercise: Lemma: Let π be a supercuspidal representation of GLn (K) and denote by ωπ its central character. For an integer m ≥ 1 let δ be the interval [π((1 − m)/2), π((m − 1)/2)]. Let η be a representation of WK × SL2 (C) associated to (r, N ) = rec(Q(∆)) (3.1.10). Then for w ∈ WK all absolute 1/n values of eigenvalues of η(w) are equal to |ωπ (Art−1 . In particular K (w))| all eigenvalues of η(w) have the same absolute value. Hint : First show the result for m = 1 where there is no difference between η and r. Then the general result can be checked by making explicit the Jacobson-Morozov theorem in the case of GLnm (K).

4.4

The onstru tion of the orresponden e after Harris and Taylor

¯ ℓ of (4.4.1) Fix a prime ℓ 6= p and an isomorphism of an algebraic closure Q Q ℓ with C. Denote by κ the residue field of OK and by κ ¯ an algebraic closure of κ. For m ≥ 0 and n ≥ 1 let ΣK,n,m be the unique (up to isomorphism) one-dimensional special formal OK -module of OK -height n with Drinfeld ¯ level pm K -structure over k. Its deformation functor on local Artinian OK algebras with residue field κ ¯ is prorepresented by a complete noetherian local OK -algebra RK,n,m with residue field κ ¯ . Drinfeld showed that RK,n,m is regular and that the canonical maps RK,n,m −→ RK,n,m+1 are finite and flat. The inductive limit (over m) of the formal vanishing cycle sheaves of ¯ ℓ gives a collection (Ψi Spf(RK,n,m ) with coefficients in Q K,ℓ,h ) of infinite¯ dimensional Q ℓ -vector spaces with an admissible action of the subgroup of GLh (K) × D × × WK consisting of elements (γ, δ, σ) such that K,1/n |Nrdδ|| det γ|−1 |Art−1 K σ| = 1. For any irreducible representation ρ of D × K,1/n set ΨiK,ℓ,n (ρ) = HomD×

K,1/n

i ). (ρ, ψK,ℓ,n

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This is an admissible (GLn (K) × WK )-module. Denote by [ΨK,ℓ,n (ρ)] the virtual representation (−1)

n−1

n−1 X

(−1)i [ΨiK,ℓ,n (ρ)].

i=0

Then the first step is to prove: Construction theorem: Let π be an irreducible supercuspidal representation of GLn (K). Then there is a (true) representation ¯ ℓ ) = GLn (C) rℓ (π): WK −→ GLn (Q such that in the Grothendieck group [ΨK,ℓ,n (JL(π)∨ )] = [π ⊗ rℓ (π)] where JL denotes the Jacquet-Langlands bijection between irreducible representations of D × K,1/n and essentially square integrable irreducible admissible representations of GLn (K). Using this theorem we can define recn = recK,n : A0n (K) −→ Gn (K) by the formula recn (π) = rℓ (π ∨ ⊗ (| |K ◦ det)(1−n)/2 ). That this map satisfies (1.2.2) (1) - (5) follows from compatibility of rℓ with many instances of the global Langlands correspondence. The proof of these compatibilities and also the proof of the construction theorem follow from an analysis of the bad reduction of certain Shimura varieties. I am not going into any details here and refer to [HT]. (4.4.2) In the rest of this treatise we explain the ingredients of the construction of the collection of maps (recn ).

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5 Explanation of the correspondence 5.1

Ja quet-Langlands theory

(5.1.1) We collect some facts about skew fields with center K (see e.g. [PR] as a reference). Let Br(K) be the Brauer group of K. As a set it can be identified with the set of isomorphism classes of finite-dimensional division algebras over K with center K. For D, D ′ ∈ Br(K), D ⊗ D ′ is again a central simple algebra over K, hence it is isomorphic to a matrix algebra Mr (D ′′ ) for some D ′′ ∈ Br(K). If we set D · D ′ := D ′′ , this defines the structure of an abelian group on Br(K). This group is isomorphic to Q/ZZ where the homomorphism Q/ZZ −→ Br(K) is given as follows: For a rational number λ with 0 ≤ λ < 1 we write λ = s/r for integers r, s which are prime to each other and with r > 0 (and we make the convention 0 = 0/1). Then the associated skew field Dλ is given by Dλ = Kr [Π] where Kr is the (unique up to isomorphism) unramified extension of K of degree r and where Π is an indeterminate satisfying the relations Πr = πks and Πa = σK (a)Π for a ∈ Kr . We call r the index of Dλ . It is the order of Dλ as an element in the Brauer group and we have dimK (Dλ ) = r 2 . If B is any simple finite-dimensional K-algebra with center K, it is isomorphic to Mr (D) for some skew field D with center K. Further, B ⊗K L is a simple L-algebra with center L for any extension L of K. In particular ¯ is isomorphic to an algebra of matrices over K ¯ as there do not B ⊗K K exist any finite-dimensional division algebras over algebraically closed fields ¯ except K ¯ itself. K Conversely, if D is a skew field with center K which is finite-dimensional ¯ is isomorover K we can associate the invariant inv(D) ∈ Q/ZZ: As D ⊗K K ¯ we have dimK (D) = r 2 . The valuation phic to some matrix algebra Mr (K), vK on K extends uniquely to D by the formula 1 vD (δ) = vK (NrdD/K (δ)) r

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for δ ∈ D. Moreover D is complete in the topology given by this valuation. It follows from the definition of vD that the ramification index e of D over K is smaller than r. Set OD = { δ ∈ D | vD (δ) ≥ 0 },

PD = { δ ∈ D | vD (δ) > 0 }.

Clearly PD is a maximal right and left ideal of OD and the quotient κD = OD /PD is a skew field which is a finite extension of κ, hence it is finite and has to be commutative. Let L ⊂ D be the unramified extension corresponding to the extension κD of κ. As no skew field with center K of dimension r 2 can contain a field of K-degree bigger than r we have for the inertia index f of D over K f = [κD : κ] = [L : K] ≤ r. Hence the formula r 2 = ef shows that e = f = r. Further we have seen that D contains a maximal unramified subfield. The extension L/K is Galois with cyclic Galois-group generated by the Frobenius automorphism σK . By the Skolem-Noether theorem (e.g. [BouA] VIII, §10.1), there exists an element δ ∈ D × such that σK (x) = δxδ −1 for all x ∈ L. Then inv(D) = vD (δ) ∈

1 ZZ/ZZ ⊂ Q/ZZ r

is the invariant of D. (5.1.2) For every D ∈ Br(K) we can consider its units as an algebraic group over K. More precisely, we define for every K-algebra R D × (R) = (D ⊗K R)× . This is an inner form of GLn,K if n is the index of D. (5.1.3) Let D ∈ Br(K) be a division algebra with center K of index n. Let {d} be a D × -conjugacy class of elements in D × . The image of ¯ ∼ ¯ is a GLn (K)-conjugacy ¯ {d} in D ⊗K K class {d}′ of elements in = Mn (K) ¯ which does not depend on the choice of the isomorphism D ⊗K K ¯ ∼ GLn (K) = ¯ ¯ Mn (K) as any automorphism of Mn (K) is an inner automorphism. Further, ¯ {d}′ is fixed by the natural action of Gal(K/K) on conjugacy classes of

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¯ GLn (K). Hence its similarity invariants in the sense of [BouA] chap. 7, §5 are polynomials in K[X] and it follows that there is a unique GLn (K)¯ is conjugacy class of elements α({d}) in GLn (K) whose image in GLn (K) ′ {d} . Altogether we get a canonical injective map α from the set of D × conjugacy classes {D × } in D × into the set of GLn (K)-conjugacy classes {GLn (K)} in GLn (K). The image of α consists of the set of conjugacy classes of elliptic elements in GLn (K). Recall that an element g ∈ GLn (K) is called elliptic if it is contained in a maximal torus T (K) of GLn (K) such that T (K)/K × is compact. Equivalently, g is elliptic if and only if K[g] is a field. We call a conjugacy class {g} in GLn (K) semisimple if it consists of ¯ or, equivalently, if K[g] is a prodelements which are diagonalizable over K uct of field extensions for g ∈ {g}. A conjugacy class {g} is called regular semisimple if it is semisimple and if all eigenvalues of elements in {g} in ¯ are pairwise different. Note that every elliptic element is semisimple. K We make the same definitions for conjugacy classes in D × , or equivalently we call a conjugacy class of D × semisimple (resp. regular semisimple) if its image under α: {D × } −→ {G} is semisimple (resp. regular semisimple). (5.1.4) Denote by A2 (G) the set of isomorphism classes of irreducible admissible essentially square integrable representations of G. We now have the following theorem which is due to Jacquet and Langlands in the case n = 2 and due to Rogawski and Deligne, Kazhdan and Vigneras in general ([Rog] and [DKV]): Theorem: Let D be a skew field with center K and with index n. There exists a bijection, called Jacquet-Langlands correspondence, JL: A2 (D × ) ↔ A2 (GLn (K)) which is characterized on characters by (5.1.4.1)

χπ = (−1)n−1 χJL(π) .

Further JL satisfies the following conditions: (1) We have equality of central characters ωπ = ωJL(π) .

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(2) We have an equality of L-functions and of ε-functions up to a sign L(π, s) = L(JL(π), s),

ε(π, ψ, s) = ε(JL(π), ψ), s)

(for the definition of L- and ε-function of irreducible admissible representations of D × see e.g. [GJ]). (3) The Jacquet-Langlands correspondence is compatible with twist by characters: If χ is a multiplicative quasi-character of K, we have JL(π(χ ◦ Nrd)) = JL(π)(χ ◦ det).

(4) It is compatible with contragredient: JL(π ∨ ) = JL(π)∨ .

(5.1.5) Remark: Note that for G = D × every admissible representation is essentially square integrable as D × /K × is compact.

5.2

Spe ial p-divisible O-modules

(5.2.1) Let R be a ring. A p-divisible group over R is an inductive system G = (Gn , in )n≥1 of finite locally free commutative group schemes Gn over Spec(R) and group scheme homomorphisms in : Gn → Gn+1 such that for all integers n there is an exact sequence in−1 ◦···◦i1

p

0 −→ G1 −−−−−−→ Gn −→ Gn−1 −→ 0 of group schemes over Spec(R). We have the obvious notion of a homomorphism of p-divisible groups. This way we get a ZZp -linear category. As the Gn are finite locally free, their underlying schemes are by definition of the form Spec(An ) where An is an R-algebra which is a finitely generated locally free R-module. In particular, it makes sense to speak of the rank of An which we also call the rank of Gn . From the exact sequence above it follows that Gn is of rank pnh for some non-negative locally constant function h: Spec(R) −→ ZZ which is called the height of G.

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(5.2.2) Let G = (Gn ) be a p-divisible group over some ring R and let R′ be an R-algebra. Then the inductive system of Gn ⊗R R′ defines a p-divisible group over R′ which we denote by GR′ . (5.2.3) Let G = (Gn ) be a p-divisible group over a ring R. If there exists some integer N ≥ 1 such that pN R = 0, the Lie algebra Lie(Gn ) is a locally free R-module for n ≥ N whose rank is independent of n ≥ N . We call this rank the dimension of G. More generally, if R is p-adically complete we define the dimension of G as the dimension of the p-divisible group GR/pR over R/pR. (5.2.4) Let R be an OK -algebra. A special p-divisible OK -module over R is a pair (G, ι) where G is a p-divisible group over R and where ι: OK −→ End(G) is a homomorphism of ZZp -algebras such that for all n ≥ 1 the OK action induced by ι on Lie(Gn ) is the same as the OK -action which is induced from the R-module structure of Lie(G) via the OK -module structure of R. In other words the induced homomorphism OK ⊗ZZp OK −→ End(Lie(Gn )) factorizes through the multiplication OK ⊗ZZp OK −→ OK . The height ht(G) of a special p-divisible OK -module (G, ι) is always divisible by [K : Q p ] and we call htOK (G) := [K : Q p ]−1 ht(G) the OK height of (G, ι). (5.2.5) If (G = (Gn ), ι) is a special p-divisible OK -module over an OK algebra R and if R −→ R′ is an R-algebra, we get an induced OK -action ι′ on GR′ and the pair (GR′ , ι′ ) is a special p-divisible OK -module over R′ which we denote by (G, ι)R′ . (5.2.6) Let k be a perfect extension of the residue class field κ of OK . Denote by W (k) the ring of Witt vectors of k. Recall that this is the unique (up to unique isomorphism inducing the identity on k) complete discrete valuation ring with residue class field k whose maximal ideal is generated by p. Further W (k) has the property that for any complete local noetherian ring R with residue field k there is a unique local homomorphism W (k) −→ R inducing the identity on k.

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In particular, we can consider W (κ). It can be identified with the ring of integers of the maximal unramified extension of Q p in K (use the universal property of the ring of Witt vectors). Set WK (k) = W (k) ⊗W (κ) OK . This is a complete discrete valuation ring of mixed characteristic with residue field k which is a formally unramified OK -algebra (i.e. the image of pK generates the maximal ideal of WK (k)). There exists a unique continuous automorphism σK of W (k) which induces the automorphism x 7→ xq on k. We denote the induced automorphism σK ⊗ idOK again by σK . (5.2.7) Proposition: The category of special p-divisible OK -modules (G, ι) over k and the category of triples (M, F, V ) where M is a free WK (k)-module of rank equal to the OK -height and F (resp. V ) is a σ- (resp. σ −1 -) linear map such that F V = V F = πK idM are equivalent. Via this equivalence there is a canonical functorial isomorphism M/V M ∼ = Lie(G). We call (M, F, V ) = M (G, ι) the Dieudonn´e module of (G, ι). Proof : To prove this we use covariant Dieudonn´e theory for p-divisible groups as in [Zi1] for example. Denote by σ the usual Frobenius of the ring of Witt vectors. Covariant Dieudonn´e theory tells us that there is an equivalence of the category of p-divisible groups over k with the category of triples (M ′ , F ′ , V ′ ) where M ′ is a free W (k)-module of rank equal to the height of G and with a σ-linear (resp. a σ −1 -linear) endomorphism F ′ (resp. V ′ ) such that F ′ V ′ = V ′ F ′ = p idM ′ and such that M ′ /V ′ M ′ = Lie(G). Let us call this functor M ′ . Let (G, ι) be a special p-divisible OK -module. Then the Dieudonn´e module M ′ (G) is a W (k) ⊗ZZp OK -module, the operators F ′ and V ′ commute with the OK -action and the induced homomorphism OK ⊗ZZp k −→ End(M ′ /V ′ M ′ ) factors through the multiplication OK ⊗ZZp k −→ k. We have to construct from these data a triple (M, F, V ) as in the claim of the proposition. To do this write Y W (k) ⊗ZZp OK = W (k) ⊗ZZp W (κ) ⊗W (κ) OK = WK (k). Gal(κ/IFp )

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By choosing the Frobenius σ = σQ p as a generator of Gal(κ/IFp ) we can identify this group with ZZ/rZZ where pr = q. We get an induced decomposition M ′ = ⊕i∈ZZ/rZZ Mi′ where the Mi are WK (k)-modules defined by Mi = { m ∈ M ′ | (a ⊗ 1)m = (1 ⊗ σ −i (a))m for all a ∈ W (κ) ⊂ OK }. The operator F ′ (resp. V ′ ) is homogeneous of degree −1 (resp. +1) with respect to this decomposition. By the condition on the OK -action on the Lie ′ ′ = Mi′ = M ′ /V ′ M ′ and hence that V Mi−1 algebra we know that M0′ /V Mr−1 for all i 6= 0. We set M = M0′ and V = (V ′ )r |M0′ . It follows that we have M/V M = M ′ /V ′ M ′ . Further the action of πK on M/V M = M ′ /V M ′ equals the scalar multiplication with the image of πK under the map OK −→ κ −→ k but this image is zero. It follows that V M contains πK M and hence we can define F = V −1 πK . Thus we constructed the triple (M, F, V ) and it is easy to see that this defines an equivalence of the category of triples (M ′ , F ′ , V ′ ) as above and the one of triples (M, F, V ) as in the claim. (5.2.8) Let (G, ι) be a special p-divisible OK -module over a ring R. We call it ´etale if it is an inductive system of finite ´etale group schemes. This is equivalent to the fact that its Lie algebra is zero. If p is invertible in R, (G, ι) will be always ´etale. Now assume that R = k is a perfect field of characteristic p and let (M, F, V ) be its Dieudonn´e module. Then (G, ι) is ´etale if and only if M = V M. In general there is a unique decomposition (M, F, V ) = (M´et , F, V ) ⊕ (Minf , F, V ) such that V is bijective on M´et and such that V N Minf ⊂ πK Minf for large N (define M´et (resp. Minf ) as the projective limit over n of S T m m n n m Ker(V |M/πK M ))). We call the WK (k)m V (M/πK M ) (resp. of rank of M´et the ´etale OK -height of (M, F, V ) or of (G, ι). We call (G, ι) formal or also infinitesimal if its ´etale OK -height is zero. (5.2.9) Proposition: Let k be an algebraically closed field of characteristic p. For all non-negative integers h ≤ n there exists up to isomorphism exactly one special p-divisible OK -module of OK -height n, ´etale OK -height h and of dimension one. Its Dieudonn´e module (M, F, V ) is the free WK (k)-module

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with basis (d1 , . . . , dh , e1 , . . . , en−h ) such that V is given by V di = di , V ei = ei+1 ,

i = 1, . . . , h i = 1, . . . , n − h − 1

V en−h = πK e1 . This determines also F by the equality F = V −1 πK . The key point to this proposition ist the following lemma due to Dieudonn´e: Lemma: Let M be a free finitely generated WK (k)-module and let V a be a σK -linear bijection where a is some integer different from zero. Then there exists a WK (k)-basis (e1 , . . . , en ) of M such that V ei = ei . A proof of this lemma in the case of K = Q p can be found in [Zi1] 6.26. The general case is proved word by word in the same way if one replaces everywhere p by πK . Proof of the Proposition: Let (G, ι) be a special p-divisible OK -module as in the proposition and let (M, F, V ) be its Dieudonn´e module. We use the decomposition (M, F, V ) = (M´et , F, V ) ⊕ (Minf , F, V ) and can apply the lemma to the ´etale part. Hence we can assume that h = 0 (note that Minf /V Minf = M/V M ). By definition of Minf , V acts nilpotent on M/πK M . We get a decreasing filtration M/πK M ⊃ V (M/πK M ) ⊃ · · · ⊃ V N (M/πK M ) = (0). The successive quotients have dimension 1 because dimk (M/V M ) = 1. Hence we see that V n M ⊂ πK M . On the other hand we have lengthWK (k) (M/V n M ) = n lengthWK (k) (M/V M ) = n = lengthWK (k) (M/πK M ) which implies V n M = πK M . Hence we can apply the lemma to the operator −1 n πK V and we get a basis of elements f satisfying V n f = πK f . Choose an element f of this basis which does not lie in V M . Then the images of ei := V i−1 f for i = 1, . . . , n in M/πK M form a basis of the k-vector space M/πK M . Hence the ei form a WK (k)-basis of M , and V acts in the desired form. (5.2.10) Definition: We denote the unique formal p-divisible OK -module

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of height h and dimension 1 over an algebraically closed field k of characteristic p by Σh,k .

′ (5.2.11) Denote by DK,1/h the ring of endomorphisms of Σh,k and set ′ DK,1/h = DK,1/h ⊗ZZ Q. Then this is “the” skew field with center K and invariant 1/h ∈ Q/ZZ (5.1.1). This follows from the following more general proposition:

Proposition: Denote by L the field of fractions of WK (k) and fix a rational number λ. We write λ = r/s with integers r and s which are prime to each other and with s > 0 (and with the convention 0 = 0/1). Denote by Nλ = (N, V ) the pair consisting of the vector space N = Ls and of the −1 σK -linear bijective map V which acts on the standard basis via the matrix

0 1 0

0

0 0 1 ...

... 0 ... 0

0 ... ... 1

r πK 0 0 . 0

Then End(Nλ ) = { f ∈ EndL (N ) | f ◦ V = V ◦ f } is the skew field Dλ with center K and invariant equal to the image of λ in Q/ZZ (cf. (5.1.1)). s

Proof : We identify IFqs with the subfield of k of elements x with xq = x. This contains the residue field κ of OK and we get inclusions OK ⊂ OKs := W (IFqs ) ⊗W (κ) OK ⊂ WK (k) and hence K ⊂ Ks ⊂ L. These extensions are unramified, [Ks : K] = s, and Ks can be described as s the fixed field of σK in L. To shorten notations we set Aλ = End(Nλ ). As Nλ does not have any non-trivial V -stable subspaces (cf. [Zi1] 6.27), Aλ is a skew field and its center contains K. For a matrix (uij ) ∈ End(Ls ) an easy explicit calculation

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shows that (uij ) ∈ Aλ if and only if we have the relations u11 ui+1,j+1 u1,j+1 ui+1,j

= = = =

−1 σK (uss ), −1 σK (uij ), r −1 πK σK (usj ), −r −1 πK σK (uis ),

1 ≤ i, j ≤ s − 1, 1 ≤ j ≤ s − 1, 1 ≤ i ≤ s − 1.

s It follows that σK (uij ) = uij for all i, j, and hence uij ∈ Ks . Further, sending a matrix (uij ) ∈ Aλ to its first column defines a Ks -linear isomorphism Aλ ∼ = Kss , hence dimKs (Aλ ) = s. The Ks -algebra homomorphism

ϕ: Ks ⊗K Aλ −→ Ms (Ks ),

α ⊗ x 7→ αx

is a homomorphism of Ms (Ks )-left modules and hence it is surjective as the identity matrix is in its image. Therefore ϕ is bijective. In particular, Ks is the center of Ks ⊗K Aλ and hence the center of Aλ is equal to K. Now define r 0 0 . . . 0 πK 1 0 ... 0 0 . . . 0 ∈ Aλ . Π = 0 1 ... 0

...

0

1

0

r Then we have the relations Πs = πK and Πd = σK (d)Π for d ∈ Aλ . We get an embedding Dλ = Ks [Π] ֒→ Aλ by

Π 7→ Π

Ks ∋ α 7→

−1 σK (α) −2 σK (α)

··· α

∈ Aλ ⊂ Ms (Ks )

which has to be an isomorphism because both sides have the same Kdimension. (5.2.12) Over a complete local noetherian ring R with perfect residue field k we have the following alternative description of a special formal p-divisible

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OK -module due to Zink [Zi2]. For this we need a more general definition of the Witt ring. Let R be an arbitrary commutative ring with 1. The Witt ring W (R) is characterized by the following properties: (a) As a set it is given by RIIN0 , i.e. elements of W (R) can be written as infinite tuples (x0 , x1 , . . . , xi , . . .). (b) If we associate to each ring R the ring W (R) and to each homomorphism of rings α: R −→ R′ the map W (α): (x0 , x1 , . . .) 7→ (α(x0 ), α(x1 ), . . .), then we obtain a functor from the category of rings into the category of rings. (c) For all integers n ≥ 0 the so called Witt polynomials wn : W (R) −→ R n

(x0 , x1 , . . .) 7→ xp0 + px1p

n−1

+ . . . + pn xn

are ring homomorphisms. For the existence of such a ring see e.g. [BouAC] chap. IX, §1. If we endow the product RIIN0 with the usual ring structure the map x 7→ (w0 (x), w1 (x), . . .) defines a homomorphism of rings W∗ : W (R) −→ RIIN0 . The ring W (R) is endowed with two operators τ and σ which are characterized by the property that they are functorial in R and that they make the following diagrams commutative W (R) W∗ y RIIN0

W (R) W∗ y RIIN0

τ

−−−−−→ x7→(0,px0 ,px1 ,...)

−−−−−−−−−−→ σ

−−−−−→ x7→(x1 ,x2 ,...)

−−−−−−−→

W (R) W y ∗ RIIN0 ,

W (R) W y ∗ RIIN0 .

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The operator τ can be written explicitly by τ (x0 , x1 , . . .) = (0, x0 , x1 , . . .) and it is called Verschiebung of W (R). It is an endomorphism of the additive group of W (R). If R is of characteristic p (i.e. pR = 0), σ can be described as (x0 , x1 , . . .) 7→ (xp0 , xp1 , . . .). For an arbitrary ring, σ is a ring endomorphism and it is called Frobenius of W (R). There are the following relations for σ and τ : (i) σ ◦ τ = p · idW (R) , (ii) τ (xσ(y)) = τ (x)y for x, y ∈ W (R), (iii) τ (x)τ (y) = pτ (xy) for x, y ∈ W (R), (iv) τ (σ(x)) = τ (1)x for x ∈ W (R), and we have τ (1) = p if R is of characteristic p. We have a surjective homomorphism of rings w0 : W (R) −→ R,

(x0 , x1 , . . .) 7→ x0 ,

n and we denote its kernel τ (W (R)) by IR . We have IR = τ n (W (R)) and W (R) is complete with respect to the IR -adic topology. If R is a local ring with maximal ideal m, W (R) is local as well with maximal ideal { (x0 , x1 , . . .) ∈ W (R) | x0 ∈ m }.

(5.2.13) Now we can use Zink’s theory of displays to give a description of special formal p-divisible groups in terms of semi-linear algebra. Let R be a complete local noetherian OK -algebra with perfect residue field k. We extend σ and τ to W (R) ⊗ZZp OK in an OK -linear way. Then we get using [Zi2]: Proposition: The category of special formal p-divisible OK -modules of height h over R is equivalent to the category of tuples (P, Q, F, V −1 ) where • P is a finitely generated W (R) ⊗ZZp OK -module which is free of rank h over W (R), • Q ⊂ P is a W (R) ⊗ZZp OK -submodule which contains IR P , and the quotient P/Q is a direct summand of the R-module P/IR P such that the induced action of R ⊗ZZp OK on P/Q factorizes through the multiplication R ⊗ OK −→ R, • F : P −→ P is a σ-linear map,

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• V −1 : Q −→ P is a σ-linear map whose image generates P as a W (R)module, satisfying the following two conditions: (a) For all m ∈ P and x ∈ W (R) we have the relation V −1 (τ (x)m) = xF (m). (b) The unique W (R) ⊗ZZp OK -linear map V # : P −→ W (R) ⊗σ,W (R) P satisfying the equations V # (xF m) = p · x ⊗ m V # (xV −1 n) = x ⊗ n for x ∈ W (R), m ∈ P and n ∈ Q is topologically nilpotent, i.e. the homomorphism V N # : P −→ W (R) ⊗σN ,W (R) P is zero modulo IR + pW (R) for N sufficiently large. (5.2.14) To define the notion of a Drinfeld level structure we need the following definition: Let R be a ring and let X = Spec(A) where A is finite locally free over R of rank N ≥ 1. For any R-algebra R′ we denote by X(R′ ) the set of R-algebra homomorphisms A −→ R′ (or equivalently of all R′ -algebra homomorphisms A ⊗R R′ −→ R′ ). The multiplication with an element f ∈ A ⊗R R′ defines an R′ -linear endomorphism of A ⊗R R′ . As A is finite locally free we can speak of the determinant of this endomorphism which is an element Norm(f ) in R′ . We call a finite family of elements ϕ1 , . . . , ϕN ∈ X(R′ ) a full set of sections of X over R′ if we have for every R′ -algebra T and for all f ∈ A⊗R T an equality in T N Y ϕi (f ). Norm(f ) = i=1

(5.2.15) Let R be an OK -algebra and let G be a special p-divisible OK module over R. We assume that its OK -height h is constant on Spec(R),

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e.g. if R is a local ring (the only case which will be used in the sequel). The m OK -action on G defines for every integer m ≥ 1 the multiplication with πK m [πK ]: G −→ G.

This is an endomorphism of p-divisible groups whose kernel is a finite locally m free group scheme G[πK ] over Spec(R) of rank q mh . ′ Let R′ be an R-algebra. A Drinfeld pm K -structure on G over R is a homomorphism of OK -modules m α: (p−m /OK )h −→ G[πK ](R′ ) h such that the finite set of α(x) for x ∈ (p−m K /OK ) forms a full set of sections.

(5.2.16) It follows from the definition (5.2.14) that if α: (p−m /OK )h −→ ′ ′ m G[πK ](R′ ) is a Drinfeld pm K -structure over R then for any R -algebra T the composition m m αT : (p−m /OK )h −→ G[πK ](R′ ) −→ G[πK ](T ), α

where the second arrow is the canonical one induced by functoriality from R′ −→ T , is again a Drinfeld pm K -structure. (5.2.17) Being a Drinfeld pm K -structure is obviously a closed property. More −m h m precisely: Let α: (p /OK ) −→ G[πK ](R′ ) be a homomorphism of abelian groups. Then there exists a (necessarily unique) finitely generated ideal a ⊂ R′ such that a homomorphism of OK -algebras R′ −→ T factorizes over R′ /a if and only if the composition αT of α with the canonical homomorphism m m G[πK ](R′ ) −→ G[πK ](T ) is a Drinfeld pm K -structure over T . It follows that for every special formal p-divisible OK -module (G, ι) over some OK -algebra R the functor on R-algebras which associates to each Ralgebra R′ the set of Drinfeld pm K -structures on (G, ι)R′ is representable by an R-algebra DLm (G, ι) which is of finite presentation as R-module. Obviously DL0 (G, ι) = R. m (5.2.18) Let α: (p−m /OK )h −→ G[πK ](R′ ) be a Drinfeld pm K -structure over ′ ′ R . As α is OK -linear, it induces for all m ≤ m a homomorphism ′

′

′

m m ](R′ ). α[πK ]: (p−m /OK )h −→ G[πK

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Proposition: This is a Drinfeld pm K -structure. ′

For the proof of this non-trivial fact we refer to [HT] 3.2 (the hypothesis in loc. cit. that Spec(R′ ) is noetherian with a dense set of points with residue field algebraic over κ is superfluous as we can always reduce to this case by [EGA] IV, §8 and (5.2.17)). If R′ is a complete local noetherian ring with perfect residue class field (this is the only case which we will use in the sequel) the proposition follows from the fact that we can represent (G, ι) by a formal group law and that in this case a Drinfeld level structure as defined above is the same as a Drinfeld level structure in the sense of [Dr1]. (5.2.19) Let (G, ι) be a special formal p-divisible OK -module over an OK algebra R. By (5.2.18) we get for non-negative integers m ≥ m′ canonical homomorphisms of R-algebras DLm′ (G, ι) −→ DLm (G, ι). It follows from [Dr1] 4.3 that these homomorphisms make DLm (G, ι) into a finite locally free module over DLm′ (G, ι). (5.2.20) Example: If R is an OK -algebra of characteristic p and if G is a special formal p-divisible OK -module of OK -height h and of dimension 1, then the trivial homomorphism h m αtriv : (p−m K /OK ) −→ G[pK ],

x 7→ 0

is a Drinfeld pm K -structure. If R is reduced, this is the only one.

5.3

Deformation of p-divisible O-modules

(5.3.1) In this paragraph we fix an algebraically closed field k of characteristic p together with a homomorphism OK −→ k. Further we fix integers h ≥ 1 and m ≥ 0. By (5.2.9) and by (5.2.20) there exists up to isomorphism only one special formal p-divisible OK -module Σh of height h and dimension triv 1 with Drinfeld pm over k. We denote the pair (Σh , αtriv ) by K -structure α Σh,m .

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Let C be the category of pairs (R, s) where R is a complete local noetherian OK -algebra and where s is an isomorphism of the residue class field of R with k. The morphisms in C are local homomorphisms of OK -algebras inducing the identity on k. (5.3.2) Definition: Let (R, s) ∈ C be a complete local noetherian OK algebra. A triple (G, α, ϕ) consisting of a formal special p-divisible OK module G over R, of a Drinfeld pm K -structure α of G over R and of an isomorphism ∼

ϕ: Σh,m −→ (G ⊗R k, αk ) is called a deformation of Σh,m over R. ˜ h,m , ϕ) consisting of a complete local noetherian ring A triple (Rh,m , Σ ˜ h,m , ϕ) of Σh,m is called Rh,m with residue field k and of a deformation (Σ universal deformation of Σh,m if for every deformation (G, α, ϕ) over some ˜ h,m , ϕ)R R ∈ C there exists a unique morphism Rh,m −→ R in C such that (Σ is isomorphic to (G, α, ϕ). A universal deformation is unique up to unique isomorphism if it exists. (5.3.3) Proposition: We keep the notations of (5.3.2). ˜ h,m , ϕ) of Σh,m exists. (1) A universal deformation (Rh,m , Σ (2) For m = 0 the complete local noetherian ring Rh,0 is isomorphic to the power series ring WK (k)[[t1 , . . . , th−1 ]]. (3) For m ≥ m′ the canonical homomorphisms Rh,m′ −→ Rh,m are finite flat. The rank of the free Rh,0 -module Rh,m is #GLh (OK /pm k ). (4) The ring Rh,m is regular for all m ≥ 0. Proof : Assertion (1) follows from a criterion of Schlessinger [Sch] using rigidity for p-divisible groups (e.g. [Zi1]) and the fact that the canonical functor from the category of special p-divisible OK -modules over Spf(Rh,m ) to the category of special p-divisible OK -modules over Spec(Rh,m ) is an equivalence of categories (cf. [Me] II, 4). The second assertion follows easily from general deformation theory of p-divisible groups (for an explicit description of the universal deformation and a proof purely in terms of linear algebra one can use [Zi2] and (5.2.13)). Finally, (3) and (4) are more involved (see [Dr1] §4, note that (3) is essentially equivalent to (5.2.19)).

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(5.3.4) Let D1/h be “the” skew field with center K and invariant 1/h. The ring Rh,m has a continuous action of the ring of units O × D1/h of the integral ˜ closure OD of OK in D1/h : Let Σg,m = (G, α, ϕ) be the universal special 1/h

formal p-divisible OK -module with Drinfeld pm K -structure over Rh,m . For × δ ∈ O D1/h the composition ϕ

δ

Σh,m −→ Σh,m −→ (G, α) ⊗Rh,m k is again an isomorphism if we consider δ as an automorphism of Σh,m (which is the same as an automorphism of Σh,0 ) by (5.2.11). Therefore (G, α, ϕ ◦ δ) is a deformation of Σh,m over Rh,m and by the definition of a universal deformation this defines a continuous automorphism δ: Rh,m −→ Rh,m . (5.3.5) Similarly as in (5.3.4) we also get a continuous action of GLh (OK /pm K) ˜ on Rh,m : Again let Σg,m = (G, α, ϕ) be the universal special formal pm divisible OK -module with Drinfeld pm K -structure over Rh,m . For γ ∈ GLh (OK /pK ), m α ◦ γ is again a Drinfeld pK -structure, hence (G, α ◦ γ, ϕ) is a deformation of Σh,m and defines a continuous homomorphism γ: Rh,m −→ Rh,m .

(5.3.6) By combining (5.3.4) and (5.3.5) we get a continuous left action of × −→ GLh (OK /pm GLh (OK ) × O × K ) × O D1/h D1/h −→

on Rh,m . Now we have the following lemma ([HT] p. 52) Lemma: This action can be extended to a continuous left action of GLh (K) × D × 1/h on the direct system of the Rh,m such that for m2 >> m1 and for (γ, δ) ∈ GLh (K) × D × the diagram 1/h Rh,m x 1

W (k)

commutes.

(γ,δ)

−−−−−→ v

σKK

(det(γ))−vK (Nrd(δ))

−−−−−−−−−−−−−→

Rh,m x 2

W (k)

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5.4

309

Vanishing y les

(5.4.1) Let W be a complete discrete valuation ring with maximal ideal (π), residue field k and field of fractions L. Assume that k is algebraically closed (or more generally separably closed). The example we will use later on is the ring W = WK (k) for an algebraically closed field k of characteristic ¯ and s = Spec(k). p. Set η = Spec(L), η¯ = Spec(L) We will first define vanishing and nearby cycles for an algebraic situation (cf. [SGA 7] exp. I, XIII). Then we will generalize to the situation of formal schemes. (5.4.2) Let f : X −→ Spec(W ) be a scheme of finite type over W and define Xη¯ and Xs by cartesian diagrams X s fs y s

i

−→ −→

¯ j

X ←− fy Spec(W ) ← η ←

X η¯ f y η¯ η¯.

The formalism of vanishing cycles is used to relate the cohomology of Xs and of Xη¯ together the action of the inertia group on the cohomology of Xη¯. Fix a prime ℓ different from the characteristic p of k and let Λ be a finite abelian group which is annihilated by a power of ℓ. For all integers n ≥ 0 the sheaf Ψn (Λ) = i∗ Rn ¯j∗ Λ is called the sheaf of vanishing cycles of X over W . Via functoriality it ¯ ¯ carries an action of Gal(L/L). Note that by hypothesis Gal(L/L) equals the inertia group of L. (5.4.3) If f : X −→ Spec(W ) is proper, the functor i∗ induces an isomorphism (proper base change) ∼

i∗ : H p (X, Rq ¯jΛ) −→ H p (Xs , i∗ Rq ¯j∗ Λ) and the Leray spectral sequence for ¯j can be written as H p (Xs , Ψn (Λ)) ⇒ H p+q (Xη¯, Λ).

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¯ This spectral sequence is Gal(L/L)-equivariant. This explains, why the vanishing cycles “measure” the difference of the cohomology of the special and the generic fibre. (5.4.4) Let Λ be as above. If X is proper and smooth over W , there is no difference between the cohomology of the generic and the special fibre. More precisely, we have ([SGA 4] XV, 2.1): Proposition: Let f : X −→ W be a smooth and proper morphism. Then we have for all n ≥ 0 a canonical isomorphism ∼

H n (Xη¯, Λ) −→ H n (Xs , Λ). ¯ This isomorphism is Gal(L/L)-equivariant where the action on the rightn hand side is trivial. Further ψ (Λ) = 0 for n ≥ 1 and ψ0 = Λ. (5.4.5) Now we define vanishing cycles for formal schemes: We keep the notations of (5.4.1). We call a topological W -algebra A special, if there exists an ideal a ⊂ A (called an ideal of definition of A) such that A is complete with respect to the a-adic topology and such that A/an is a finitely generated W -algebra for all n ≥ 1 (in fact the same condition for n = 2 is sufficient ([Ber2] 1.2)). Equivalently, A is topologically W -isomorphic to a quotient of the topological W -algebra W {T1 , . . . , Tm }[[S1 , . . . , Sn ]] = W [[S1 , . . . , Sn ]]{T1 , . . . , Tm }, in particular A is again noetherian. Here if R is a topological ring, R{T1 , . . ., Tm } denotes the subring of power series X cn T n n∈IINn 0

in R[[T1 , . . . , TM ]] such that for every neighborhood V of 0 in R there is only a finite number of coefficients cn not belonging to V . If R is complete Hausdorff with respect to the a-adic topology for an ideal a, we have a canonical isomorphism ∼

R{T1 , . . . , Tm } −→ lim(R/an )[T1 , . . . , Tm ]. ←− n

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311

Note that all the rings Rh,m defined in (5.3.3) are special WK (k)algebras. (5.4.6) Let A be a special W -algebra and let X = Spf(A). Denote by X rig its associated rigid space over L. It can be constructed as follows: For A = W {T1 , . . . , Tm }[[S1 , . . . , Sn ]] we have X rig = E m × D n where E m and D n are the closed resp. open polydiscs of radius 1 with center at zero in Lm resp. in Ln . If now A is some quotient of the special W -algebra A′ = W {T1 , . . . , Tm }[[S1 , . . . , Sn ]] with kernel a′ ⊂ A′ , X rig is the closed rigid analytic subspace of Spf(A′ )rig defined by the sheaf of ideals a′ OSpf(A′ )rig . This defines a functor from the category of special W -algebras to the category of rigid-analytic spaces over L (see [Ber2] §1 for the precise definition). (5.4.7) Let A be a special W -algebra and let a ⊂ A be its largest ideal of definition. A topological A-algebra B which is special as W -algebra (or shorter a special A-algebra - an abuse of language which is justified by [Ber2] 1.1) is called ´etale over A if B is topologically finitely generated as A-algebra (which is equivalent to the fact that for every ideal of definition a′ of A, a′ B is an ideal of definition of B) and if the morphism of commutative rings A/a −→ B/aB is ´etale in the usual sense. The assignment B 7→ B/aB defines a functor from the category of ´etale special A-algebras to the category of ´etale A/a-algebras which is an equivalence of categories and hence we get an equivalence of ´etale sites (A)´et ∼ (A/a)´et . Note that for every ideal of definition a′ of A the ´etale sites (A/a)´et and (A/a′ )´et coincide. We just chose the largest ideal of definition to fix notations. On the other hand, if we write X = Spf(A) and Y = Spf(B) for an ´etale special A-algebra B we get a (quasi-)´etale morphism of rigid analytic spaces Y rig −→ X rig .

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By combining this functor with a quasi-inverse of B 7→ B/aB we get a morphism of ´etale sites ∼ ¯ ´et −→ (X rig )´et −→ (X )´et −→ s: (X rig ⊗L L) (Xred )´et

with Xred = Spec(A/a). Let Λ be a finite abelian group which is annihilated by a power of ℓ. For n ≥ 0 the sheaves ψn (Λ) := Rn s∗ Λ are called vanishing cycle sheaves. (5.4.8) Let A be a special W -algebra with largest ideal of definition a, X = Spf(A). Then the group AutW (X ) of automorphisms of X over W (i.e. of continuous W -algebra automorphisms A → A) acts on ψn (Λ). Further we have the following result of Berkovich [Ber2]: Proposition: Assume that ψi (ZZ/ℓZZ) is constructible for all i. Then there exists an integer n ≥ 1 with the following property: Every element g ∈ AutW (X ) whose image in AutW (A/an ) is the identity acts trivially on ψi (ZZ/ℓm ZZ) for all integers i, m ≥ 0.

5.5

Vanishing y les on the universal deformation of spe ial p-divisible O-modules

(5.5.1) Let k be an algebraic closure of the residue field κ of OK . Then ˆ nr , the completion of the maximal W = WK (k) is the ring of integers of K unramified extension of K. Further denote by IK the inertia group and by WK the Weil group of K. (5.5.2) Consider the system P of special formal schemes . . . −→ Spf(Rm,h ) −→ Spf(Rm−1,h ) −→ . . . −→ Spf(R0,h ). By applying the functor ( )rig we get a system P rig of rigid spaces . . . −→ Spf(Rm,h )rig −→ Spf(Rm−1,h )rig −→ . . . −→ Spf(R0,h )rig .

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313

these systems P and P rig have an action by the group GLh (OK ) × O × D1/h (5.3.4) and (5.3.5). Denote by Ψim (Λ) the vanishing cycle sheaf for Spf(Rh,m ) with coefficients in some finite abelian ℓ-primary group Λ and set ¯ ℓ. Ψim = (lim Ψim (ZZ/ℓn ZZ)) ⊗ZZℓ Q ←− n

Note that we have ˆ¯ ZZ ), lim Ψim (ZZ/ℓn ZZ) = H i ((Spf Rm )rig ⊗Kˆ nr K, ℓ ←− n

ˆ¯ K ˆ nr ). Furin particular these ZZℓ -modules carry an action by IK = Gal(K/ ther write Ψi = lim Ψim . −→ m

¯ℓ ∼ Via our chosen identification Q = C we can consider Ψim and Ψi as C-vector spaces which carry an action of GLh (OK ) × O × D1/h × IK .

(5.5.3) Lemma: We have the following properties of the (GLh (OK ) × i i O× D1/h × IK )-modules Ψm and Ψ . (1) (2) (3) (4)

The Ψim are finite-dimensional C-vector spaces. We have Ψim = Ψi = 0 for all m ≥ 0 and for all i > h − 1. The action of GLh (OK ) on Ψi is admissible. i The action of O × D1/h on Ψ is smooth.

(5) The action of IK on Ψi is continuous. Proof : For the proof we refer to [HT] 3.6. We only remark that (3) – (5) follow from general results of Berkovich [Ber2] and [Ber3] if we know (1). To show (1) one uses the fact that the system of formal schemes P comes from an inverse system of proper schemes of finite type over W (cf. the introduction) and a comparison theorem of Berkovich which relates the vanishing cycles of a scheme of finite type over W with the vanishing cycle sheaves for the associated formal scheme.

314

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(5.5.4) Let Ah be the group of elements (γ, δ, σ) ∈ GLh (K) × D × 1/h × WK such that vK (det(γ)) = vK (Nrd(δ)) + vK (Art−1 K (σ)). The action of GLh (K) × D × 1/h on the system (Rh,m )m (5.3.6) gives rise to i an action of AK on Ψ . Moreover, if (ρ, Vρ ) is an irreducible admissible representation of D × 1/h over C (and hence necessarily finite-dimensional (2.1.20)) then we set Ψi (ρ) = HomO× (ρ, Ψi ). 1/h

This becomes naturally an admissible GLh (K) × WK -module if we define for φ ∈ Ψi (ρ) and for x ∈ Vρ ((γ, σ)φ)(x) = (γ, δ, σ)φ(ρ(δ)−1 x) −1 where δ ∈ D × 1/h is some element with vK (Nrd(δ)) = vK (det(γ))−vK (ArtK (σ)).

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