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Hom c . (
*
V, V' ).
By general functorial results it follows that is a right exact functor and rp ' a left exact functor. V ' !I'G' • (b) e V !I'c V' = rp1 V' V' !I'G' · In order to get a counterexample to Frobenius reciprocity, it is enough to find a smooth representation V') of G' such that V'oo # V' and to consider the group G= since in this case. Consider the left translations acting on =
125
REPRESENTATIONS OF pADIC GROUPS
.Yt'(G'). The identity map is a generalized vector in .Yt'(G')  oo and is not of the form */0 for a fixed fo in £'( G') unless G' is discrete. Hence £'( G') # .Yt'(G')oo in >+ this case. G" (c) G'
ff 0, let A(e) consist of the elements in A such that l a ) l F � e for every root a in L1\8. Moreover, let P be the parabolic subgroup of G opposite to P. The roots associated to P are the roots  a where a runs over the roots associated to P. Let fit be the unipotent radical of P. Let now be any admissible finitely generated representation of G. Let (it, V) be the representation contragredient to Using first Jacquet's functor we get representations and of M.
(n, V)
(nN, VN) (7rfl> VR) (n, V) .
REPRESENTATIONS OF pADIC GROUPS
129
The following theorem is due to Casselman [17] . It plays a crucial role in the representation theory of reductive padic groups.
2.3. Therewithexithsetfoll s a ouniwinqguepropert Minvariy : ant nondegenerate pairing < ·, · )N between V Nand in with canoni thereGivexien svtsina Vrealandnumber such thcatal images u in VN and in respectively, holds for every a in (ii:N, (1CN, VN). any irreducible admissible representation of G. The followi(7C,ng V)assertis absolut ioLetns are(1C,elyequiV)cuspivbealent: dal. P G of G, with unipotent radical we have For every paraboli c subgroup VN = Z Z(G) G. lO Z £\(Gt Jf'/G) G. (a)fbel f beloongsngs ttoo Yl'x;.:;(Gt; (G) and THEOREM
f1fl
v
V,
c
>
u
0
f1fl
(4)
A(c).
As a matter of fact, the previous pairing identifies f1R) to the representation of M contragredient to The following criterion is due to Jacquet [32] and results easily from Theorem
2.3.
THEOREM 2.4.
(1)
i=
(2)
0.
N,
Note that 'absolutely cuspidal' is with reference to the maximal split torus in the center of An alternate formulation of Theorem 2.4 is as follows. Choose a character X of and recall that is the subspace of generated by the coefficients of the irreducible absolutely cuspidal xrepresentations of Then the following condi tions are equivalent : (b)
Jlt'
(5)
holdsfor g in G and every subgroup as in Theorem G/Z. Letpair(1C,(P,V) bewianyth associ irreduciatebdleLeviadmidecomposi ssible represent at=ion of· G.andTherean eirxreduci ist a paraboli c t i o n P phibyc ble absolutatieolyn ocuspi dali\represent a1tiiosnth(i\e,extW)ensiofon ofsuchi\ tothPat=1CM·is isomor ti\o1(mn) a subrepresent f lnd� 1 where i\ gi v en ( = i\(m)). G M (P,G G. N 2.4. The next result is again due to Jacquet [32] . It is easily proved by induction on the split rank of THEOREM 2.5.
A)
M N
M
N
In principle, the classification problem for irreducible admissible representations is split into two problems : (a) Find all irreducible absolutely cuspidal representations for and for the groups which occur as centralizer of A for some parabolic pair A) in of
10 I t is well known that Z(G)/Z i s a compact group. I t makes no essential difference to take Z o r Z(G) a s central subgroup. The choice o f Z is quite convenient however.
P. CARTIER
130
il1
(b) Study the decomposition of the induced representations lnd� as above, in particular look for irreducibility criteria. Needless to say, a general answer to these problems is not yet in sight. The con struction of some absolutely cuspidal representations has been given by Shintani [40] for GLn(F) and in more general cases by Gerardin in his thesis [21]. The idea is to induce from some compact open subgroups. As to problem (b), let us mention the notion of Two parabolic pairs (P, A) and (P ', A') are associated iff A and A' are conjugate by some element in G. Let f/(P, A) be the set of irreducible admissible representa tions of G which occur in a composition series of some induced representation Ind� il1 where il is an irreducible absolutely cuspidal representation of M and M is the centralizer of A in G. If (P, A) and (P', A') are associated, then f/(P, A) = f/(P', A'). 2.4. A Take for instance the group G = GLn(F). For any (ordered) partition = of let Pn�o ... , n, consist of the matrices in block + ··· + form = (Gk1)1�k. t� r where Gk1 is an nk x matrix and Gk1 = if > I. Let for =1 I An�o ... . n, be the set of matrices which in block form are such that Gk1 and Gkk is a scalar matrix · lnk · For Mn �o · · ·, n, take the matrices in diagonal block form, i.e., Gk1 = for =1 I and let finally Nn �o ... , n, be the subgroup of Pn�o .. . n, defined by the conditions G11 = lnl · · · , G, = In, · Then (Pn�o .... n,• An�. .... n) is a para ' bolic pair, with associated Levi decomposition Pn�o ···, n, = Mn�o ···, n, · Nn�o ···, n, · Up to conjugation, the pairs (Pn 1 ; , n,• An�o · ·· , n) comprise all parabolic pairs in GLn (F). The pairs corresponding to partitions = n1 + · · · + and = + ··· + are associated iff = and · · ·, is a permutation of · · ·, Notice that the group Mn�o ···, n, is isomorphic to GLn1(F) x · · · x GLn,(F) . The wellknown operation a o {3 o n characters o f the various finite groups GLn{q ), introduced by Green [24], may now be generalized to the case of a local field. Namely, let = + be an admissible representation of GLn,(F) let (n ' , " and (n , an admissible representation of GLnn(F). Define a representation of Pn ' , nn acting on the space by
associated parabolic subgroups.
g
nnexampln1 e.
n, n, 0 ka
n1
0 0k k =
.
.•.
r s nl > n, n
V")n n' n", V' ® V"V') il1 ( � g��) G
=
n'(Gu)
n, ml >n mm1 •.
n
"
m. il1
(G ) .
® zz
By induction from Pn ' , nn to GLn{F) one gets a representation n ' o n " of GLn(F). This product is commutative and associative. In particular, given characters a1 o · · · , an of px one gets a representation a1 o • • • o an of GLn(F) ; these representations com prise the Given a partition = + · · · + the associated to the parabolic group Pn�o · · ·, n, consists of the representations of the form n 1 o • • • o 12: , where ni is an irreducible absolutely cuspidal representation of 1 , · , Finally the is the family of irreducible GLn/F) for absolutely cuspidal representations of GLn(F). From the general results summarized in §2.3, one infers that any irreducible admissible representation of G Ln(F) is contained in some representation of the form = n1 + · · · + n o • • • o 12:, where and ni belongs to the discrete series of GLn/F). The question of the irreducibility of the representations n 1 o • • • o 12:, has now been completely settled by Bernshtein and Zhelevinski [3].
principal series. j r. =
1
n
n n1 n, intermediate series discrete series
· ·
n,
131
REPRESENTATIONS OF pADIC GROUPS
Squareintegrable representations. G. rff (G, (n, V)2 G n(gz) = x(z) n(g) z g G, S j (u \ n(g) v) \ 2 dg (ui v) V. l 1 j (u \ n(gz)·v) \ 2 = j (u \ n(g)·v) \ 2 ; (7) d(n) li uu l 2 l v l v.2 d(n) 0, formal degree ofn, squareintegrable. G (n, V) G, (n: nGiven and as above, one gets d(n) (n: and in particular (n: for every squareintegrable irreducible representation n of G. (n, V) rff2(G, VK (n: V n( K). Voo = U K VVK G. V (n00, V G.n(G). oo Jlt'(G), n(f) = Vfcf(g)n( g) dg V K ntegrable irreduci f) = (n(8",f))a J't'(G,ble representation hasEMa character every squarei distributiTheonfundament on G. al estimate. (P, G. N G; P
2. 5 . For the results expounded in this section, see BarishChandra and van Dijk [28, especially part 1] . Let again Z be the maximal split torus contained in the center of Fix a unitary character X of Z. We let x) denote the (equivalence classes of) irreducible unitary representations of which satisfy the following two conditions :
for
(6)
(7)
GI Z
Here
·
E
E
Z,
the is independent from and These representations are called as usual Fix now a compact open subgroup K of and a class b of irreducible continuous representations of K. Every representation of class b acts on a space of finite dimen sion, to be denoted by deg b. Moreover, for any unitary representation of denote by b) the multiplicity of b in the restriction of to K. The following theorem is easy to prove (see [28, p. 6] ) . K, b
THEOREM 2. 6.
(8)
�
1r E C2 (G,
(9 )
X)
X
b) � deg b/meas(K/(Z n K))
deg b b) � d(n) · meas(K/(Z n K))
Let be in x). If b is the unit representation of K, the integer b) is the dimension of the space of vectors in invariant under Let where K runs over the compact open subgroups of It is easy to check that is dense in and stable under By the previous theorem we get an admissible xrepresentation 00) of Moreover for any function/ in the operator in the Hilbert space has a finitedimensional . range (contained in if / belongs to Tr K)) , hence a trace To sum up, 2.6 . Fix a parabolic pair A) where P is a minimal parabolic subgroup of hence A is a maximal split torus in Let be the uni potent radical of and N the unipotent radical of the parabolic subgroup P op11
The scalar product (u J v) is assumed to be linear in the second argument
v.
P. CARTIER
1 32
posite to P. Hence there exists a basis L1 of the root system of G w.r. t. A such that N (resp. is associated with the roots a with positive (resp. negative) coefficients when expressed in terms of LJ. Let be the set of elements in such that i a ) i F � I for every root a in LJ. According to Bruhat and Tits [13], there exist a compact subgroup of A and a finitely generated semigroup in such that Moreover there exist finitely many elements , in G and a compact open subgroup of G such that G ('Cartan decomposition'). Let be a compact open subgroup of G. We make the following assumptions : (a) (b) n n N)
N) A
a A (a L S A A=ZLS. gio K0 gm K0 =K U 1;;;;;,;m K0 SZg, Koneis hasinvariK a=nt (Kin K0 ;P)·(K and a1(K N)a K N, a(K P)a1 K P for every a in K P). g aga1 K K. openthsubgroup Kmofple module is as above. Then tK)hereeitexiherstiss aitificonstniteadntimensi=onalThe(overcompact K) C) orsuch at every si over else has a dimension bounded by Z Kt K. (n, V) VK K); VK There[ is an i!nteger p such that the higher commutator ft . · · · . = ·· vanishesfor arbitrary elementsf�o in Kt. Properties of unitary representations. Letrepresent K be aaticompact openTheresubgroup ofconstandant be a classK,of irredu cisuchble tconti n uous o ns of exi s t s a hat (n: for every irreducible unitary representation V) of KK. 12 · · ·
c
n
c
n
n
n
s (hence the inner automorphism f> of G expands n N and contracts n It is known that every neighborhood of the unit element in G contains such a subgroup The following estimate is due to Bernshtein [1]. THEOREM 2.7. N
N(G,
>
G
0
Yf'(G,
N.
be the subalgebra of Yf'x (Gt Let X be a unitary character of and Yf'.jG, consisting of the functions invariant under right and left translation by an element of If is an irreducible absolutely cuspidal x representation of G, then � N by Theorem 2.7. By a well is a simple module over Yf'(G, hence dim known argument due to Godement [23], one infers from this bound the following corollary : CoROLLARY 2.2.
�2
� sgn{o)/.. m
p]
( 1 0)
aESp
· · ·
,
Jp
·
1. for =f. in V and
X·
( 1 1)
0. preuni t a ry tll(v, v) 0 v 0 tll( g · v, g v') tll(v, ' n( )
n( ) ·
=
v)
for in V and g in G. We can then complete V to a Hilbert space V and extend by continuity n(g) to a unitary operator fr(g) in V. Then (fr, V) is a unitary representa tion of G. If (n, V) is admissible, V is exactly the set of smooth vectors in V, that is V U K VK where K runs over the compact open subgroups in G. It follows from Theorem 2.8 and these remarks that G G.
v, v'
=
theofclathssifie preuni cationtaoryf therepreir b l e uni t a ry represent a t i o ns o f amount s t o t h e search reduci sentations among the irreducible admissible representations of 3.1. Preliminaries about tori. multiplicative group in one variable, k k. k,
. III. Unramified principal series of representations. For this section only, we denote by a (com mutative) infinite field. Let Gm be the considered as an algebraic group defined over To a connected algebraic group H defined over we associate two finitely generated free Zmodules, namely X * (H) = Homz(X * (H), Z).
In these formulas, Hom kgr (resp. Homz) means the group of homomorphisms of algebraic groups defined over (resp. of Zmodules). We denote by (q>, .1.) (for
k
P. CARTIER
134
in X* (H) and it in X*(H)) the pairing between X* (H) and X * (H). We can as well use this pairing to identify X * (H) to Homz (X * (H), Z) . Let now S be a split torus defined over k. A sequence (ii. l > · · · , An) is a basis of the free Zmodule X * (S) iff the mapping s ...... ( ii. 1 ( s), · · · , An (s)) is an isomorphism from S onto the product (Gm)n Gm x · · · x Gm (n factors). Moreover we may identify X * (S) to Hom kgr (Gm , S) in such a way that the following relation holds rp
=
(1 )
A(rp(t))
=
t X* (H) characterized by =
3. 2. (2)
for h in H and it in X * (H). In the righthand side of this formula, ordF (ii.(h)) is the valuation of the element it(h) of p x . We denote by o H the kernel and by A (H) the image of the homomorphism ordH. By construction, one gets an exact sequence 1
(S)
_____,
o H _____, H � A(H)
_____,
1.
We can also describe o H as the set of elements h in H such that ii.(h) e D� for any rational homomorphism it from H into F x . Therefore o H is an open subgroup of H. A character X of H is called unramified if it is trivial on o H. Otherwise stated, an unramified character is of the form u o ordH where u is a homomorphism from A(H) into ex . Introduce the complex algebraic torus T = Spec C[A(H)]. By definition, one has A(H) = X * (T) and T(e) = Hom(A(H), ex). Thus, there exists a well defined isomorphism ...... X t between the group T(e) of complex points of the torus T and the group of unramified characters of H. If H is a torus, one has
t
(3)
X e(rp(wF))
=
rp( t )
for t in T (e), rp in X * (H) = X * (T) and any prime element w F of the field F. Let again be a connected reductive algebraic group defined over the local field F. We fix a maximal split torus A in and denote by M its centralizer in We let N(A) be the normalizer of A in and W = N(A)/M be the corresponding Weyl group. We choose also a parabolic group such that A) is a parabolic pair. Hence is a minimal parabolic subgroup of and = M · N where N is the uni potent radical of We denote by (/) the set of roots of w.r.t. A and by A the group A(M). Hence (/) is a subset of X*(A) and defines a basis L1 of (/). Let LJ be the set of roots opposite to the roots in L1. The basis LJ of (/) corresponds to a parabolic subgroup
G
P
GG P (P, G P P. G P
G.
REPRESENTATIONS OF jJADIC GROUPS
1 35
P  = M · N of G. Let n be the Lie algebra of For any m in M, the adjoint re presentation defines an automorphism Ad n ( ) of n. We set (4)
A AtA
o(m)
m N.
= j det Ad n ( ) j F for m in M.
m A
A
A
The group o (resp. oM) is the largest compact subgroup of (resp. M) and o is equal to oM n A . Thus the inclusion of into M gives rise to an injective homo morphism of into M;oM. We do not know in general if this map is surjective (see however Borel [5, 9. 5]). More precisely, the inclusion of A into M gives rise to a commutative diagram with exact lines
(D)
l
+
l
+
o
+
A A 1
1
ord A X (A) *
1
oM + M ordM A
A A
+
1
+ 1
A
Indeed, since is a split torus, ordA is surjective. The inclusion of into M enables us to identify XA ) to a subgroup of finite index in X* (M), hence the relation X* ( ) c A c X* (M). We denote by X the group of unramified characters of M. We may (and shall) introduce as before a complex torus such that X * (T) = A and an isomorphism �> Xt of onto This isomorphism enables us to consider X as a complex Lie group. The subgroup of acts on M, oM, o via inner automorphisms. Using diagram (D) above, we may let the Weyl group W = N(A)/M operate on X* (M) so as to leave invariant the subgroups X * (A) and A of X* (M). The group W acts there fore on X and by automorphisms of complex Lie groups. For instance, if X is any unramified character of M and w any element of the Weyl group W, the trans formed charaCter wx is given by
A
t
T(C) X. N(A) G T
T
A, A
(5) where is any representative of w in The unramified character X of M is called if wx of. X for every element w of. I of W. For the applications to automorphic functions, one has to examine the case where G is that is the following hypotheses are fulfilled : (a) G (b)
xw ar N(A). regul F,F. iTheres unramifi quasiexistssplaneidt over over splits over F' . unramified extension F' ofF, of.finitF.e degree d, such that G T', X. g' g'" T' t' h aconjugate gi gz g2 h1 gih". Any semisimple element is aconjugate to an element of T'. Let a denote the Frobenius transformation of F' over In this situation, the Lgroup associated to G is defined. It is a complex connected reductive algebraic group L Go endowed (at least) with a complex torus an automorphism ,_. such that T'" = and a homomorphism �> x ;, of T' onto We say two elements and of L Go are if there exists in L Go such that = The following theorem has been proved by Gantmacher [20] and Langlands [35]. THEOREM 3 . 1 . (a)
in L Go
P. CARTIER
1 36
� and t� of T' are aconjugate iff the unramified characters x'1,1 Two element s t and X;'z of Mare conjugate under the action of the Weyl group W. L G0• The unramifi 2. er of We define the re P Leted priinncipalbeserianyes.unramifieod1 1charact presentTheatiospace n I(x)I(x))consiof Gstass offoltlhoews:locally constant functions/: G > C such that f(mng) o(m)1 12 x(m)f(g) form in M, n in N, gin G. The group G acts by right translations on I(x), namely (v/g) f)(g') f(g'g) for fin I(x), g, g' in G. (7) Yl'(G). Px : Yl'(G)I(x)> J(x) Pxf(g) J J o1 12(m)x1(m)f(mng) dm dn. dm M dnYl'(G) and N P I(x). (b)
Otherwise stated, the orbits of W in the group X of unramified characters of M are in a bijective correspondence to the aconjugacy classes of semisimple elements in For more details, we refer the reader to Borel's lectures [5, §6, 9.5] in these PRO CEEDINGS. This series shall presently be defined via in 3 .3 . duction from with the slight adjustment of DEFINITION 3 . 1 . X X M. (vx, (a)
(6)
=
(b)
=
It is important to give an alternate description of Indeed one defines a surjective linear map (8)
=
M
as a factor space of by
N
(See formula ( 37) in § 1 . 8 .) The groups M and are unimodular, hence the Haar measures on and on N are left right invariant. The map x intertwines the right translations on with the representation vx acting on the space From the general results described in §II, one gets immediately the following theorem . THEOREM 3 .2. (a) (b) (c)
X
s admiofssiI(x).ble. Fortion every isn isomorthe prepresent aetiocont n ragredi I (x))eontf G(/i(x))1 The represent a I(x) i hi c t o t h preuniIfxtary.is a unitary unramified character of M, the representation I(x)) ofG is o1 12 J(x) J /(o1 12) G J(P0vz f) L f(g) dg f Yl'(G) f I(x) f' J(x 1) I(ol l2) f') J(ff ') . f�>I(x) J(x) /1 /2 J(o1 12) X (vx,
X,
(vx,
One of the reasons for inserting the factor in the definition of is to get assertions (b) and (c) above. We state them more precisely : there exists a linear form on invariant under the right translations by the elements of and characterized by
(9)
=
for in
(see Bourbaki [6, p. 4 I ] for similar calculations) . For in function//' belongs to and the pairing is given by ( 1 0)
(f,
Similarly, for unitary and the unitary scalar product in
.h in the functi on is given by
( I I)
and
in

, the
=
belongs to
and
REPRESENTATIONS OF pADIC GROUPS
1 37
We now state one of the main results about irreducibility and equivalence (see also Theorem 3. 1 0 below). X
Letry andbereanygulunrami.fi ed charactateiornofM. is irreducible. i s uni t a a r, t h e represent atiareons irreducible.and have the same charactTheLeter,.Yf(G)modul hencebe inareW.equieThevalentrepresent if t h ey is of.finite length. semisimplijied W, Structure of Jacquet's module M (m, z) ...... m any smootsmh representation of G. For any unramified character of M,Letone(g:n:et, s anbeisomorphi THEOREM 3.3. (a) If x (b) w
(vx,
(vx,
(c)
I(x))
I(x))
(vwx•
l(wx))
I(x)
In general, if V is a module of finite length over any ring, with a Jordan Holder series 0 V0 c V1 c . . · c Vn  1 c Vn V, the semisimple module V cs> 83 7= 1 V;/ V;_ 1 is called the form of V. According to JordanHolder theorem, it is uniquely defined by V up to isomorphism. Let J(x) cs> be the semisimplified form of I(x). It exists by Theorem 3.3(c) above. the It is clear that I(x) and J(x) cs> have the same character. Hence for any w in representations l(x) cs l and J(wx) cs l are semisimple and have the same character. By the linear independence of characters, they are therefore isomorphic. I(x)N. For any unramified character X of M, 3.4. let Cx denote the onedimensional complex space C 1 on which acts via X • viz. by x( ) z. Frobenius reciprocity takes here a simple form, namely (see §2.2) : =
=
=
·
THEOREM 3.4. x
V)
Homc ( V, I(x)) ""> HomM( VN, Cx0 1 12).
The proof is obvious. Indeed the relation ·
W(v) (g) (rp, :n:(g) v) =
( 1 2)
v
(for g in G, in V)
expresses an isomorphism (/j +> rp of Homc( V, I(x)) with the space of linear forms on V such that
rp
(rp, :n:(mn) . v) v m:n:(n)Mvv n n v
( 1 3)
=
o 1 12 ( )x( )
m m (rp, v) rp
for in V, in and in N. Recall that VN VI V(N) where V(N) is generated by the vectors · for in N and in V. Any solution of ( 1 3) vanishes on V(N), hence factors through VN· The previous theorem exemplifies the relevance of Jacquet's module I(x)N in the study of the intertwining operators between representations of the unramified principal series. We know by Theorems 2. 1 , 3.2 and 3. 3(c) that I(x)N is a finite dimensional complex vector space. The following basic result is due to Casselman [17] . =
X
er oftM,he gtroup he semiMsimactplifis terdivform the Mmodule For ianys unrami.fied charact Moreover, ial y onof Weyl group The dimension of over is equal to the order I WI of the Assume regular. Then as an Mmodule is isomorphic to THEOREM 3 . 5 .
I(x)N
I(x)N ·
EB w e w Ccw r.J ·o l /2 ·
CoROLLARY 3. 1 .
I(x)N
W.
CoROLLARY 3.2.
EB w e W C (w r.) · ol l2 ·
X
0
C
I(x)N
P. CARTIER
1 38
For the proof of Corollary 3.2, notice that M acts on through the commu tative group isomorphic to A. By Schur's lemma, the semisimplified form of is therefore of the form EB :=l Cx; for some sequence of unramified of by a wellknown lemma, each representation Cx; occurs characters as a subrepresentation of Hence for w in W, Ccw xl · o i/2 occurs as a subrepre sentation of by Theorem 3 . 5 . Corollary 3.2 follows at once from thi s remark as well as the following corollary (use Frobenius reciprocity in the form of Theorem 3 .4) :
I(x)N M/" M I(x)NXI > · ··, X M; s l(X)N. I(x)N LetThereX beexianysts aunramifi edicharacter ofoperator M and w be anyI(x)element of tIfheX Weyl n tertwi n i n roup W. nonzero > l(wx). g g is regular this operator is unique up to a scalar. (n, I(x), wE (n, I(w· x) T COROLLARY 3 . 3 .
Tw :
Tw
A similar argument shows that if V) is an irreducible subquotient of then there exists W such that V) is isomorphic to a subrepresentation of (see 6.3.9 in [17]). We shall describe more explicitly the operators w in §3.7. We sketch now a proof of Theorem 3 . 5, a streamlined version of Casselman's proof of more general results in [17]. Basically, it is Mackey's double coset tech nique extended from finite groups to padic groups. The case of real Lie groups has been considered by Bruhat in his thesis [7], it is much more elaborate. (A) We remind the reader of Bruhat's decompositi on = w where means For w in for any g in N(A) representing the element w in W = N(A)/ W, the set is irreducible and locally closed in the Zariski topology, hence has a welldefined dimension. Set  dim(P). For any integer � 0, = dim let be the union of the double cosets such that < The Zariski closure of any double coset is a union of double cosets of smaller dimension, hence is Zariski closed, hence closed in the padic topology. We let be the subspace of consisting of the functions vanishing identically on We have then a decreasing filtration
Fr
PgPPwP
Fr I(x)
Ir
d(w) (PwP)PwP PwP
G U PwP,M. PwP d(w) Pw'r.P r Fr.
( 1 4)
I(x)The Pnextstablstepe subspaces. is to prove that anyfunction on Fr+l which satisfies the relation f(mng) ol l2(m)x(m)f(g) m n Fr+l is the restriction of someG function belonP\G) ging to I(x). f(g) IM IN o1 12(m)xl(m)q>(mng) dm dn q> Frf+l· q> q>' G). P .,_ q> ' I(x) Fr+ l· Irllr+l Fr+l by
of
(B)
( 1 5)
fo r
=
local crosssections of
fibered over
in M,
in N, g in
Indeed, one proves easily (using that such a function is of the form
=
for a suitable locally constant and compactly supported function on Extend to a function in Jll'( Then belongs to and restricts to
(C) From this it follows that
satisfy the following conditions : (a) f is locally constant ; (b)/ vanishes on
Fr;
is the space of functions f on
in
which
REPRESENTATIONS OF pADIC GROUPS
1 39
relation (15). d(w) r, open (16) Jw f(mng) o1 12(m)x(m)f(g) n (16) .Yt(M)modules I(x)N and ffiwEW(Jw)VN haveVN isomorphic semisimplifiedforms. M CJw)N. w N(A)N(w) w) (w) M·N(w)Jw (mn) wI(ol i2X) (m) (17) M, n N(w). M4, a (mn) m M n 1. aAaw · ow) (18) V V a)*(aw·ow).o a* o a)a* P(w) M CJw)N(w)N o awowi M. o N(w) (24c) o M ol l2(wIo)1 12 Cial y on I(x)N. I(x)N oM actE8 wEW s t r i v compact M. I(x)N M I(x)N, o) I(x)N. 3. 5 . Buildings and Iwahori subgroups. 3. 5 . (c) Moreover, F,+J is the union of F, and the various double cosets PwP such that = which are in F,+ J ; hence one gets an isomorphism
Here
is the space of functions/ on PwP such that
for m in M,
=
in N, g in PwP,
and which vanish outside a set of the form PQ where Q is compact. Since Jacquet's functor =:> is exact, one infers from that the
Here we (D) It remains to identify the representation of on the space are paid off the dividends of our approach to induced representations via tensor products. Choose a representative Xw of in and put P( = p n x;;} Pxw ; hence P = with a suitable subgroup of N. lt is then easy to show that, as a Pmodule, carries the representation cInd� (wl aw where the character aw of P(w) is defined by aw
for m in
=
in
Consider the group homomorphisms P(w) ..!!., P L and � = for m in and in N. By Theorem where the character Ow of P(w) is defined by
where is the injection one gets cInd� (wl aw =
Since � * is Jacquet's functor =:> N and �* one gets that (Jw)N is = (� the carrier of the representation (� Since � is the projection of onto with kernel and the characters aw and Ow are trivial on one It remains to prove the gets an isomorphism � c, where A = to be able to conclude formula (see formula below) ;;;1 1 = ( 1 9)
<w xl1!112 have (E) From (C) and (D) above, we know that and isomorphic semisimplified forms. It remains to show that But oM is a subgroup of Hence its action on and its semisimpli fied form are equivalent. Since any unramified character of (including is trivial on oM, this group acts trivially on the semisimplified form of hence on This concludes our proof of Theorem
Our aim i n this section is mainly t o fix notations. For more details, we refer the reader to the lectures by Tits in these PROCEEDINGS [42] or to the book by Bruhat and Tits [13]. Let P4 be the building associated to and let .9/ be the apartment in P4 associated to the split torus We choose once for all a special vertex x0 in d. Among the conical chambers in .9/ with apex at x0 there is a unique one, say, enjoying the following property : for every in N, the intersection n contains a translate
A.
G
n
({? n({? ({?
1 40
P.
CARTIER
for one of its of 'tl . There is then a unique chamber C contained in 'tl having it is called by vertices (see Figure 1). The stabilizer of in shall be denoted by Bruhat and Tits a of The interior points of C all have the same stabilizer in called the attached to c.
x 0 G K; x 0 special, good, maxi groupsubgG.roup of G B mG,al compact sublwahori
FIGURE 1
Any element takes the apartment d to itself, it fixes every point of d iff it belongs to We may therefore identify the group = called the to a group of affine linear transformations in d. The group can be represented in two different ways as a semi direct product : (a) Let Q be the subgroup of consisting of the w's taking the chamber C to itself. The walls in d are certain hyperplanes and to each of them is associated a certain reflection. These reflections generate the invariant subgroup w.ff of Since acts simply transitively on the set of chambers contained in .s:J, the group is the semidirect product Q · = (b) Any element w of has a representative in and We may therefore identify the Weyl group = to the stabilizer with the group of transla The intersection of of x0 in tions in d is which we identify to A by means of the exact sequence (D), p. 1 35. Then is the semidirect product A where A is an invariant subgroup. The fundamental structure theorems may now be formulated as follows.
gioM.n N(A) W1 modified Weyl group,
W1 N(A WM,
W1
W1 . WWa1H WaH· w(w) K n N(A) oM W K(Knn M.N(A))j(K n M) W1 . WW1 N(A)/M WM/0M W· 1 and, more the sets Pw(w)Bfor runningGover tPKhe Weyl W. sely, G is the disjoint union of grouppreci over the modified Weyl group WG1 • is the disjoint union of the sets Bw1Bfor w1 running Let of beopposi the subset of intconsi selftingThenof thGe element ssjoiofnt tuniakionngofthteheconisetscalK·chamber ·Kfor). t e to o i t s i s t h e di running over tion). Moreover one has B n1 MB (BoMnandN)·(B n M)·(B n N) (unique factoriza m(B n N)m B n N, for any m in M such that G A. a(x0 ) , a) =
lWASAWA DECOMPOSITION.
w
BRUHATTITS DECOMPOSITION. CARTAN DECOMPOSITION.
ordA}().)
'tl
Ad
'tl
A
A
A.
=
lWAHORI DECOMPOSITION.
=
c:
ord M (m) E A.
As before, we denote by fP the system of roots of w.r.t. We identify the ele ments of 1> to affine linear functions on sif in such a way that + ). = ( ).
REPRESENTATIONS OF pADIC GROUPS
a
141
for in r!J and ). in A. The conical chamber '?? is then defined a s the set o f points x in s1 such that (x) > 0 for every root in the basis L1 of r!J associated to the para bolic group We denote by r!J0 the set of affine linear functi ons a on d, vanishing at x0 and such that the hyperplane is a wall in s1 iff the real number is an integer. An affin e root is a functi on on d of the form + where a belongs to r!J0 and is an integer ; their set is denoted by r/Jaff · For any affine root the reflection in the wall is denoted by Sa. The reflections Sa for running over r!J0 (resp. r!J.H) generate the group W (resp. w. H) . For in r/J0, there exists a unique vector ta in A such that
P. a
a1(0) )
a
a1(r)
a
Sa(X) = x 
(20
a(x)
a k a
·
r
a,
k
ta for any x in d.
We denote by aa any element in M such that ta = ordM(aa). The set r/J 1 is obtained by adjoining to r!J 0 the set of functions /2 for in r/J 0 such that (Bw(Sa)B : B) # o(aa) I IZ . The sets r/J, r!J0 and r/J 1 are root systems in the customary sense (see for instance [37, p. 14 sqq .]). When the group G is split, the sets r/J, r!J0 and r/J 1 are identical. In the nonsplit case, all we can assert is that, for any a in r/J, there exists a unique root in r/J0 proporti onal to and that any element of the reduced root system r!J0 is of the form for a suitable in r/J. Let w 1 be any element of W1 . Since w 1 is a coset modulo the subgroup a M of B the set Bw 1 B is a double coset modulo B. We put
.A.(a) (2 1
a a
a, a
.A.(a)
)
Since K = B WB, one gets
(K: B) = 2: q (w). E
(22)
w W
Let L1 1 be the set of affine roots in s1 which are positive on the chamber C and whose null set is a wall of C. The group W is generated by and the reflections Sa for in L1 1 • The value of q (w ) is given by
a
(23)
1
Q
1
where w 1 = wSa1 . . . sam is a decomposition of minimal length m (w in a[ , · , a m in Ll l ). To each root {3 in is associ ated a real number q� > 0. This association is characterized by the following set of properties
Q,
0 0
r/J1
qw ·� = q� for {3 in r/J 1 and w in W, IT q (w) = q�, w �>O ;  1 . �O
M.
In the previous formulas {3 is a variable element in r/J 1 and the notation {3 > 0 means that {3 takes only positive values on '?? . We make the convention that q a 1 z = I for a in r/J0 if a/2 does not belong to r/J 1 • Two corollaries of the previous relations are worth mentioning
P. CARTIER
1 42 (24.)
for any > 0 in f/)0 • When is split, q� is equal to the order of the residue field D FIPF for any root {3 i n = r/Jo = The structure of the Heeke algebra has been described by Iwahori and Matsumoto [30], [31].
a
q f/J G f/J1 . .Yt'(G, B) eristic function of the double coset BwThe1fami B. ly ForC(wW1)1}winEWhw1 isleta basiC(ws lof) bethethecomplcharact ex vector space .Yt'(G, B) (Bruhat 1 Tits decomposi t i o n! ) . ... , am in L11) be Let w be any element o f W and let wSa1"· S am (w i n ah 1 1 a decomposition ofw1 ofminimal length m. Then For each a in L11 , one has whereIfq(Sa)a andhas arebeendidestifinnedct elbyementformuls inaL1h thereabove.exists an integer such that THEOREM 3.6. (a)
{
0,
(b)
(25)
(c)
(26)
(2 1)
(d)
rna� � 2
{3
(27)
factors ma{3factors Moreover the relations and are a complete set ofrelations among the C(Sa)'s. Action ofthe Iwahori subgroups on the representations. projection of V ontLeto VN defiV)nesbeananyisomoradmiphissismbleofrepresent VB onto a( tVioNntMof. G. The natural VB (VN)"M P G. M VB (VN)" , v a V(N)A VB, l a(a)I F 0 a f/J C(a)) . v P.0 v q(Sa) en any I(x)(N). unramified character of M, one has a direct sum decomposition I(x) GivI(x)B oM I(x)N. G I(x)B. w(w) openM B Pw(w)B (woM W), GM xo1 12. mnw(w)b, f/Jw. x(g) 0o1 12(m)x(m) g Pw(w)B. rna �
(26)
(27)
3.6. due to Casselman [18] and Borel [4].
Here is the main result,
(n,
THEOREM 3.7.
One proves first that maps onto using Iwahori decomposition of B and the methods used in the proof of Theorem 2.3. There are some simplifications due to the fact that is a minimal parabolic subgroup of To prove that maps injectively in one first proves that, for any given = vector in n there exists a real number e > such that n( for every in satisfying � e whenever the root e is positive on But this relation implies = 0 by Theorem 3 . 6, since one has > 0 there. CoROLLARY 3.4.
=
X
®
This follows from Theorem 3 . 7 since A direct proof acts trivially on can also be obtained using the methods used in the proof of Theorem 3 . 5 . Using the decomposition of into the pairwise disjoint subsets Indeed, normalizes and n = in one gets easily a basis for Hence the following function is well defined on lies in the kernel of (28)
=
=
if = if g rt
REPRESENTATIONS OF pADIC GROUPS
143
The family {W.,,x}w EW is the soughtfor basis. The next result is due to Casselman [18] (see also Borel [4]).
(n , V)
bealeanynt: admissible irreducible representation of The follo wiThere ng assertare iionLetns arenonequi v ero vecteodrscharact invariaentr under Bsuch(thatthiats (7r, is isomorphic zunrami.fi There exi s t s some o f to a subrepresentation of THEOREM 3 . 8 .
G.
V
(a) (b)
X
(vx, I(x)).
VB
M
i= 0).
V)
B y Theorem 3 . 7 , assertion (a) means that ( VNrM i= 0. B y Frobenius reciprocity (Theorem 3.4), assertion (b) means that there exists in the space dual to VN a non zero vector invariant under oM which is an eigenvector for the group M. Since oM is compact, acts continuously on V N and M;oM is commutative, the equivalence follows immediately. X
)B
I(x) as a Gmodule.Let be any unrami.fied character of The space I(x generates Ix B I(x). I x I x = u I(x B) = Ix I x)) I x I(x1) w we assumew theW,W,unrami.fied character to be Intregulertawr,ining operators. Ix mn · f) 1 1 2 m wx m (29) m n I(x), J f(w(w)1n) dii l)). w \N w F, (rI x d(....w. I(wx) Ix w, w(w) w. PK K= I(x) K WKjmnk) (j1 12(m)x(m) n CoROLLARY
3.5.
M.
We prove Corollary 3 . 5 by reductio ad absurdum . Assume that ( ) does not generate Since ( ) is finitely generated, there exist an irreducible admissible representation (n, V) of G and a Ghomomorphism u : ( ) > V which is nonzero 0. and contains ( )B in its kernel. Since B is compact, one gets VB ( ) By duality, one gets an injective Ghomomorphism u : V > ( (  . Since ( ( )) by Theorem 3 . 2, it follows from Theorem 3 . 8 that VB i= 0. is isomorphic to the finitedimensional spaces VB and VB are dual to each other and this is But clearly impossible. In this section, 3.7. that is the characters wx, for running over are all distinct. X Corollary 3.2 may be reformulated as follows : given in there exists a linear form Lw i= 0 on ( ), such that
Lw(vx (
)
for in M, in N and f in normalize Lw by (30)
Lw(f) =
= o
( )
( ) Lw(f)
unique up to multiplication by a constant. We
N (w) \N
= The Haar measure for a function / whose support does not meet is chosen in such a way that N( ) ( ) · (N n B) be of measure 1 . To Lw we associate an intertwining operator Tw : ( ) defined by (3 1 )
Twf(g) = Lw(vx(g) · f) for f in ( ) and g in G.
It is easily checked that Lw, hence Tw , depends only on not on the representative for Since G = (Iwasawa decomposition) and the character o112 X of M is trivial on M n 0 M, there exists in a unique function WK .x invariant under and normalized by W K . /I ) = 1 . Explicitly, one has
(32)
=
for m in M,
in N and k in K.
P. CARTIER
144
If the Haar measures on and N are normalized by JM nK = J NnK = 1, one may also define (/)K x a s Px(h) where Px i s defined a s o n p. 1 36 and IK is the characteristic function of K. The space I(x ) K consists of the constant multiples of (/)K.r The next theorem again is due t o Casselman [18] .
,M
dn
dm
I(x) K
The operator Tw takes into where the constant cw(X) is defined by cw(X) (34) THEOREM 3 . 9 .
l(wx) K .
(33)
More precisely, one has
C ( ), a aX
= IT
(35) ljq,
a
(/)0
Thebut product such thatin is ines gextatievnded e overover the affine roots in which are positive o1•er   a (34)
ljq,
wa
When G is split, formula (3 5) takes the simpler form 1 C a) ca(x )   I qx1 x(aa)  · _
(36)
·

The bulk of the proof of Theorem 3.9 rests with the case where w is the reflection associated to a simple root {3 in (/)0. In this case, there exists exactly one positive root a in (/)0 taken by w into a negative root, namely a {3. The general case follows is equal to when the lengths of w1 and w2 add to the length then since of w1w2•
= Tw1 Twz Tw1wz tertwining map Tw is an isomorphism from onto cw(x) and Theareinnonzero. cw(Xkwi(wx) Tw 1 Tw Assume Xisiisrreduci any unrami.fi ed regular character offor erery Thenposithe treire present a t i o n of b l e root in positive over where is the unique element in which takes into I(x )
COROLLARY 3 .6.
Uf
C wi( wx)
I(wx )
Indeed is multiplication by by Theorem 3.9. One may strengthen the irreducibility criterion (Theorem 3 . 3) . THEOREM 3 . 1 0 .
ljq.
a
(/)0
(v x , I(x))
G
ijq,
w0
iff ca (X)
=1
0, ca(w0x)
=1
M.
0 W
CC
IV. Spherical functions.
Some integrationformulas.
We keep the notation of the previous part. 4. 1 . If F is any of the groups G, M, N, K, then F is unimodular. We normalize the (left and right invariant) Haar measure on F by Jr nK dr = I . The group P M · N is not unimodular. One checks immediately that the formulas (I)
(2)
J f( ) dip = JM J J d,p J J f( P
P
p
f(p)
N
=
define a left invariant Haar measure
M
dip
N
=
dm dn, ) dm dn
f(mn) nm
and a right invariant Haar measure
REPRESENTATIONS OF pADIC GROUPS
(P K)P. (3) d,p on n
145
Since aM = M n K is the largest compact subgroup of M, one gets (M n K) (N n K), hence the normalization
=
·
The Haar measures are related to the Iwasawa decomposition by the formulas
sG f(g) dg = JK JPj(p k) dk dip, L f(g) dg = JK L f(kp) dk d,p
(4) (5)
for f in Cc(G). Let us prove for instance formula into Cc(G) by the rule h >+ uh from Cc(K x
P)
(6)
ui pk 1 ) =
(4).
One defines a linear map
JPnK h(kp1 , PP1) d1PJ .
is then a left invariant Haar measure The linear form h >+ Jc uh(g) dg on Cc(K x on K x hence by our normalizations, one gets
P)
P;
(7)
sG uh(g) dg = s K sPh(k, p) dk d1p.
It suffices to substitute f(pk 1 ) for h(k, p) in formulas From the definition of a (see p. one gets
135),
(8)
(6)
and (7) to get formula (4) !
for any function / in Cc(N). As a corollary, we get the alternate expressions for the Haar measures on
P
JPj(p) dip s fN o(m) 1/(nm) dm dn, JP f(p ) d,p = s sN o(m)j(mn) dm dn .
(9)
)
(10
=
M
M
Otherwise stated, the modular function of
Llp(mn) = o(m)  1
(1 1)
P
is given by
for m in M, n in N.
The group N is unipotent and M acts on N via inner automorphisms. It is then easy to construct a sequence of subgroups of N, say N = N0 ::J N1 ::J • • • ::J N,_ 1 ::J N, = 1 , which are invariant under M and such that Nj_I fNj is isomorphic (for j = 1, to a vector space over on which M acts linearly. Putting . . .
( 1 2)
, r)
F
for m in M, one gets by induction on ( 13)
r
the integration formula
JN f(n) dn = Ll(m) JN f(nmn 1 m1 ) dn
for f in Cc(N) and any m in M such that Ll(m) # 0 (see [28, Lemma 22]). Let us define now the socalled Let m be any element of M
orbital integrals.
P. CARTIER
146
such that # 0, let be the centralizer of in G and Gm the conjugacy class of in G. Then A c M and n M has finite index in since is a compact group, the same is true of Moreover, the conjugacy class Gm is closed in G; hence the mapping >+ is a homeomorphism from onto Gm . Therefore, the mapping A from G/A into G is proper and we may set after HarishChandra
m Ll(m)
Z(m) Z(m) m 1 m gZ(m)Z(m)/A. g g g ...... gmg1 Fj(m) LJ(m)J f(gmg1) dg
G/Z(m) (14)
=
Z(m); M/A
GI A
for any function / in Cc(G). The Haar measure on A is normalized in such a way that JM I A = }.
dffz f beisanyequalfunctioNn f(innm) dn. K) and m an element of such that LJ(m) ThenLetFj(m) J u(g) dg J JN u(nm1) dm 1 dn Fj(m) u(g)u f(gmg1), F1(m) J h(m1) dmb LEMMA 4. 1 . # 0.
to
Yf(G,
J
M
From formulas (2) and (5), one gets
( 1 5)
=
G
M
for any function in C,(G) which is invariant under left translation by the elements = one gets the following representation for in K. Putting =
( 1 6)
MI A
( 1 7)
= and set We get therefore
(m)m1 L1(m2). m2 m1mm11 . h(m1) L1(m2) JN j((nm2 n1 m21)m2) dn (13). JN f(nm2) dn m2 mK K, f(nm2 f(fnm) K) n Nf(nm) dn / m1 a ake isomorphism. Fix Ll
=
From the definition ( 1 2) of Ll, one gets
=
by ( 1 7)
=
by
Notice that the group Mt M = M/(M n is commutative. Hence we get E and since the function is invariant under right translation by the is elements of one gets ) = for any in N. It follows that equal to J for any in M . The contention of Lemma 4. 1 follows from formula ( 1 6) since M A is of measure 1 . 4.2. S t The construction we are going to expound now is due to Satake [38]. It is the padic counterpart of a wellknown construction of Harish Chandra in the setup of real Lie groups. For any A in A, let ch(A) be the characteristic function of the subset ordA}(A) of = I , one gets M. Since JMn K (1 8)
dm
ch(i!) * ch(A')
=
ch(A
+
h(m1)
A')
for A, A' in A. Moreover the elements ch(A) (for A in A) form a basis of the complex algebra Yf(M, aM), which may be therefore identified to the group algebra C[A]. We define now a linear map S : Yf(G, Yf(M, 0M) by the formula
(19)
K) > Sf(m) o(m)1 12 JN f(mn) dn o(m)1 12 JN f(nm) dn =
=
147
REPRESENTATIONS OF pADIC GROUPS
f .Yf'(G, K)
for in and m in M. The two integrals are equal by formula (8). It is im mediate that the function on M belongs to Cc (M). That it is hiinvariant under oM follows from the fact that is hiinvariant under K and that oM M n K. The following fundamental theorem is due to Satake [38].
Sf f f ionconsiS sistinang ofalgthebrae invariisomor antspofhisthme from Weyl .Yf'groupTHEOREM (G, K)W. onto4.1.the ThesubalSatgebraake transfoormat S S is a homomorphism of algebras. .Yf'(G, K) .Yf'(P) G {3 P. ({3u)(m) u(mn) dn f)(m) f(m)o(m)1 12 ; The image of Sis contained in W Sf(xmx1) Sf(m) m M mx n(m  In) F M; m (21) Sf(m) D(m) J f(gmg1) dg 1 1 2 ; D(m) .d(m)o(m) D(m)2 I  1nF) D(m) (m) D(xmx1) D(m) m x G A. G A, =
C[A]W
C[A]
Here are the main steps in the proof. By construction, (A) three linear maps �
is the composition of
L .Yf' (M) L .Yf'(M).
Here a is simply the restriction of functions from to It is compatible with convolution by an easy corollary of (4). The map is given by JN and one checks easily that it is compatible with convolution. The map r is given by (r = since o is a character of M, it is compatible with convolution. (B) C[A]W . Since (N(A) n K)rM, this pro perty is equivalent to =
=
(20)
=
for in and in N(A) n K. The function ,_. det ( Ad ) from M to is polynomial and nonzero. The elements of M which do not annihilate this function are therefore dense in they are called the regular elements. Hence by continuity it suffices to prove (20) for regular. From Lemma 4. 1 , one gets =
G/A
hence = j det(Adn(m)  l n) j}  j det Adn(m) j:F1 = jdet(Adn(m)  1 n) l p · l det(Adn(m l ) 1 n)
for m regular in M. Here
is equal to

= jdet(Adn(m)  l n) lddet(Adn(m)
F
) IF ·
The last equality follows for instance from the fact that the weights in 11 ® F P (F an algebraic closure of are the inverses of the weights in 11 ® F P of any maximal torus of M. Since g = 11 ffi m ffi 11, one gets = !det(Ad gt nt (22)  l gt m) l ¥ 2 and therefore
(23)
=
for
in M,
in N(A).
On the other hand, the compact group N(A) n K acts by inner automorphisms on and It leaves therefore invariant the Haar measures on and hence
P. CARTIER
148
G/A. m M G;
the Ginvariant measure on Let in be regular, X in f in One hasf(xgx1) = f(g) for any g in hence
.Yt'(G, K). J
GI A
J = J
f(g(xmx1)g1) dg =
GI A
GI A
N(A) K n
and
f((x1 gx)m(x1 gx)1) dg f(gmg1) dg.
The invariance property (20) follows from the representation (2 1), the invariance property and the justestablished invariance. (C) > C[A]W We let as before A  denote the subset of A consisting of the translations in d which take '?! into itself. For ). in A  , let lfJ;. be the characteristic function of the double coset ordA{().) by Cartan decomposition, the family { lfJ;.};."',� is a basis of Moreover, any element of A is conjugate under to a unique element in A  . We get there fore a basis {ch'().)}J.r,.1 of C[A]W by putting . I ch'().) = ch( w . ).), (J.)i W
The(23)linear mapS: .Yt'(G, K)
is bijective.
W
(24)
where (25)
W().)
.Yt'(G,KK). ·
·
K;
w�W
I
is the stabilizer of ). in W. Define the matrix { ).' ) ch'().)
= I: ).
S!p;. c().,
·
where )., ).' run over A  . To calculate c(A, A'), choose representatives Then we get (J..t is the Haar measure on
c().,
).')} by
and m' respectively of A. ).
in M.
G) 1 m S!p.,(m) o(m) 12 f..t(Km'K NmK). mK NmK mK; 1 (27) o(m) 12. non negative realKmcoeKfficieNmK nts ofisthempt e posiytiunlve erss s is linear combination with (27), S!p;. , 4.1. The algebra .Yt' G is generated over Z. .Yt'(G, K) Mf(m)x(m) Mo odmM) .Yt'(G, K) c(). ,
(26)
).')
n
It is clear that K
=
n
=
=>
'
hence
c(A, A1) �
Moreover
'
n
oo
).'
A
a
Using a suitable lexicographic order ing � , we conclude that c(A, A' ) = 0, unless 0 � ).' � A. Since c(A, ).) =f. 0 by this remark shows that the elements for ).' in A, form a basis of C[A]W, hence our contention (C) . CoROLLARY
C.
(
t
, K)
.
commutative an d finitefy
The algebra C[A] is commutative and generated by ch(). 1 ) , · · · ch().m), ch(A11 ), Since the group W is finite, it follows from wellknown results in commutative algebra 1 3 that C[A]W is finitely generated as an algebra over C, and that C[A] is finitely generated as a module over C[A]W. We determine now the algebra homomorphisms from to C. Let X be dm any unramified character of M. Since J = I, the map / >> J to C, and we get in this way all is a unitary homomorphism from .Yt'(M, such homomorphisms. Define a linear map w x : > C by ,
· · ·, ch().;;;1 ) if {A 1 . · · · , Am} is a basis of A over
13 See for instance N. Bourbaki, Commutative algebra, Chapter V, p. 3 2 3 , Addison Wesley , 1 972.
149
REPRESENTATIONS OF pADIC GROUPS
(J.Jx(f) f Sf(m) . x(m) dm. 4. . Anyed charact unitaryerhomomor pMoreover, hism fromoneJft'(hasG, K)(J.Jx into(J.JCix' s otfheretheexiformsts some unramifi o f M. (J.Janx forelement in such that = 4.1 Jft'(G, K) M. * T) T 4.3. DetJft'ermi(G,nK)atiKongKof the sphericalfunctions. Jft'(G, K), Jft' ( G, K) G, K: Jft'(G, K), (J.J(f) L t(g)F(g) dg (30) F(g) (J.J(IKgK) jfKgK dg1 g G. (JJOneis ahashomomorphism ofalgebrasfrom Jft'(G, K) to C. (31) for gbForg2 anyin G.function fin Jft'(G, K), there exists a constant such that (28)
=
COROLLARY w
2
x
W
'
X
w
M
=
· X·
iff
This corollary follows from Theorem and the classical properties of invariants of finite groups acting on polynomial algebras (see previous footnote). Otherwise stated, the set of unitary homomorphisms from to C is in a bijective correspondence with the set X/ W of orbits of W in the set X of unramified characters of We refer the reader to §3.2 for a discussion of this set X/ W. Notice that since X is isomorphic to a complex torus such that X ( = A, then X/ W is a complex algebraic affine variety and Satake isomorphism defines an isomor phism of with the algebra of polynomial functions on X/ W. Since the characteristic functions of the double cosets form a basis of the complex vector space one de fines as follows an isomorphism of the dual to the space onto the space of functions on hiinvariant under (29)
=
for f in
=
for
in
I claim that the following conditions are equivalent : (a) (b)
(c)
}..(f)
(32)
The equivalence of (a) and (b) follows from the following calculation
(JJ(/1 * /2)
ff F(g1g2) /1 (g1) /2(g2) dg1 dg2 = f /1 (g 1 ) dg 1 f f2(g2) dg2 f K F(g 1 kg2) dk G G =
(gh g2) JKF(gK1kg2) dk G G (JJ(/1 * /2) f/2(g)(j1*F)(g) dg f f1(g)(F*f2)(g) dg (J.J(j). /1 , /2 Jft'(G, K) f f(g 1) .
and the fact that the function ,_. on x under left and right translations by elements of x K. The equivalence of (a) and (c) follows from the following formula : =
for
in
where (g) =
=
Hence A.(f) =
is invariant
P. CARTIER
1 50 DEFINITION 4. 1 .
n on Gngw.thre. t.equiivsalent a .functipropert on ronies G, hiinandvariant above. under A such thatspherical functandioenjoyi M. mn ) x(m)o1 12(m) M, n g) J G. Theunramifi spheriecdalcharact functioensrs onof M.G w.Ther.t. spheriare ctalhefuncti functioonsns r and Let and x' be are equal iff there exists an element win the Weyl group W such that x' (zonal)
K,
=
K
I
(a), (b) (c) We may now translate our previous results in terms of spherical functions. Let X be any unramified character of Recall that the function (/JK . x is defined by
(/J K , x(
(33)
F(I)
k
=
for m in
in N and k in K.
We put
Fx(
( 34)
=
(/J x ( for g in K K, kg) dk
THEOREM 4.2. (a) (b) X
x· Fx = W · X·
K
Fx·
It is clear that Fx is hiinvariant under K, and Fx( l ) = I . Theorem 4.2 follows from Corollary 4.2 and the formula w .x_(.f) =
(35)
J
c
Fx ( ) ·
This in turn is proved as follows :
s
G
F x( ) ( )
g f g dg
= = = =
g f(g) dg
for f in
Ye(G,
K).
s (/JK, ig ).f(g) dg JK J J (/JK , x (mnk)f(mnk) dk dm dn s x(m)o(m) 1 1 2 dm sN .f(mn) dn s x(m) S.f(m) dm = (f) G
M
N
M
w 'X
M
We used the integration relations ( 1 ) and (4). 4.4. DEFINITION 4.2.
The sphericalA prirepresent ncipal aseritionesofofGrepresent aspheri tions. cal i s called irreducible and contains a nonzero Gvector invariant under f G g ) cl > ..
.
K) if it is smooth,
(w.r.t. K. Let F be a spherical function on (w.r.t. K). We denote by Vr the space of func tions f on of the form (g) = 1: 7=1 c;F( g; for , cn in C and g l > . . · , gn in G. From the functional equation of the spherical functions (formula one de duces ·
(31)),
(36) dk J G (37) ion ant multiipslesspheriofr.cal and that the elements of Vr invariant under the represent are theatconst K
We let
.f(gkg')
= F(g ) · f(g ')
for f in Vr and g , g ' i n G.
operate on Vr by right translations, namely
I claim that
nr(K)
(nr, Vr)
Indeed , it is clear that for any
REPRESENTATIONS OF pADIC GROUPS
151
function/in Vr there exists a compact open subgroup K1 o f G such thatfis invariant under right translation by the elements of K1 ; hence the representation (11:r, Vr) is smooth. Let / "# 0 in Vr and choose an element g' in G such that f(g') "# The functional equation (36) may be rewritten as
0.
(38) Any vector subspace of Vr containing / and invariant under 11:r(G) contains there fore hence is identical to Vr. Finally, if a function in Vr is invariant under 11:r(K), one gets/ = f(I ) · Fby substituting g' I in the functional equation (36).
F, f = Let ion Fsuchbe tanyhat spheriicsalisomorphi representc taotion of There exists unique sphericalfunct 11:(/) simple : v v' = v, v' C= 11:(/) v. K) C. C C 11:(/)v =w(f)v v v ii (ii, v) = = (ii, 11:(g)v) f w( = F ) = cF(g) f(g) F v' THEOREM 4.3.
(11:, V)
G. (11:r, Vr).
(11: , V)
·
a
As usual, let V K denote the subspace of V consisting of the vectors invariant under 11:(K). If/ is any function in .1t'(G, K), the operator takes VK into itself; hence VK is a module over .1t'(G, K). I claim that this module is indeed, let "# 0 and v' be two elements of VK. Since the £(G)module V is simple, there exists a function f in .1t'(G) such that · The function fx lx * f •lx be longs to .1t'(G, K) and 11:(/K ) · substantiating our claim. The algebra .1t'(G, K) over the field is commutative and of countable dimen sion. By the reasoning used to prove Schur's lemma (see p. I I 8) (or by Hilbert's Zero Theorem), one concludes that any simple module over .1t'(G, is of dimension I over Hence VK is of dimension I over and there exists a unitary homomor phism w : .1t'(G, K) > such that =
(39)
for any f in .1t'(G, K) and any
in VK.
Let (it, V) be the representation of G contragredient to (11:, V). The space VK of vectors in V invariant under it(K) is dual to VK, hence of dimension 1 . Choose a vector in VK and a vector in VK such that I and define the function r on G by F(g)
(40)
for g in G.
From (39) and (40) one deduces dg for any fin .1t'(G, K). It is J obvious that F( l ) I and that is hiinvariant under K. Hence is a spherical function. The map which associates to any vector in V the coefficient 11:v ' , ii defines an iso morphism of (11:, V) with (11:r, Vr). Moreover, for any spherical function F' on G, the representation (11:, V) is isomorphic to (11:r ' • Vr, ) iff the following relation holds (4 I )
F' F
This holds for only. Q.E.D. The definition of spherical functions as well as the results obtained so far in this section depend only on the fact that K is a compact open subgroup of G and that the Heeke algebra .1t'(G, K) is commutative. We use now the classification of sphe rical functions on G afforded by Theorem 4.2. Let X be any unramified character of M; when the spherical function F is set equal to Fx , we write (11:x , Vx) instead of =
15 spherical principal series ofrepresentations of G. represent ationare the constis irareduci blipe,leadmi ssible, and the only functions in invariAny a nt under nt mult s o f Let areandisomorphi be unramifi ed exicharact erseloefment Thein represent agroup tions such thatand c iff t h ere s t s an t h e Weyl x' = w·x e represent antiI(x)on defineI t))heifunct n theiounram!fi ed principalf(serikg)esdk.is iThen rreducitAssume bhelmap e. ForfthatanyPthfunct i o n! i nP by P(g) with the repre sentatIniongeneral, let is an= isomorphism of the representV,ati=onI(x) I(x)) be a thatJordanHolderr seriandesthatof tthhee .Yt'represent (G)modulatioen I(x).of G There exi s t s a uni q ue i n dex such in is spherical. Then this representation is isomorphic to I(x)= I )I)( ) (w provided it is irreducible. 144). bounded G P. CARTIER
2
(nor, Vr). The family of representations {(nx , Vx) } x ccx is called the
We summarize now the main properties of the spherical principal series ; they are immediate corollaries of the results obtained so far. (a) (nx, Vx) Vx
(b)
(nx , , Vx,)
X
x
nx(K) '
Fx .
M.
(nx , Vx) W
w
·
(c)
(x
().! x ,
=
,_.
(vx,
JK
(nx, Vx).
(d)
0
V0
c
V1
c
·· ·
c
V,_1
c
j
Vj/ Vj I
(nx, Vx).
I �j �
The last two statements come from the fact that (j)K.x is, up to constant multiples, the unique function in invariant under vxCK) and from the relation (34) which can be expressed as rx (j)k x · REMARKS. ( 1 ) It is easy to show without recourse to Satake's Theorem 4. 1 that the representation (vx, x) is spherical Since then the representations ().!x, (x and (vw · x• l · x ) ) are equivalent for any w in W, this provides another proof of step (B) in Satake's theorem (see criterion 3 . 1 0, p. (2) We refer the reader to Macdonald [37, p. 63] for a characterization of the spherical functions, that is the spectrum of the Banach algebra of inte grable functions on which are hiinvariant under K. It does not seem to be known which spherical functions are positivedefinite, or stated in other terms, which spherical representations are preunitary. The following result is due to Macdonald [36], [37].
4.5. The explicitformulafor the sphericalfunctions. he unramifi in the val4.u4e. ofSuppose the spherithatcaltfuncti onI eisd gicharact ven asefolr loofws: is regular. For any m = o(m) Iz w x) x(m) where q(w)i, THEOREM
M,
(42)
(43)
X
Fx
Fx( )
Q I
I; c (
wEW
·
M
111
w ·
Q = I; wEW
a
forwhicanyh areregulposiartivunramifi er of (product extended over the roots in (j)0 e on theedconicharact cal chamber A
M
'6').
We sketch the proof given by Casselman [18], which rests on the properties of the intertwining operators. We use the notations of §3.7.
REPRESENTATIONS OF j.JADIC GROUPS
1 53
For each w in the linear form Lw on I(x) transforms by M according to the character o l 1 2(w · and is trivial on I(x)(N) Since X is regular, these characters of M are distinct ; hence the linear forms Lw are linearly independent. The vector space I(x ) 8 is supplementary to I(x )(N) in I(x) and its dimension is equal to Hence there exists i n I(x) B a basis Uw .x}w E W characterized by
x)W,
.
I WI .
LwUw• , z) = I if w' = w, = 0 if w' #: w .
(45)
As a corollary of Theorem one gets WK x = .E wEW cJx) ·fw. ·;: On the other hand, one has Fz (m) = fK WK. z (km) dk. Sincefw. z is invariant under ).i z (B ) and B is a subgroup of one gets from these remarks the relation
3. 9
K,
(46)
.
where 1 4 gw = fJ.(BmB)  1 n(I8m8) fw. z · By the methods used in Theorem 2. 3, one proves, in general, that, for any admissible representation (n, of G and any m in M, the operator n(lB mB) fJ.(BmB)n(m) maps into (N) From (45), one infers (47) gw = o (m) 1 1 2 w · x (m) w . •
V V V) .

f z From (46) and (47), one deduces that, on M, the spherical function Fz agrees 2
with a linear combination of the characters o l l (w · ) of M. Taking into account the invariance property rw· x = rx , i t suffices t o calculate one o f these coefficients, for instance the coefficient of o 1 2(Wo · X ) where Wo is the (unique) element of which takes the conical chamber C(l into its opposite ((5 (or any positive root in r/)0 to a negative root) . In this case, one proves without difficulty that fwo. x is equal to the function r/)wo. x defined by formula (28) in § 3 . 6 . The soughtfor coefficient is obtained by multiplying cwo<wo · x) by
x
1
sK fwo,
x
(k) dk
=
JK Ww0, x
(k) dk
=
W
f.J.(Bw0B) .
It remains to show that the measure of Bw0B, that is the index (Bw0B : is equal to Q 1 . This follows easily from the formulas (2 1) to (24) in § 3 . 5 . Q.E.D. Note added i n proof As I was told by the editors, my conventions about alge braic groups differ slightly from those of other authors in these PROCEEDINGS. The local field F being infinite, the set G(F) of Fpoints of any of the algebraic groups G used in the previous paper is Zariskidense in G and I allowed myself to identify G to G(F). " The Haar measure fJ. on G is normalized by fJ.
(K)
K),
=
1.
REFERENCES 1. I. N. Bernshtein , All reductive padic groups are tame, Functional Anal . Appl. 8 ( 1 974), 9 1 93. 2. I . N. Bernshtein and A.V. Zhelevinskii, Represelltations of the group GL(n, F), Uspehi Mat. Nauk 31 (3) ( 1 976), 570 Russian Math . Surveys 31 (3)(1 976), 1 68. 3. , Induced representations of the group GL(n) over a padic field, Functional Anal. Appl. 1 0 ( 1 976), 747 5 . 4. A. Borel, Admissible representations of a semisimple group over a local field with vectors fixed under an lwahori subgroup, Invent. Math. 35 (1 976), 233259. =

1 54
P.
CARTIER
5. A. Borel, Automorphic Lfunctions, these PROCEEDINGS, part 2, pp. 276 1 . 6. N. Bourbaki, Mesure de Haar ; Convolutions et representations, Integration, Chapitres 7, 8 , Actu. Sci . Indust. n o . 1 306, Hermann , Paris, 1 963. 7. F. Bruhat, Sur les represen tations induites des groupes de Lie, Bull. Soc. Math. France 84 ( 1 956), 97205 . 8.  , Sur les represenlalions des groupes classiques 'padiques. I, II, Amer. J. Math. 83 ( 1 96 1 ) , 3 2 1 3 3 8 ; 343368. 9.  , Distributions sur un groupe localement compact et applications a ! 'etude des represenla tions des groupes 'padiques, Bull. Soc. Math. France 89 ( 1 9 6 1 ) , 437 5 . 1 0 .  , Sur une classe de sousgroupes compacts maximaux des groupes de Chevalley s u r u n corps 'padique, I n s t . Hautes Etudes Sci . Publ . Math. 2 3 (1 964), 4674. 1 1 .  , 'padic groups, Algebraic Groups and Discontinuous Subgroups, Proc. Sympos. Pure Math . , vol . 9, Amer. Math. Soc . , Providence, R.I., 1 966, pp. 6370. 12. F. Bruhat and J. Tits, Groupes algebriques simples sur un corps local, Proc. Conference on Local Fields (Driebergen , 1 966), SpringerVerlag, Berlin, 1 967. 13.  , Groupes reductifs sur un corps local. Chapitre I , Donnees radicie/les valuees, Inst. Hautes Etudes Sci . Publ. Math. 41 (1 972), 525 1 . 14. P. Cartier, Sur les representations des groupes reductifs 'padiques et leurs caracteres, Semi naire Bourbaki, 28e annee, 1 975/76, expose 47 1 , Lecture Notes i n Math . , vol . 567, Springer Verlag, Berlin, 1 977. 15.  , Representations ofGL, o ver a local field, lectures at the I . H . E . S . , Spring term 1 979. 16. W. Casselman, An assortment of results on representations of GL , (k) , Modular Functions of One Variable, II. Lecture Notes in Math., vol. 349, SpringerVerlag, Berlin, 1 97 3 , pp. 1 54. 17.  , Introduction to the theory of admissible representations of 'padic reductive groups (to appear) . 18.  , The unramified principal series of padic groups. I (to appear) . 19. J. Dixmier, Les C*algebres et leurs representations, GauthiersVillars, Paris, 1 969. 20. F. Gantmacher, Canonical representation of automorphisms of a complex semisimple Lie group, Mat . Sb. 47 ( 1 939), 1 0 1  1 46. 2 1 . P . Gerardin, Construction de series discretes padiques, Lecture Notes i n Math . , vol . 462, SpringerVerlag, Berlin, 1 97 5 . 2 2 .  , Cuspidal unramified series/or central simple algebras o ver local fields, these PROCEED INGS, part 1 , pp. 1 5 7 1 69. 23. R . Godement, A theory of spherical functions. I, Trans. Amer. Math . Soc. 73 (1 952), 496556. 24. J. Green, The characters of the finite general linear groups, Trans . Amer. Math . Soc. 80 ( 1 955), 402447. 25. BarishChandra, Harmonic analysis on reductive Padic groups, Harmonic Analysis on Homogeneous Spaces, Proc. Sympos . Pure Math., vol . 26, Amer. Math. Soc . , Providence, R . I . , 1 973, p p . 1 671 92. 26.  , The characters of reductive 'padic groups (to appear). 27.  , The Plancherelformula for reductive 'padic groups (to appear) . 28.  (Notes by G. van Dij k), Harmonic analysis on reductive 'padic groups, Lecture Notes in Math . , vol . 1 62, SpringerVerlag, Berlin, 1 970. 29. R . Howe, The Fourier transform and germs of characters (case of GL. o ver a padic field), Math. Ann. 208 ( 1 974), 305322. 30. N. Iwahori , Generalized Tits systems (Bruhat decomposition) on 'padic semisimple Lie groups, Algebraic Groups and Discontinuous Subgroups, Proc. Sympos. Pure Math. , vol. 9, Amer. Math. Soc., Providence, R . I . , 1 966, pp. 7 1 8 3 . 31. N. Iwahori a n d H. Matsumoto, On some Bruhat decompositions and the structure of the Heeke ring of the 'padic Chevalley groups, Inst. Hautes Etudes Sci . Pub I. Math. 25 (1 965), 548. 32. H . Jacquet, Representations des groupes lineaires P adiques, Theory of Group Representa tions and Harmonic Analysis, (C. I . M . E . , II Cicio, Montecatini terme, 1 970), Edizioni Cremonese, Roma, 1 97 1 , pp. 1 1 9220. 33.  , Sur les representations des groupes reductifs Padiques, C. R. Acad. Sci . Paris Ser. A 280 ( 1 975), 1 27 1  1 272.
REPRESENTATIONS OF pADIC GROUPS
1 55
34. H. Jacquet and R. P. Langlands, A utomorphic forms on G L(2), Lecture Notes in Math. , vol. 260, SpringerVerlag, Berlin, 1 972. 35. R. P. Langlands, Problems in the theory of automorphic forms, Lectures i n Modern Analysis and Applications. III, Lecture Notes in Math . , vol. 1 70, SpringerVerlag, Berlin, 1 970, pp. 1 861 . 36. I. G. Macdonald, Spherical /unctions on a padic Chevalley group, Bull. Amer. Math. Soc. 74 ( 1 968), 520525. 37.  , Spherical functions on a group of padic type, Ramanujan Institute, Univ. of Madras Pub! . , 1 97 1 . 38. I. Satake, Theory of spherical functions on reductive algebraic groups over Padic fields, Inst. Hautes Etudes Sci . Pub! . Math. 1 8 ( 1 963), 1 69. 39. G. Shimura, Introduction to the arithmetic theory of automorphic functions, Princeton Univ. Press, Princeton, N.J., 1 97 1 . 40. T. Shintani, On certain square integrable irreducible unitary representations of some padic linear groups, J. Math. Soc. Japan, 20 ( 1 968), 522565. 41. T. A. Springer, Reductive groups, these PROCEEDINGS, part 1, pp. 327. 42. J. Tits, Reductive groups o ver local fields, these PROCEEDINGS, part 1, pp. 2969. JNSTITUT DES HAUTES ETUDES SCIENTIFIQUES, 9 1 440 BURES SUR YVETTE, FRANCE
Proceedings of Symposia in Pure M athematics Vol. 33 ( 1 979), part I , pp. 1 571 69
CUSPIDAL UNRAMIFIED SERIES FOR CENTRAL SIMPLE ALGEBRAS OVER LOCAL FIELDS
PAUL GERARDIN
1. Statement of the theorem.
1 . 1 . Let A be a central simple algebra over a nonarchimedean local field F. Call
E an unramified splitting field for A, and r the Galois group of E over A quasi character of Ex is called regular if all its conjugates by the action of r are distinct. For any quasicharacter X of Fx , let XA I F and XEI F be respectively the quasi characters of Ax and EX defined by : XA I F = x o NAI F and XEI F = x o NEI F• with NAI F the reduced norm of A over and NE 1 F the norm of E over F. For any nontrivial character ¢ of F, the characters rf;AI F = rf; o TAI F of A and ¢E1 F = rf; o TEl F of E are nontrivial : here, TA 1 F is the reduced trace of A over F, and TE 1 F the trace of E over F. Let be the rank of A , that is the degree of E over F. 1 .2. We have the following theorem :
F.
F,
n the above notastsiiobns,le forrepresent any reagtiulonar quasiofcharactsucher()that:of there is a welthlederepresent finedWitihrreduci b l e admi ationsif and onlyandif0 andassoci aconjugate ted to twounder regulF;ar quasicharacters and 0tharee restequiricvtalent 0 are ion of () to the center of is given by the quasicharacter anyto quasicharacter of the twisted representation ® is equivalforthente cont ragredion oefnt() represent ation of () is equivalent to i s 1; tthhee Ltffauncti Moreover, whenctor oisfnot() a diisvision algebra, the representation() is cuspidal. THEOREM.
0
(b)
X E Fx
(c)
(d) (e) (f)
AI F
0AI F
(a)
I>
0
AI F (  1 ) Cnl) val x ()(x) ; (() X EI F) A I F ;
A
()AI F
X
Fx
EX,
Ax
Ax
()A I F
FX ,
AI F AI F t( 8 AI F• cp )
AI F
=
XA I F
(()1)A 1 F ;
(  1 ) Cn l ) ord ¢ c ( 8 , rf;EI F) . AI F
Here, cuspidal means that the coefficients of the representation are compactly supported modulo the center. 1 . 3 . Recall how the £function and tfactor are defined [6]. The reduced trace A I F defines a nondegenerate bilinear form on A by its value on the product of two the mapping elements in hence, for any nontrivial additive character rf; on
T
A;
A MS (MOS) subject classifications ( 1 970) . Primary 22E50, 20G25.
F,
© 1979, American M athematical Society
1 57
158
PAUL GERARDIN
a, b e A
,_.
¢A1 p(ab) = cp (TA 1 p(ab)) ,
identifies A with its Pontryagin dual group ; the corresponding selfdual measure on A will be written dA .
1
a quasicharacter () of E x defines an ndimensional representation of WE1 F by in duction ; but WE1 F is a quotient of Wp , so this gives an ndimensional representa tion of Wp, denoted Indk 0. Moreover, this representation is irreducible if and only if all the transforms of () by F are distinct, that is for () being regular. In general, let F0 be the stabilizer of () in r, and E0 be the corresponding subfield of E; there
1 59
CUSPIDAL UNRAMIFIED SERIES
are = [E : £0] quasicharacters 0' of E0x such that 0 = OF:tEo , and each of them is regular with respect to F; so the representation Indk 0 is the direct sum of the n0 representations Indf0 0', each one being irreducible. This gives part (a) of the fol lowing proposition. For part (e), we use the property of inductivity of the £..func tion ; for part (f), there is only a degree zero inductivity for the efactor, and the factor ).(E/F, ¢) is computed from the identity e(IndfO, ¢) = ).(E/F, cp)e(O, ¢E 1 F) by taking for 0 the trivial character, so that Indfl is the regular representation of F, that is the sum of the n unramified characters �+ 'val x where ' are the n different nth roots of 1 ; so :
n0
IT e('val , ¢) c
=
x
).(E/F, ¢) e(l , ¢E l F) ;
but e('val , ¢) = (' q l iZ)ord ¢ and e(l , ¢E I F) = n ord ¢12 , if q is the order of the residue field F of F. This gives the formula }.(E/F, ¢) = (  l) l is the group ( l + Pk)/( 1 + A, :;6 A ,_1 where it is the central extension : =
r
p£)
� 'Pk/P£ , unless
r
=
E r")
is odd and
0  Pk/P£  K,/Kt  A �1 (p•"ft;J")  0 ,
defined by the 2cocycle
TA,;E(ab) E PF1/p£. r O,x'£J8,. r r (} £,_1 PF1 x) f/l'i18,jx) x E p 1 £ f/l'iJ8,_1( TA,tEab) r', r" a, bE Forl on eachand such thattheretheis resta uniricqtueioncltoass of irreduciis isbotleypirepresent aby(},; tions ,;de�fiofLEMMA t r i v i a c , v en gi a representonlyaton(}, ion andofis.fixbyed by!C,(txhe) actix5fon:toEf,(anyx),;;(x),s Er. then, the classneofth,;en, depends r" r r a, bE
A �1(p •'j p•")
�+
3.3. For each � l , choose a character (}, of Ex with conductor equal to Max7 e r,a(r), such that (}j(}, is fixed by the group r, : this means that there is a For l , the character (} 1 defines a quasicharacter x
l,
A:,
E x n K,
K,
=
K,
X
K, ;
this is clear. In the remaining cases, we use r ' and from as above ; we introduce the subgroup H(r) of K,/Kt defined by the central extension by p£ 1/p£ of A �1(p• 'Jp•") given by the same cocycle ; it is a Heisen berg group, and if K(r) is its pullback in K., then 1 + Pk' normalizes K(r) and we have K, = (1 + Pk)K(r) . The first part of the lemma comes from a wellknown property of Heisenberg groups ; for the second, we observe that the restriction of =
CUSPIDAL UNRAMIFIED SERIES
161
� to K(r) factors through any group H5(r ) , 1 � s < r, central extension by !JE:;1 /PE:, of A�1 (p''/p'") obtained from H(r) by the map TE!E, on its center :
11:
0 > p�;;J PE:, > H, (r) > A � 1 (p''/p'") > 0
defined by the 2cocycle a, b E A �1 (p''/p'")
>+
TA,!E,TA,!A,(ab) E PE:;1/ PE:,
c
p£';/ !JE:, ;
it suffices now to observe that the action of A; by conjugation on A, fixes the form u E A, >+ TA,IE,TA,IAP · 3 .4. More precisely, for r > s > 1 , the action of on gives a trivial action on H.(r) . For s = 1 , the group H1(r) is a central product of the group's H1(r) inverse images of the subspaces rA1(p''/p'") + r 1 A1(p''/p'"), the group H1(r) is a cen tral extension of this subspace by lJk/lJE: 1, the 2cocycle being given by a, b E rAl( P'' /p'") ...... TA 11E1(ab) for r rl = r 1 rh a, b E rAl(lJ''fp'") + r 1 Al(J.1''/p'") >+ 1;E1(a'b" + a"b')
K. K, TA
, with a = a' + a", b = b' + b", a', b' E
rA1(p''/p"'), a", b" E r 1 Al( P''/p'"), in general ; the action of K1 on K, gives an action of KdKt = A [ on each of these subgroups H1(r) ; for rF1 = r 1 rh the above 2cocycle is a unitary form on the £1 vector space rA1(p'' f p'") with respect to the field E� of fixed points of r in E1 o and A 1 preserves this form ; from [5, Corollary 4. 8 .2] , there is a canonical extension to A[ >4 H1(r) of any irreducible representation ��:; of H1(r) which is given on the cen ter b� c/J'k{IE r  1 ; for a r E r;1 with rr1 '# r 1 rh the group A [ leaves stable each of the A1 submodules of the quotient of H1(r) ; hence any irreducible representation 11:7 of this group which is given on lJE:;1/pE;1 by ¢)!{1E , _ 1 has a canonical extension to the group A1 � H1(r) ([5, Proposition 1 .4], in fact, the character of the restriction to A [ is positive). The product of the representations 11:� defines a representation 11: � of as in the lemma, and the product of the extensions defines an extension of 11: � to the group Now, the representation of this group obtained from by twisting with x < r l as in the lemma is, up to equivalence, independent of the choice of cp < r l . For r = 1 , there is a unique class of irreducible representations 11:� of K1 which is trivial on and such that its tensor product with the pullback of the Steinberg representation of A [ is the representation induced by the onedimensional repre santation tx >+ {} 1 (t), t E it is the representation x E Kt, of the subgroup RE1(01) of [2, Theorem 8 . 3], up to a sign (  l)n 1\ n1 = [E : Ed, and it is a cuspidal representation. Define then ��:1 as the twist of 11:� by x m :
i�i; K,
K1K,.
Kt
i,
@'i;Kt:
@�,
its class does not depend on the choice of x m . Finally, the representation ��:0 of = Fx is the restriction of {} to
Fx . PROPOSITION 2. A regular quasicharacter {} of Ex defines canonically a class of ir reducible representations of by the formula Ko(xox1 · · · x, · · · ) = Ko(xo) ® 11:1(x1) ® · ® i rCx 1 )��: , (x,) ® · · ·
K8
K0
· ·
PAUL GERARDIN
1 62
for x, in Kn where K , is defined as abo ve, ii: , is its canonical extension to K1 for r odd > 1 with A, of. A ,_J, and if, is the trivial representation for the other r > 1 .
4. Commuting algebra.
4. 1 . Let 'JCo be the representation of A x induced by the representation teo of K0 in functions on Ax with compact support modulo the center px . The commuting al gebra is identified with the set of applications from Ax to the space of teo which transform on both sides under the action of K0 according to the representation te0 . For a quasicharacter (} of E x such that any character (}j(}r, for r E r, r # I ' has conductor I , the representation teo is obtained by twisting from a cuspidal ir reducible representation te ' of A. x ; if T is a maximal diagonalizable subgroup of A x such that A = K0 TK0, then a function ¢ in the commuting algebra i s determined by its values on T ; for a t E T not in K0, there exists a unipotent radical U of a proper parabolic subgroup of A x such that t(U n K)tl is contained in the kernel of K0 > A x ; we have then, for u E n K: rp(tu) = Ko ( t )rp(u ) = rp(tut 1t) = rp(t) ; hence, from the cusp property of te ' , ¢(t) 0. This shows that the commuting al gebra is reduced to the scalars, and .,.8 is irreducible ; its coefficients being com pactly supported modulo the center, the representation .,.8 is cuspidal too. Let now r be such that r = M axr ,a( r) > 1 . Let ¢ be a nonzero function in the commuting algebra of the representation 'JCo ; choose a in A x with ¢(a) of. 0. Sup pose we know that a E K0a,K0 for some a, in A;' ; as the values of ¢ are determined by its values on a set of representatives modulo K0 on both sides, we replace a by a,. From the definition of te0, its restriction to the subgroup 1 + A ,(p'") of K, (r " is the smallest integer with 2r " ;;;;: r ) , is a scalar operator, defined by the restriction of K n that is
4.2.
U
=
4. 3 .
1 + X E 1 + A ,(p'")
I+
x5r/;E,( l + x) (} ,( l + TA,!Ex) .
As ¢(a) is nonzero, we have, for X E A,(p'") n a1A,(p'") a, the identity : (} ,( 1 + TA,!Ex)
=
(} , ( 1 + TA,;E (axa 1) ) .
Choose a character a of E such that a( t) = 8 , (1 + t ) for t E pjf' ; then the character of A, defined by x �+ aA,IE (axa 1 )aA ,;E (x)  1 is trivial on the intersection a1 A ,(p'")a n A,(p'") so it can be written as IJ(axa1 ) 1 �(x) with two characters � and 1J of A , trivial on A,(p'"), so that aA,IE (x) � (x) = aA ,;E (axa 1 )1J(axa1 )
for any x E A , .
4.4. Fi x a character ¢ of E, with order  r ; the restriction to PE l of the character a is trivial on PE, and its stabilizer in r, is r,_ 1 ; so, there is an element r in the multiplicative pullback of £x in Ex such that E,(r) = £,_ 1 and a(t) = ¢E;E, (r t) for t in p£1 . LEMMA 2 . With these notations, then
(a) any character of A, equal to aA,IE , on A,(p'") has a conjugate by an element of
K, which is equal to
x �+
¢A,;E, (rx) on A ,_ 1 (p'") + A�1 ;
(b) if an element of A;:' conjugates two characters of A , equal to x on A,_ 1 (p• ") + A�1 , then this element belongs to A;:'_ 1 .
�+
aA,IE, (rx)
The proof uses the identification of A, with its group of characters obtained from
CUSPIDAL UNRAMIFIED SERIES
I63
is the conjugation. the application a, b E A , ,__. a A tE (ab) , where the action o , , Part (a) is easy, and for part (b), we have to show that an element b E such that b(1: + X) b 1 = 1: + X' with X, Y in , 1 (l.Y' ) , r ' + r " = r, is in fact in as X and Y centralize 1:, taking the successive powers qn m of both sides, we get at the limit m infinite, the equality b 1: b 1 = 1:, so b centralizes the center of hence is itself in A;_1 • We apply this lemma to the above situation : the characters of a Ar!E r ' and aA tE r;, are conjugate by suitable elements from K, to characters equal to x ..... , , part (b) of ¢ A ,td7: x) on A,_1 ( p'") + A;1 ; as they were conjugate by a E Lemma 2 shows that a belongs to K,A;_1 K,. As for r large, = A , we have shown that the relation ¢(a) # 0 implies a E KAtK. If A1 = E, then a is in K0, so ¢(a) = Ko (a) ¢ ( I ) , and ¢ ( 1 ) is an intertwin ing operator on the representation Ko ; the irreducibitity of Kn implies that this operator is scalar, and the representation is irreducible, and cuspidal for its coef ficients are compactly supported modulo the center. Suppose now that A1 is not £. Let a be an element of At which is not in K0, i.e. not in K1 Fx .Then, there is a unipotent radical U of a proper parabolic subgroup of such that a(U n K1)a 1 is in Kt, and even in the kernel of any "' for any r � I , for the determinant i s trivial on any unipotent element. For each r E F with a(r) r, an odd number > 1 , the group U n K1 acts on the£1 vector space r A1(�r' / p'") + r 1 A 1 (p''f p'") ; its fixed points form a subspace which can be lifted as a commuta tive subgroup Wr of H1 (r) (notations as in and there is a subgroup Wr of unipotent elements in K, with projection Wr on H(r). For r even, or without r such that a (r) = r , define W, = I ; for r = I, put W1 = U n K1 ; for the other r ' s , W, is the product of Wr for r with a(r) = r, taken in a chosen order, and without repetition. Then, for any r, a w, a 1 is in K;; and made of unipotent elements so is in the kernel of "o · For x = w 1 • · · w, · · · , w, E W, we have :
A, A: A:_1: £,_1 A,h A, A;,
A_
4. 5 .
A,
A{ =
3.4),
From the next lemma, it results that the operators Jw, K,(w,) dw, are fixed by
K,(w 1 ), so that the average
is equal to
Sw r
..
W,
··
K 1( w1 ) ®
·· ·
® IC ,(wi)K, ( w,) ®
·· ·
dw1
•..
dw, . . .
which is from the cusp property of K 1 . The lemma is the following, with the same notations as in [5] :
0 3. Let be afinite dimensional vector space) nontrivial overfinite field, and be an irreducible representation of the Heisenberg group on the center; let U be the unipotent radical of a proper parabolic subgroup of G( and W the space of its fixed points in then W has a lift in as a subgroup, and the LEMMA
(a)
r;
V
H( V
V
x
V* ;
V),
H( V)
operator L: w r;(w) isfixed by the action of U through the Wei/ representation wv ; (b) let F be a K/kunitary vector space over a finite filed, and r; be an irreducible
164 represent at tUionbeofthethunie associ atradi ed Heicalsofenberg groupparaboli H(F, ci)subgroup which is ofnontriU (tvhiealuniontathery cent e r; l e p ot e nt a proper group i)),asanda subgroup, W the corres pthondie operat ng tootrally is7Jot(rw)opiiscfixed subspaceunderin tF;he actthenionWofhasU athlrough ift inU(F,H(V) and the Wei/ representation W (i!ij7l:Li!i) i x
1\
Fx
=
w}.
The fibres of XL.h + XL,h1 have a property similar to that in the Lemma in §2. This allows us to define H,{XL) = inj limhH;(XL,h) as in §2. We also define H;(X) = EBL H;(XL). Similar arguments should apply in the general case.
4. Let G, T, B, U be as in § 1 . Assume that G comes from a Chevalley group over Z by extension of scalars so that G((!J), G(i!i) are well defined. Let Gh = G(iii/7l: hi!i).
Assume that T(K ) c: G(i!i). Let th, Oh be the images of T(K ), U(K ) n G(i!i) under + Gh. F: G( K ) + G(K ) induces F: G h + Gh. This gives a k rational struc ture on the kalgebraic group Gh. Define
G(i!i)
xh = {g e Gh 1 g1 F(g) e Oh}l oh n F1 oh.
The finite group G f x Tf acts on Xh, as before, by left and right multiplication. For each character () : Tf + Q/1 we form R0 = I; (  l )•H;(Xh)o. i
± R8
It is independentIf()ofisthsueffichoicientcelyofregular, the virtual Gfmodule is irreducible. h THEOREM.
In the case
B.
= 1 , this follows from [1].
REFERENCES
1. P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. (2) 103 ( 1 976), 1 031 6 1 . 2 . P . Gerardin, Cuspidal unramified series for central simple algebras over local fields, these PROCEEDINGS, part 1 , pp. 1 57169. UNIVERSITY OF WARWICK, COVENTRY, GREAT BRITAIN
Proceedings of Symposia in Pure Mathematics Vol . 33 ( 1 979), part 1 , p p . 1 79 1 8 3
DECOMPOSITION OF REPRESENTATIONS INTO TENSOR PRODUCTS
D. FLATH I. In this paper some generalizations of the classical theorem which classifies the irreducible representations of the direct product of two finite groups in terms of those of the factors will be discussed. The first generalization consists in expanding the class of groups considered.
1. Let be locally compact total y disconnected groups and let If ssibisleanirreduci admisbsilbelrepresentati e irreducibleonrepresentati on of 2, then is an(1.1.1)admi of If ibsleanrepresent admissibalteioinsrreduciofble represent ation of then tThehereclexiassesst admi s sibarele 2ir)detreduci such that o f the ermined by that of e ke algebra K K). K Irreduci bilitbyleCriterioK)modul n. smoote forh all modul e i s i r reduci b l e if and onl y if is an i r reduci compact open subgroups K of G
THEOREM = G 1 x G2 . ( 11:1
Gb G2
G;,
G.
71:;
11:.
=
I,
11: 1 ® 11:2
G, 11: � 11: 1 ® 11:2 •
11:
71:;
i
G;
We recall some notation. For a locally compact totally disconnected group G, the H e H(G) of G is the convolution algebra of locally constant complex valued functions on G with compact support. For a compact open subgroup K of G, let ex be the function (meas K) l . chx, where chx is the characteristic function of and meas is the Haar measure on G which has been used to define convolution in H(G). Then ex is an idempotent of H(G). The subalgebra exH(G)ex of H(G) will be denoted H(G, A smooth Gmodule W is in a natural way an H(G) module, and for every compact open subgroup c G the space wx = ex W is an H(G, K)module. Before proving Theorem I we state an A G W wx H(G, G. PROOF. This follows from the fact that if U is an H(G, K)submodule of w x , then ( H(G) · U)X = U. 0 Remark that in applying the irreducibility criterion it is sufficient to check that wx is an irreducible H(G, K)module for a set of which forms a neighborhood base of the identity in G.
K of suchThenthat K) is com mutative, and letLet Kbebeanaadmicompact ssible open irreducisubgroup ble Gmodule. CoROLLARY.
G
W
H(G, dim wx � I .
A MS (MOS) subject classifications ( 1 970). Primary 22E55 .
© 1 9 79, American Mathematical Society
1 79
D . FLATH
1 80
PROOF OF THEOREM 1 . It is straightforward that (i) H(Gt x G2) � H(Gt) ® H(G2), (ii) H(G1 x G2, K1 x K2) � H(G�o K1 ) ® H(Gz, K2) and (iii) ( W1 ® W2) K 1 x K2 � W1Kt ® Wj ,u2 are described by ,u 1 (p), ,u2(p), where p is the generator of the prime ideal Let be an integer � I . We can define local factors of the form
kj;. p. n
n
Ln (np , s) = ( I  ,u�/P/ s) 1(1  .url.uz /P/') 1 . . . ( 1  ,u�/P/')1 . Here we are writing !1 1 > .uz for ,U l (p), ,uz(p) . For n, as above we can attach to prime p a local factor Ln (np , s ) , agreeing with the definition above at the unramified primes, such that the function Ln (n, s) = IT Ln (np , s) exists, and is "nice". For = I this conjecture is proved in the book by JacquetLanglands. For = 2, it has been proved by GelbartJacquet and Shimura. For = 4 it can be proved that L n (n, exists and has a meromorphic continuation. Beyond these cases, the situation is unresolved. Notes of this talk were made by Lawrence Morris (I. H. E. S.) and Ben Seifert (U. C. Berkeley).
Conjecture. n
n
every
n 3,
s)
n
REFERENCES I. E. Heeke, Mathematische werke, Vandenhoek and Ruprecht, Gottingen , 1 959. 2. H . Jacquet and R.P. Langlands, Automorphic forms on GL(2) , Lectu re No tes i n Math . , vol. 1 1 4, Springer, 1 970. 3. A. Borel, Introduction aux groupes arithemetiques, Hermann, Paris, 1 969 . 4. G. Harder, Minkowskische Reduktionstheorie uber Funktionen korpern, Inven t . Math. 7 ( 1 969), 3354. 5. A. Borel, Automorphic Lfunctions, these PROCEEDINGS, part 2, pp. 2761 .
UNIVERSITY OF TEL AVIV
Proceedings of Symposia in Pure Mathematics Vol. 33 ( 1 979), part I, pp. 1 89202
AUTOMORPHIC FORMS AND AUTOMORPHIC REPRESENTATIONS
A. BOREL AND H. JACQUET Originally, the theory of automorphic forms was concerned only with holomor phic automorphic forms on the upper halfplane or certain bounded symmetric domains. In the fifties, it was noticed (first by Gelfand and Fomin) that these au tomorphic forms could be viewed as smooth vectors in certain representations of the ambient group G, on spaces of functions on G invariant under the given discrete group r. This led to the more general notion of automorphic forms on real semisimple groups, with respect to arithmetic subgroups, on adelic groups, and finally to the direct consideration of the underlying representations. The main purpose of this paper is to discuss the notions of automorphic forms on real or adelic reductive groups, of automorphic representations of adelic groups, and the relations between the two. We leave out completely the passage from automorphic forms on bounded symmetric domains to automorphic forms on groups, which has been discussed in several places (see, e.g., [2], or also [5], [6], [15] for modular forms) . 1. Automorphic forms on a real reductive group. 1 . 1 . Let G be a connected reductive group over Q, the greatest Qsplit torus of the center of G and K a maximal compact subgroup of G(R). Let g be the Lie algebra of G(R), U(g) its universal enveloping algebra over C and (g) the center of U(g) . We let .Yf or .Yf(G(R), K) be the convolution algebra of distributions on G(R) with support in K [4] and A K the algebra of finite measures on K. We recall that .Yf is isomorphic to U(g) ®u(!) A K · An idempotent in .Yf is, by definition, one of A K , i.e., a finite sum of measures of the form d(0") 1 . Xu · where O" is an irreducible finite dimensional representation of K, d(O") its degree, Xu its character, and dk the normalized Haar measure on K. The algebra .Yf is called the Heeke algebra of G(R) and K. 1 .2. A norm 11 11 on G(R) is a function of the form llgll = (tr O"(g)* · O"(g)) 1 12, where O" : G(R) 4 GL(E) is a finite dimensional complex representation with finite kernel and image closed in the space End(£) of endomorphisms of E and * denotes the adjoint with respect to a Hilbert space structure on E invariant under K. It is easily seen that if r is another such representation, then there exist a constant > 0 and a positive integer n such that
Z
Z
dk,
C
A MS (MOS) subject classifications ( I 970). Pri mary J OD20 ; Secondary 200 3 5 , 22E5 5 . © 1 9 79, American Mathematical Society
1 89
A . BOREL AND H. JACQUET
1 90
ll xll a
(1)
�
C
·
l l x l l � , for all x
slowlny increasing
A function / on G(R) is said to be and a positive integer such that
G(R) , a constant (2)
C
I J(x) I
�
C
E G(R) .
· ll x l l n ,
if there exist a norm I I on
for all x E G(R) .
n
In view of ( l ) this condition does not depend on the norm (but does). REMARK. If (J : G + GL(E) has finite kernel, but does not have a closed image in End E, then we can either add one coordinate and det (J (g) 1 as a new entry, or consider the sum of (J and (J * : g ,__. (J(g 1 ) * . The associated square norms are then l det (J(g) 1 1 z + tr((J(g) * (J (g)) and tr((J (g) * (J (g)) + tr((J(g 1) (J (g 1 ) * ) . 1 . 3. Let F be a n arithmetic subgroup of G(Q). A smooth complex valued func tion f on G(R) is an automorphic form for (F, K), or simply for F, if it satisfies the following conditions : (a) f(r · x) = f(x) (x E G(R) ; r E F). (b) There exists an idempotent � E Jft'(cf. 1 . 1) such that f * � = .f (c) There exists an ideal of finite codimension of Z(g) which annihilates f: f * X = 0 (X E (d) f is slowly increasing ( 1 .2). An automorphic form satisfying those conditions is said to be of type (�, We let d (F, �. K) be the space of all automorphic forms for (F, K) of type (�, 1 .4. ExAMPLE. Let G = GLz , F = GLz(Z), K = 0(2, R), � = 1 , the ideal of Z(g) generated by ( C  A), where is the Casimir operator and A E C. Then f is an eigenfunction of the Casimir operator which is left invariant under right inva riant under K, and invariant under the center Z of G(R). The quotient G(R)/Z · K may be identified with the Poincare upper halfplane H. The function f may then be viewed as a Finvariant eigenfunction of the LaplaceBeltrami operator on It is a "waveform," in the sense of Maass. See [2], [ 5] , [6] for a similar interpreta tion of modular forms. 1 . 5. REMARKS. (1) Condition 1 . 3(b) is equivalent to : fis Kfinite on the right, i.e., the right translates off by elements of K span a finite dimensional space of func tions. These functions clearly satisfy 1 . 3(a), (c), (d) iff does. (2) Let r : K + GL( V) be a finite dimensional unitary representation of K. One can similarly define the notion of a Vvalued automorphic form : a smooth func tion ({! : G + V satisfying (a), (c), (d), where I I now refers to the norm in V, and
J).
J). J,
J
J).
J F,
C
X.
(b')
f(x · k) =
r(k1) • f(x)
(x E G(R) ; k E K) .
For semisimple groups, this is HarishChandra's definition (cf. [2], [11]). For E V, the functions x ,__. (({!(x), v) are then scalar automorphic forms. Conversely, a finite dimensional space E of scalar automorphic forms stable under K yields an £valued automorphic form. (3) A similar definition can be given if G(R) is replaced by a finite covering H of an open subgroup of G(R) and r by a discrete subgroup of H whose image in G(R) is arithmetic. For instance, modular forms of rational weight can be viewed as automorphic forms on finite coverings of SL2(R). v
191
AUTOMORPHIC FORMS AND AUTOMORPHIC REPRESENTATIONS
GrowtG,h condition.
1 . 6. ( 1 ) Let A be the identity component of a maximal Qsplit torus S of and gi/J the system of Qroots of with respect to S. Fix an ordering on gi/J and let L1 be the set of simple roots. Given > 0 let =
At
(1)
E A:
{a G
i a(a) i
Gt t, (a
;;:::
E J)} .
Let f be a function satisfying 1 .3(a), (b), (c). Then the growth condition (d) i s equivalent to : ( d') Given a compact set R c (R), and > 0, there exist a constant > 0 and a positive integer such that
m i · a) C :$;
if(x
(2)
t
· i a(a)!m, for all
a
E At,
a E J,
x
E R.
C
G' G'(R).
This follows from reduction theory [11, §2]. More precisely, let be the derived group of Then A is the direct product of Z(R)" and A' A n For a function satisfying 1 .3(a), (b), the growth condition (d') is equivalent to (d) for a E A ; ; but says nothing for E Z(R)". However condition (c) implies thatf depends polynomially on E Z(R), and this takes care of the growth condition on Z(R). (2) Assume f satisfies 1 . 3(a), (b), (c) and
G.
=
z
(3)
a
f(z .
=
x) x(z)f(x) (z
E
Z(R),
X E
G(R))
· F\G(R).
where X is a character of Z(R)/(Z(R) n r) . Then /// is a function on Z(R) If /// E LP(Z(R)F\G(R)) for some p � 1 , then fis slowly increasing, hence is an auto morphic form. In view of the fact that Z(R)F\G(R) has finite invariant volume, it suffices to prove this for = 1 . In that case, it follows from the corollary to Lemma 9 in [11], and from the existence of a Kinvariant function a E ";' ( (R) ) such that f = f * a (a wellknown property of Kfinite and Z(g)finite elements in a dif ferentiable representation of G(R), which follows from 2. 1 below) .
p
1 . 7. THEOREM [11, THEOREM 1 ] .
The space
.sd(F, �,
CG J, is finite dimensional. K)
This theorem i s due to BarishChandra. Actually the proof given in [11] is for semisimple groups, but the extension to reductive groups si easy. In fact, it is im plicitly done in the induction argument of [11] to prove the theorem. For another proof, see [13, Lemma 3.5]. At any rate, it is customary to fix a quasicharacter K ) x of elements in X of Z(R)/(F n Z(R)) and consider the space .sd(F, �, .sd(F, �, K) which satisfy 1 .6(3). For those, the reduction to the semisimple case is immediate. Note that since the identity component Z(Rt of Z(R) (sometimes called the split component of has finite index in Z(R) and Z(Rt n F = { 1 } , i t is substantially equivalent t o require 1 . 6(3) for a n arbitrary quasicharacter of Z(R)". The space .sd(F, �, K) is acted upon by the center of by left or right translations. Since it is finite dimensional, we see that C(G(R))finite. 1 .8. A continuous (resp. measurable) function on is cuspidal if
J,
J,
is Cusp forms.
G(R))
J,
(1)
for all (resp. almost all)
S
(r n N (R) ) \ N (R)
x G(R), in
f(n·x)dn
=
C(G(R))any automor G(R), phic form G(R)
0,
where N is the unipotent radical of any proper
A. BOREL AND H. JACQUET
1 92
parabolic Qsubgroup of G. It suffices in fact to require this for any proper maxi mal parabolic Qsubgroup [11, Lemma 3]. A is a cuspidal automorphic form. We let o d(F, �. J, K) be the space of cusp forms in d(F, �. J, K). Let f be a smooth function on G(R) satisfying the conditions (a), (b), (c) of 1 . 3 . Assume that/is cuspidal and that there exists a character X of Z(R) such that 1 .6(3) is satisfied. Then the following conditions are equivalent : (i) / is slowly increasing, i.e. , / is a cusp form ; (ii) f is bounded ; (iii) I f I i s squareintegrable modulo Z(R) (cf. [11, §4]). In fact, one has much more : Ill decreases very fast to zero at infinity on Z(R)F\G(R), so that if g is any automorphic form satisfying 1 .6(3), then 1 / · g l is integrable on Z(R) F\G(R) (loc. cit.). The space o d(F, �. J, K ) x of the functions in °d(F, �. J, K) satisfying 1 .6(3) may then be viewed as a closed subspace of bounded functions in the space V(F\G(R))x of functions on F\G(R) satisfying 1 . 6(3), whose absolute value is squareintegrable on Z(R)F\G(R) . Since Z(R)F\G(R) has finite measure, this space is finite dimensional by a wellknown lemma of Godement [11, Lemma 1 7]. This proves 1 .7 for d(F, �. J, K) x when Z(R)F\G(R) is compact, and is the first step of the proof of 1 . 7 in general. is an arithmetic subgroup of G(Q), 1 .9. Let E G(Q). Then ar = and the left translation a by induces an isomorphism of d(F, �. J, K) onto d(ar, �. J, K) . Let .E be a family of arithmetic subgroups of G(Q), closed under finite intersection, whose intersection is reduced to { I } . The union d(.E, �. J, K) of the spaces d(F, �. J, K) (F E .2) may be identified to the inductive limit of those spaces :
cusp form
·r
·
1 a·F·al a
a
d (2, � . J, K ) = ind limr "'I d (F, � . J, K),
(1)
where the inductive limit is taken with respect to the inclusions ( 2)
j
rnr ' : d (F', �. J, K)
+
s:Y(F", �. J, K)
(F" c F')
associated to the projections F \ G(R) + F '\G(R). Assume .2 to be stable under conjugation by G(Q) . Then G(Q) operates on d(.E, �. J, K) by left translations. Let us topologize G(Q) by taking the elements of .2 as a basis of open neighborhoods of I . Then this representation is admissible (every element is fixed under an open subgroup, and the fixed point set of every open subgroup is finite dimensional). By continuity, it extends to a continuous ad missible representation of the completion G(Qh of G(Q) for the topology just de fined. For suitable .2, the passage to d(2, �. J, K) amounts essentially to considering all adelic automorphic forms whose type at infinity is prescribed by �. J, K; the group G(Q)z may be identified to the closure of G(Q) in G(A1) and its action comes from one of G(A1). See 4.7. 1 . 10. Finally, we may let � and J vary and consider the space d(.E, J, K) spanned by the d(.E, �. J, K) and the space d(4, K) spanned by the s1'(4, J, K). They are G(Q)zmodules and (g, K)modules, and these actions commute. Again, this has a natural adelic interpretation (4. 8) . Let Je"(G(Q), F ) b e the Heeke algebra, over C , o f G(Q) 1.11.
Heeke operators.
"
1 93
AUTOMORPHIC FORMS AND AUTOMORPHIC REPRESENTATIONS
G(Q)
mod F. It is the space of complex valued functions on which are hiinvariant under F and have support in a finite union of double cosets mod F. The product may be defined directly in terms of double cosets (see, e.g., [17]) or of convolution (see below). This algebra operates on d(F, t;, J, K). The effect of FaF e is given by ...... I: bE (rn•rJ 1r More generally, let Z) be the Heeke algebra spanned by the characteristic functions of the double cosets F'aF" (F', F" e Z, e [17, Chapter 3]. It may be identified with the Heeke algebra of locally constant compactly supported functions on This identification carries F) onto f), where f is the closure of F in [12] . The product here is ordinary convolution (which amounts to finite sums in this case). Since d(Z, t;, J, K) is an admissible module for the action of extends in the standard way to one of The space d(F, t;, J, K) is the fixed point set of f, and the previous operation of £'( F) on this space may be viewed as that of f). For an adelic interpretation, see 4.8.
.n"(G(Q), G(Q)z. Jf'(G(Q)z, G(Q)z, .n"(G(Q)z). G(Q), .n"(G(Q)z, G(R).
f a G(Q)) Jf"(G(Q),
ld
(a G(Q)) Jf'(G(Q)z) G(Q)z G(Q)z
2. Automorphic forms and representations of The notion of automorphic form has a simple interpretation in terms of representations (which in fact sug gested its present form). To give it, we need the following known lemma (cf. [18] for the terminology).
(n, V)
be s finisible.te and )fiLetnite Thenbethae diff smalerenti lest ablK)e represent submodulateioofn of G(R). containLeting is admi G(Rt G(R) Jf'oJf'o G(Rt, 2. 1 . LEMMA. Z(g
.
(g ,
V
ve V v
K
Indeed, £' · v is a finite sum of spaces · w, where £'0 is the Heeke algebra of the identity component of and K0 = K n and w is K0finite and · v is an admissible (g , K0)module. Z(g)finite. It suffices therefore to show that By assumption, there exist an ideal R of finite codimension of the enveloping algebra U(t) of the Lie algebra t of K and an ideal J of finite codimension of Z(g) which annihilate v and moreover U(t)/R is a semisimple tmodule. Then £'0 v may be identified with U(g)/ U(g) · R · J. By a theorem of HarishChandra (see [19, 2.2. 1 . 1]), U(g)/ U( g) · R is tsemisimple and its tisotypic submodules are finitely generated Z(g)modules. Hence their quotients by J are finite dimensional. 2.2. We apply this to acted upon by via right translations. Therefore, if f is automorphic form, then f * £' is an admissible £' or (g , K) module. This module consists of automorphic forms. In fact, 1 .3(a) is clear, and 1 . 3(b) follows from 2. 1 ; its elements are annihilated by the same ideal of Z(g) as v, whence (d). Finally, there exists a e such that / * a = f so tha t / * satis fies 1 .2(c) (with the same exponent as f) for all X e U(g) [11, Lemma 1 4]. Thus the spaces •
coo(F\G(R)),
C�(G)
G(R)
X
d (F, J, K) = Ze d(F, t;, J, K), are (g , K)modules and unions of admissible (g , K)modules. If f is a cusp form, then f * £' consists of cusp forms. Thus the subspace d(F, K) of cusp forms is also an £'module. If X is a quasicharacter of Z, then the space d(F, K)x of eigenfunctions for Z with character X is a direct sum of irreducible admissible (g , K)modules, with finite multiplicities. In fact, after a twist by l x l  1 , we may assume X to be unitary, and we are reduced to the Gelfand PiatetskiShapiro theorem ([7], see also [11, Theorem 2], [13, pp. 4142]) once
o
o
1 94
A.
BOREL AND H. JACQUET
we identify o d(F, K) to the space of Kfinite and Z(g)finite elements in the space
x o L 2 (F\G(R)) x of cuspidal functions in L2 (F\G(R)) x (see 1 . 8 for the latter).
3. Some notation. We fix some notation and conventions for the rest of this paper. 3 . 1 . F is a global field, OF the ring of integers of F, V or VF (resp. Voo, resp. V1) the set of places (resp. archimedean places, resp. nonarchimedean places) of F, F. the completion of F at v E V, 0 v the ring of integers of F. if v E V 1. As usual, A or AF (resp. A1) is the ring of adeles (resp. finite adeles) of F. 3.2. G is a connected reductive group over F, Z the greatest Fsplit torus of the center of G, £'. the Heeke algebra of G. = G(F.) (v E V) [4]. Thus £'. is of the type considered in § I if v E Voo and is the convolution algebra of locally constant com pactly supported functions on G(F.) if v E V 1. We set
£' = £'oo ® £'1, £'1 = ® £' v , v eV1 where the second tensor product is the restricted tensor product with respect to a suitable family of idempotents [4]. Thus £' is the global Heeke algebra of G(A) [4]. If F is a function field, then V00 is empty and £' = £' 1. If L is a compact open subgroup of G(A1), we denote by eL the associated idem potent, i.e., the characteristic function of L divided by the volume of L (relative to the Haar measure underlying the definition of Yf1). Thus f eL = f if and only iff is right invariant under L. The right translation by x E G(A) on G(A), or on functions on G(A), is denoted r" or r(x). 3.3. A continuous (resp. measurable) function on G(A) is cuspidal if (1)
Jf"oo = @V £'., vE
oa
*
J
N (F) \N (A)
f(nx) dn = 0,
for all (resp. almost all) x E G(A), where is the unipotent radical of any proper parabolic Fsubgroup of G. It suffices to cheek this condition when runs through a set of representatives of the conjugacy classes of proper maximal par abolic Fsubgroups.
P
N
4. Groups over number fields. 4. 1 . In this section, F is a number field. An element e if it is of the form
P
E
£' is said to be
� � E £'1, �oo idempotent in £'oo•
(1)
simple
We let Goo = Ti v E V oo G. and goo be the Lie algebra of Goo, viewed as a real Lie group. We recall that Goo may be viewed canonically as the group of real points H(R) of a connected reductive group H, namely the group H = RFIQ G obtained from G by restriction of scalars from F to Q. This identification is understood when we apply the results and definitions of §§ 1 , 2 to GooThe group G(A) is the direct product of Goo by G(A1). A complex valued function on G(A) is if it is continuous and, if viewed as a function of two arguments x E Goo, y E G(A1), it is c oo in x (resp. locally constant in y) for fixed y (resp. x). 4.2. Fix a maximal compact subgroup Koo of G00• A smooth function / on G(A) is a K00automorphic form on G(A) if it satisfies the following conditions :
smooth Automorphic forms.
195
AUTOMORPHIC FORMS AND AUTOMORPHIC REPRESENTATIONS
x)
(a) /(r /(x) ( r E G(F), x E G(A) ). (b) There is a simple element � E .Y"e such that/ * .; = f (c) There is an ideal J of finite codimension of Z(goo) which annihilates [ (d) For each y E G(A1), the function x >+ f(x · y) on Goo i s slowly increasing. We shall sometimes say that f is then of type (.;, J, K00). We let d(.;, J, Koo) be the space of automorphic forms of type (.; , J, K00). There exists a compact open subgroup L of G(A1) such that .;1 * .;L = .;1. Then d(.;, J, Koo) c d (.;oo * .;L, J, K00). We could therefore assume .;1 = .;L for some L without any real loss of generality. 4.3. We want now to relate these automorphic forms to automorphic forms on G=. For this we may (and do) assume .; = .; (8) �L for some compact open sub group L of G( A 1) . There exists a finite set C c G(A) such that G( A) = G(F) · C · G • [1]. We assume that C is a set of representatives of such cosets and is contained in G(A1). Then the sets G(F) · c · G= · form a partition of G(A) into open sets. For c E C, let ·
=
=
00
L
L
(1)
I t is a n arithmetic subgroup o f G00 • Given a function f on G(A) and c E C, let fc be the function x >+ f(c · on Goo . Suppose f is right invariant under Then it is immediately checked that f is left invariant under G(F) if and only if/c is left invariant under Fe for every c E C. More precisely, the map f >+ (/Jc E C yields a bijection between the spaces of functions on G(F)\G(A)/L and on llc(Fc\Goo). It then follows from the definitions that it also induces an isomorphism
x)
L.
d(.; oo (8) .;L , J, Koo) =::.... cEB d(Fc , � oo ' J, Koo) , EC
(2)
so that the results mentioned in § I transcribe immediately to properties of adelic automorphic forms. In particular, 1 . 6 , 1 .7 imply : (i) The space cc1"(�, J, K=) is finite dimensional. (ii) A smooth function / on G(A) satisfying 4.2(a), (b), (c), and f(z · x)
(3)
=
x(z) ·f(x)
(z E Z(A), x E G(A)),
for some character X of Z(A)/Z(F), such that 1/1 E LP(Z(A)G(F)\G(A)) for some ;;;; is slowly increasing. (iii) For a smooth function satisfying 4.2(a), (b), (c), the growth condition 4. 2(d) is equivalent to 1 .6(d'), where R is now a compact subset of G(A). (iv) Any automorphic form is C(G(A))finite, where C(G(A)) is the center of
p
I,
G(A) .
We note also that one can also define directly slowly increasing functions on + GLn with finite
G(A), as in 1 .2, using adelic norms : given an Fmorphism G kernel define, for x E G(A),
(4)
( or simpl y max lo (g0) 1 v if o(G) is closed as a subset of the space of n
x n matrices) . For continuous functions satisfying 4.2(a), (b), this is equivalent to 4.2(d).
196
A. BOREL AND H. JACQUET
4.4. A is a cuspidal automorphic form . We let o d(�. J, Koo) be the space of cusp forms of type (�, J, Koo). The group N is unipotent, hence satisfies strong approximation, i.e., we have for compact open subgroup Q of N(A1)
any
(2)
cuspform
N(A)
=
)
N(F) · Noo · Q ; hence N(F)\N(A � (N(F) n (Noo
G(A)
X
Q))\Noo
[1]. Let now f be a continuous function on which is left invariant under G(F) and right invariant under From (2) it follows by elementary computations that f is cuspidal if and only if the fc's (notation of 4.3) are cuspidal on Goo Hence, the isomorphism of 4.3(2) induces an isomorphism
L.
(3) so that the results of 1 . 8 extend to adelic cusp forms. In particular, assume that / satisfies 4.2(a), (b), (c), and also 4.3(3) for a character X of Z(A)/Z(F). Then the following conditions are equivalent : (i) fis slowly increasing, i.e. , / is a cusp form ; (ii) is bounded ; (iii) is squareintegrable modulo Z(A) · G(F). REMARK. In 4.3, 4.4, we have reduced statements on adelic automorphic forms to the corresponding ones for automorphic forms on G(R), chiefly for the conven ience of references. However, it is also possible to prove them directly in the adelic framework, and then deduce the results at infinity as corollaries via 4.3(2), 4.4(3). In particular, as in 1 . 8, one proves using (ii) above and Godement's lemma that o d(�, J, Koo) is finite dimensional.
fI f I
P ROPOSITION.
Let ftiobens aaresmootequihvfunct ion on satisfying Then(I)fithes folan lautomor owing condi al e nt: pe hiplacec form.v, the space ! * £. is an admissible £.module. For each i n fini t ForTheeach e space! £. is an admissible £.module. / * v is anthadmi spaceplace ssible* £module. 4.5. (2) (3) (4)
G(A)
4.2(a), (b), (d).
E V,
.Yt'
PROOF. The implications (4) => (3) => (2) => ( 1 ) are obvious ; (1) => (4) follows from 4. 3(i). 4.6. An irreducible representation of is (resp. if it is isomorphic to a subquotient of a representation of .Yt' in the space of automorphic (resp. cusp) forms on G(A). It follows from 4.5 that such a representation is always admissible. It will also be called an automorphic representation of G or G(A), although, strictly speaking, it is not a G(A)module. However, it is always a G(A1)module. More generally, a topological G(A)module E will be said to be if the subspace of admissible vectors in E is an automorphic representation of .Yt'. In particular, if X is a character of Z(A)/Z(F), any Ginvariant irreducible closed subspace of
phic Automor cuspipdhial)c representations.
.Yt' automor
automorphic
V(G(F)\G(A)) x = {I E V (G(F) ·
Z(A)\G(A)) , ,f(z x(z) z · X)
=
f(x), E Z(A), x E G(A))}
is automorphic in this sense, in view of [4, Theorem 4]. By a theorem of Gelfand and
AUTOMORPHIC FORMS AND AUTOMORPHIC REPRESENTATIONS
1 97
PiatetskiShapiro [7] (see also [8]), the subspace o V(G(F)\G(A)) x of cuspidal func tions of £2( G(F )\G(A)) x is a discrete sum with finite multiplicities of closed irreduci ble invariant subspaces. Those give then, up to isomorphisms, all cuspidal auto morphic representations in which Z(A) has character X · The admissible vectors in those subspaces are all the cusp forms satisfying 4.3(3). 4.7. Let L ::J L' be compact open subgroups of G(A1). Then �L �L ' = �L ; hence d(�oo ® �L ; J, Koo) c d(�oo ® �L' • J, Koo). The space ..91(�00, J, Koo) spanned by all automorphic forms of types (,;:00 ® ,;=1, J, Koo), with �� arbitrary in Jlt'1, may then be identified to the inductive limit d(,;=oo , J, Koo) = ind l i m d(�oo ® �L• J, Koo) , (I)
*
L
where L runs through the compact open subgroups of G(A1), the inductive limit being taken with respect to the above inclusions. The group G(A1) operates on Jlt'1 by inner automorphisms. Let us denote by x,;= the transform of � E Jlt'1 by Int x. We have in particular (2)
(x e G(A1), ,;=L as in 4. 1 ) ;
if/ is a continuous (or measurable) function on G(A), then (3) Therefore, G(A1) operates on ..91(�00' J, Koo) by right translations. It follows from 4.3(i) that this representation is admissible. In view of 4.3(3), ..91(�00, J, Koo) is the adelic analogue of the space d(Z, ,;=00, J, Koo) of 1 .9, where Z is the family of arithmetic subgroups of G(F) of the form rL = G(F) n (Goo x L), where L is a compact open subgroup of G(A1). These are the of G(F), i.e., those subgroups which, for an embed ding G 4 GLn over F, contain a congruence subgroup of G n GLn(OF). This analogy can be made more precise when G satisfies strong approximation, which is the case in particular when G is semisimple, simply connected, almost simple over F, and Goo is noncompact [16]. In that case, as recalled in 4.4(2), we have G(A) = G(F) · Goo · L for any compact open subgroup of G(A1), so that we may take C = { I } i n 4.3. Then 4.3(2) provides an isomorphism
congruence arithmetic subgroups
(4) for any L, whence (5) where Z is the set of congruence arithmetic subgroups of G(F). Moreover, the projection of G(F) in G(A1) is dense in G(A1) and G(A1) may be identified to the completion G(Fh of G(F) with respect to the topology defined by the subgroups n. It is easily seen that the isomorphism (5) commutes with G(Q), where, on the lefthand side x E G(Q) acts as in 1 .9, via left translations, and, on the righthand side, x acts as an element of G(A1) by right translations. It follows that the isomor phism (5) commutes with the actions of G(F)z = G(A1) defined here and in 1 .9. Also, the isomorphism G(F)z ...:::. G(A1) induces one of the Heeke algebra Jlt'( G(F)z) (see 1 . 1 0) onto Jlt'1 and, again, (5) is compatible with the actions defined here and in 1 . 1 0. Note also that FL is dense in L, by strong approximation, so that this
198
A.
BOREL AND
H.
JACQUET
isomorphism of Heeke algebras ind uces one of .Yt' G(F) x , fL) , which is equal to .Yt' ( G(F), FL), onto ( G(A1) , [Strictly speaking, this applies at first for F = Q, but we can reduce the general case to that one, if we replace G by RF1Q G (4. 1 ) .] In the general case, the isomorphisms 4.3(2) for various �L are compatible with the action of G(F) defined here and in 1 .9 respectively, and this extends by continu ity to the closure in G(A 1) of the projection of G(F). 4. 8. Let d(J, K=) be the span of the spaces d(�=• J, K=) and d(K=) the span of the d(J, K=). These spaces are £modules, and union of admissible .Yt'submo dules. When 4.6(5) holds, they are isomorphic to the spaces d(Z, J, K=) and d(Z, K=) of automorphic forms on G(R) defined in 1 . 1 0. Otherwise, the relation ship is more complex, and would have to be expressed by means of the isomor phisms 4. 3(2).
.Yt'
(
L).
5. Groups over function fields. In this section, F is a function field of one variable over a finite field. A function on G(A) is said to be smooth if it is locally constant. 5. 1 . Let X be a quasicharacter of Z(A)/Z(F) and K an open subgroup of G(A). We let 0i""( x, K) be the space of complex valued functions on G(A) which are right invariant under K, left invariant under G(F), satisfy
f(z · X) = x(z) ·
(1)
f(x)
(z E Z(A),
x
E G(A)),
and are cuspidal (3 .3). [These functions are cusp forms, in a sense to be defined below (5.7), but the latter notion is slightly more general.] We need the following : X
Let andof be given.hasThensupportthereinexists a compact subset o f such that every element In particular, isfinite dimensional. 5.2. PROPOSITION (G. HARDER). C G(A) 0i""( x, K)
K
0i""(X , K)
Z(A) · G(F) · C.
This follows from Corollary 1 .2.3 in [10] when G is split over F and semisimple (the latter restriction because ( I ) is not the condition imposed in [10] with respect to the center). However, since G(A) can be covered by finitely many Siegel sets, the argument is general (see [9, p. 1 42] for GLn).
X
Letthe space be a finite setm,of quasiofcuspi charactdalefunct rs of ions which areandrimghta posi t i v e i n t e ger. Then invariant under and satisfy the condition is finite dimensional. m, m, s, s s , q: (j 5.3. CoROLLARY.
0i""(X,
Z(A)/Z(F)
K)
K
IT (r (z)  x => ( 1 ) . Assume (1). Let be as in ( 1) . Then / is annihi lated by an admissible ideal J of £0• Let U = £. We have to prove that UK is finite dimensional for any compact open subgroup K of There is no harm in replacing by a smaller group, so we may assume that fixes f We may also assume that K = x is compact where is compact open in and open in the subgroup of elements in with vcomponent equal to 1 . We also have £ = £0 ® £ where Jfv is the Heeke algebra of Let �v (resp. �·) be the idempotents associated to (resp. (3.3). Then �v ® � = �K is the idempotent associated to Any element in U is a finite linear combination of elements of the form / * a * (3 (a Jfv , (3 £0). If such an element is fixed under then �K =
AUTOMORPHIC FORMS AND AUTOMORPHIC REPRESENTATIONS
20 1
hence we may assume that each summand is fixed under K, and that a * �· = a, = (3. Since / is fixed under K, it follows that f * a is fixed under K. The ele ments * a then belong to the space "f/" ( G, v, J, K), which is finite dimensional by the theorem. For each such element * a * f3 is contained in the space of K.fixed vectors in the admissible .no.module * a * .no. = * .no. * a, whence our asser tion. 5 . 8 . DEFINITIONS. An on G(A) is a function which is left in variant under G(F), right invariant under some compact open subgroup, and satis fies the equivalent conditions of 5. 7. A is a cuspidal automorphic form. Any automorphic form is Z(A)finite (as follows from 5.7(3)) . An G(A) if it is isomorphic to a sub quotient of the G(A)module .91 of all automorphic forms on G(F)\G(A). It follows from 5. 7 that it is always admissible. More generally, a topologically irreducible continuous representation of G(A) in a topological vector space is automorphic if the submodule of smooth vectors is automorphic. As in 4.6, it follows from [4, Theorem 4] that if X is a character of Z(A)/Z(F), then any Ginvariant closed irreducible subspace of V(G(F)\G(A)) x is automorphic.
g;
f3 * �.
f
ff f automorphic form cusp form irreducible representation of is automorphic
Letfbe a function on Then thefollowing conditions are(1)equival e nt: f iiss aZ(A)fini cuspform.te, cuspidal and right invariant under some compact open f subgroup of I 5.9. PROPOSITION.
G(F)\G(A).
(2)
(3 . 3),
G(A).
PROOF. That ( 1 ) � (2) is clear. Assume (2). Then /is annihilated by an ideal of finite codimension of C[Z(A)/Z(F)]. Let U be the space of functions spanned by r(G(A)) · f Every element of U is cuspidal, annihilated by /and right invariant under some compact open subgroup. If L is any compact open subgroup, then UL is con tained in the space 0"f/"{/, L) (notation of 5.4), hence is finite dimensional. Therefore U is an admissible G(A)module and ( 1 ) holds. 5. 1 0. It also follows in the same way that the space o .91(1) (resp. o .91(X, of all cusp forms which are annihilated by an ideal of finite codimension of (resp. which satisfy 5.3( 1 )) is an admissible G(A)module. Moreover, if X consists of one element X• and if = in which case we put .91(X, m) = o .91 x• then this space is a direct sum of irreducible admissible G(A)modules, with finite multiplicities. To see this we may, after twisting with lxl 1 , assume that X is unitary. Then, since o .91 x consists of elements with compact support modulo Z(A),
CfZ(A)/Z(F)]
I
m 1,
(f,
g)
=
J
Z (A) G (F) \G (A)
f(x) g(x) dx
m))
·
de fines a nondegenerate positive invariant hermitian form on o .91 x · Our assertion follows from this and admissibility. This is the counterpart over function fields of the GelfandPiatetskiShapiro theorem (4.6). We note that, by [14], every automorphic representation transforming according to X is a constituent of a representation induced from a cuspidal automorphic representation of a Levi subgroup of some parabolic Fsubgroup, for any global field F.
A. BOREL AND H. JACQUET
202
REFERENCES 1. A. Borel, Some finiteness theorems for adele groups over number fields, lnst. Hautes Etudes Sci . Pub! . Math. 16 (1 963), 1 0 1  1 26. 2. , Introduction to automorphic forms, Proc. Sympos. Pure Math . , vol. 9, Amer. Math . Soc . , Providence, R . I . , 1 966, pp. 1 99210. 3. P. Cartier, Representations of reductive Padic groups : A survey, these PROCEEDINGS, part 1 , pp. 1 1 1  1 5 5 . 4 . D . Flath, Decomposition of representations into tensor products , these PROCEFDINGS, part 1 , pp. 1 79 1 8 3 . 5 . S . Gel bart, Automorphic forms o n adele groups, Ann . o f Math. Studies, n o . 8 3 , Princeton Univ. Press, Princeton, N.J., 1 975. 6. I . M . Gelfand, M . I . Graev and I . I . PiatetskiShapiro, Representation theory and automorphic functions, Saunders, 1 969. 7. I . M . Gelfand and I. Piatetsk iShapiro, A utomorphic functions and representation theory, Trudy Moscov. Mat. Obsc. 12 ( 1 963), 3 894 1 2 . 8. R . Godement, The spectral decomposition of cusp forms, Proc. Sympos. Pure Math . , vol . 9 , Amer. Math. Soc . , Providence, R . 1 . , 1 966, p p . 225234. 9. R. Godement and H . Jacquet, Zeta functions of simple algebras, Lecture Notes i n Math . , vol. 260, SpringerVerlag, New York, 1 972. 1 0. G . Harder, Cheval/ey groups over function fields and automorphic forms, Ann . of Math. (2) 100 ( 1 974), 249306. 1 1 . HarishChandra, A utomorphic forms on semisimple Lie groups, Notes by J. G. M . Mars, Lecture Notes i n Math . , vol. 68, SpringerVerlag, New York, 1 968. 12. M . Kuga, Fiber varieties over a symmetric space whose fibers are abelian varieties, Mimeo graphed Notes, Univ. of Chicago, 1 96364. 13. R . P. Langlands, On the functional equations satisfied by Eisenstein series, Lecture Notes in Math., vol . 544, Springer, New York, 1 976. 14. On the notion of an automorphic representation, these PROCEEDINGS, part 1, pp. 203207. 15. I . P iatetskiShapiro, Classical and adelic automorphic forms . An introduction, these PROCEED INGS, part 1 , pp. 1 8 51 8 8 . 16. V . I . Platonov, The problem of s trong approximation and the Kneser Tits conjecture, lzv. Akad. Nauk SSSR Ser. Mat. 3 ( 1 969), 1 1 391 1 47 ; Addendum, ibid. 4 ( 1 970), 784786. 17. G . Shimura, Introduction to the arithmetic theory of automorphic forms, Pub!. Math. Soc. Japan 11, I . Shoten and Princeton Univ. Press, Princeton , N.J., 1 9 7 1 . 18. N. Wallach, Representations of reductive Lie groups, these PROCEEDINGS, part 1 , pp. 7 1 86. 19. G . Warner, Harmonic analysis on semisimple Lie groups. I , Grundlehren der Math. Wis senschaften, Band 1 88, Springer, Berlin, 1 972. 
 ,
INSTITUTE FOR ADV ANCED STUDY
COLUMBIA UNIVERSITY
Proceedings of Symposia in Pure Mathematics Vol. 33 (1 979), part 1 , pp. 203207
ON THE NOTION OF AN AUTOMORPHIC REPRESENTATION. A SUPPLEMENT TO THE PRECEDING PAPER
R. P. LANGLANDS The irreducible representations of a reductive group over a local field can be obtained from the squareintegrable representations of Levi factors of parabolic subgroups by induction and formation of subquotients [2], [4]. Over a global field F the same process leads from the cuspidal representations, which are analogues of squareintegrable representations, to all automorphic representations. Suppose is a parabolic subgroup of G with Levi factor and fJ = @fJv a cuspidal representation of Then Ind fJ = ®v Ind fJv is a representation of which may not be irreducible, and may not even have a finite composition series. As usual an irreducible subquotient of this representation is said to be a constituent of it. For almost all v, Ind fJv has exactly one constituent 11:� containing the trivial representation of G(O.). If lnd fJv acts on x. then 11:� can be obtained by taking the smallest G(F.)invariant subspace v. of x. containing nonzero vectors fixed by G(O.) together with the largest G(F.)invariant subspace u. of v. containing no such vectors and then letting G(F.) act on v.; u• .
G(A)
P
M
M(A).
is a constituentTheof constitanduents of forarealmostthe alrepresent l ations LEMMA 1.
lnd fJv
Ind fJ 1l:v = 11:�
v.
11:
= ®1rv
where
1l:v
That any such representation is a constituent is clear. Conversely let the con stituent 11: act on V/ U with 0 £ U £ V £ X = ®X• . Recall that to construct the tensor product one chooses a finite set of places S0 and for each v not in S0 a vector x� which is not zero and is fixed by G(O.). We can find a finite set of places S, containing S0, and a vector Xs in Xs = ®vES x. which are such that x = Xs ® (®voes x�) lies in V but not in U. Let Vs be the smallest subspace of Xs containing xs and invariant under Gs = 11 vES G(F.). There is clearly a surjective map Vs ® (®voes V.) > Vf U,
and if v0 ¢ S the kernel contains Vs ® U.0 ® (®voesu rvol V.). We obtain a surjec tion Vs ® (®vES v.; u.) + V/ U with a kernel of the form Us ® (®vES V./ U.), Us lying in V5. The representation of Gs on Vs/ Us is irreducible and, since Ind fJv has a finite composition series, of the form ®vES 1l:v, 1l:v being a constituent of lnd fJv· Thus 11: = ®1rv with 1l:v = 11:� when v ¢ S. AMS (MOS) subject classifications ( 1 970). Primary 30A58, 20E35, 1 0D20. © 1979, American Mathematical Society
203
R. P. LANGLANDS
204
The purpose of this supplement is to establish the following proposition.
an automorphic representation if and only if is a constituentrepresent of atiforon someof G(A)andis some PROPOSITION 2. A n
i nd (J
n
(J .
P
The proof that every constituent of Ind (J is an automorphic representation will invoke the theory of Eisenstein series, which has been fully developed only when the global field F has characteristic zero [3]. One can expect however that the analogous theory for global fields of positive characteristic will appear shortly, and so there is little point in making the restriction to characteristic zero explicit in the proposition. Besides, the proof that every automorphic representation is a constituent of some Ind (J does not involve the theory of Eisenstein series in any serious way. We begin by remarking some simple lemmas.
Let Z be the centre ofG. Then an automorphicform is Z(A).finite. Suppose isLeta maxibemaalparaboli compactcsubsubggrouproupofofG(A)G. andChooseangautomor pandhic form wi t h res p ect t o G(A) letP(A)K'onbeM(A). a maxiThen mal compact subgroup of M(A) containing the projection ofgKg1 (m; g) J qJ(nmg) is an automorphicform on M(A) with respect to g) M qJ(g)cj;(g) dg G(A). M(A) g. g) for all then is zero. LEMMA 3.
This is verified in [1].
LEMMA 4.
qJ
K
K.
P
=
([Jp
N (F) \ N (A)
E
n
dn
K'.
It is clear that the growth conditions are hereditary and that ([Jp( ; is smooth and K'finite. That it transforms under admissible representations of the local Heeke algebras of is a consequence of theorems in [2] and [4]. = 0 whenever We say that qJ is orthogonal to cusp forms if JoG , ,. CAl ¢ is a cusp form and [) is a compact set in If P is a parabolic subgroup we write ¢ j_ P if ([Jp( ; is orthogonal to cusp forms on for all We recall a simple lemma [3]. ·
·
LEMMA 5. If qJ
j_
P
P
qJ
We now set about proving that any automorphic representation n is a con stituent of some Ind (J. We may realize n on Vf U, where U, V are subspaces of the space .91 of automorphic forms and V is generated by a single automorphic form (/) · Totally order the conjugacy classes of parabolic subgroups in such a way that {P} < {P'} implies rank P � rank P' and rank P < rank P ' implies {P} < {P ' } . Given qJ let {PI' } be the minimum {P} for which { P } < {P '} implies qJ j_ P ' . Amongst all the qJ serving t o generate n choose one for which { P } = {PI' } i s minimal. If ¢ E .91 let cpp(g) = cpp( I , g) and consider the map ¢ > cpp o n V. I f U and V had the same image we could realize n as a constituent of the kernel of the map. But this is incompatible with our choice of ([J, and hence if Up and Vp are the images of U and V we can realize n in the quotient Vpf UP · Let .91� be the space of smooth functions ¢ on satisfying (a) ¢ is Kfinite.
N(A)P(F)\G(A)
AN AUTOMORPHIC REPRESENTATION
205
(b) For each the function + = is automorphic and cuspidal. Then Vp � sl�. Since there is no point in dragging the subscript P about, we change notation, letting 1C be realized on Vf U with U � V � sl�. We suppose that V is generated by a single function
g
m cf;(m, g) cj;(mg) rp. We may(g)forso alchoose character ofLet A)besatithsfeyicentng rp(reaog)f M. x(a)q; l g E G(A)rp andand allthaatE thereA).is a P(A)\G(A)/Krp q; a. E C {l(a)rpla E A } a E A) ( a) ) ) ( a ) (( ) ) ( a ( (a) ) (/(a)  a)rp, /(a) q; rp {bE Bl l(b)rp {jrp, {3 E C}, rp M(A) rp l x a) , a eA(A) , M(F)\M(A), x) M(Q)\M(A) m E M(A) a dmA(A), cj;(ag) x(a)cf;(g). m cj;(m, g) grp( ·,g)G(A). g E G(A).cf;( ·,g) cf; ' ( ·,g) cf;'(g) cf;'(m, m1g). cf;'(g) cf;'(l, g). m1 E M(A),{cf;' lcf;cf;'(mmM(A)l > g) N(A) M(A) v1 LEMMA 6. X
A
V
=
A(
A(
Since is finite, Lemma 3 implies that any in d� is A(A)finite. Choose V and the generating it to be such that the dimension of the span Y of A( ) is minimal. Here /(a) is left translation by If this dimension is one the lemma is valid. Otherwise there is an A( and a such that 0 < dim I(  a Y < dim Y. There are two possibilities. Either / a  a U = /( )  a V or /( )  a U # /  a V. In the second case we may replace rp by contradicting our choice. In the first we can realize 1C as a subquotient of the kernel of  a in V. What we do then is choose a lattice B in A(A) such that BA(F) is closed and BA(F)\A(A) is compact. Amongst all those and V for which Y has the minimal possible dimension we choose one for which the subgroup of B, defined as = has maximal rank. What we conclude from the previous paragraph is that this rank must be that of B. Since is invariant under A(F) and BA(F)\A(A) is compact, we conclude that Y must now have dimension one. The lemma follows. Choosing such a and V we let v be that positive character of which satis fies
v(a)
=
( j
and introduce the Hilbert space L� = L� ( of all measurable functions cf; on satisfying : (i) For all and all e = < 00 . (ii) JA (A ) M(Q) \M ( A) v Z(m)jcp(m)j 2 L� is a direct sum of irreducible invariant subspaces, and if cf; E V then + lies in L� for all e Choose some irreducible component H of L� on which the projection of is not zero for some For each cb in V define to be the projection of on H. It is easily Thus we may define by seen that, for all = = If V' = e V}, then we realize 1C as a quotient of V'. However if o2 is the modular function for on and (J the representation of (J. on H then V' is contained in the space of lnd To prove the converse, and thereby complete the proof of the proposition, we exploit the analytic continuation of the Eisenstein series associated to cusp forms. Suppose 1C is a representation of the global Heeke algebra .Yt', defined with respect to some maximal compact subgroup K of Choose an irreducible representa tion ,. of wh i ch is contained in 1e . If E. is the idempotent defined by let .Yt'k = E,.Yt'E. and let 1e. be the irreducible representation of .Yt'. on the ICisotypical sub space of 1C. To show that 1C is an automorphic representation, it is sufficient to show that 11:. is a constituent of the representation of .Yt'. on the space of automorphic
K
G(A).
K
206
R. P.
LANGLANDS
forms of type " · To lighten the burden on the notation, we henceforth denote 1T: K by 7T: and £K by £. Suppose P and the cuspidal representation u of are given. Let L be the lattice of rational characters of M defined over F and let Lc = L Each ele ment p. of Lc defines a character �I' of Let be the ICisotypical subspace of Ind �l'u and let = I0 . We want to show that if 1T: is a constituent of the representa tion on I then 1T: is a constituent of the representation of £> on the space of au tomorphic forms of type '" · If {g;} is a set of coset representations for then we may identify II' with by means of the map rp > rpl' with
M(A). II'
I
I
M(A)
® C.
P(A)\G(A)/K
rpl'(nmg;
k)
= �l'(m) rp(nmg;k).
In other words we have a trivial ization of the bundle {II'} over Lc, and we may speak of a holomorphic section or of a section vanishing at p. = 0 to a certain order. These notions do not depend on the choice of the g;, although the trivialisa tion does. There is a neighbourhood U of p. = 0 and a finite set of hyperplanes passing through U so that for p. in the complement of these hyperplanes in U the Eisenstein series E(rp) is defined for rp in To make things simpler we may even multiply E by a product of linear functions and assume that it is defined on all of U. Since it is only the modified function that we shall use, we may denote it by E, although it is no longer the true Eisenstein series. It takes values on the space of automorphic forms and thus E(rp) is a function g > E(g, rp) on It satisfies
Iw
G(A).
E(pl'(h)rp) = r(h)E(rp)
if h E £> and pi' is Ind �l'u. In addition, if rpl' is a holomorphic section of {II'} in a neighbourhood of 0 then E(g, rpl') is holomorphic in p. for each g, and the deriva tives of E(rpl'), taken pointwise, continue to be in d. Let I, be the space of germs of degree r at p. = 0 of holomorphic sections of I. Then rpl' > pl'(h)rpl' defines an action of £> on I,. If s � r the natural map > I5 i s an £homomorphism. Denote its kernel by I:. Certainly I0 = I. Choosing a basis for the linear forms on Lc we may consider power series with values in the ,._ isotypical subspace of d, I: lal � r p.a c/Ja · £> acts by right translation in this space and the representation so obtained is, of course, a direct sum of that on the Kisotypical automorphic forms. Moreover rpl' > E(rpl') defines an £homomorphism ). from to this space. To complete the proof of the proposition all one needs is the JordanHolder theorem and the following lemma.
I,
I,
LEMMA 7. For r sufficiently large the kernel of ). is contained in I�.
Since we are dealing with Eisenstein series associated to a fixed P and u we may replace E by the sum of its constant terms for the parabolic associated to P, modify ing ). accordingly. All of these constant terms vanish identically if and only if E itself does. If Qr. . . . , Qm is a set of representatives for the classes of parabolics as sociated to P let E;(rp) be the constant term along Q;. We may suppose that is a Levi factor of each Q;. Define v(m) for m E by �l'(m) = e c' . If () = () 1 * () f with () f (g) = iMg 1 ), then the right side of ( l . 1 7) is strictly positive. So by ( 1 . 1 5)and the result described by ( 1 . 1 4) 0'1 must be a subrepresentation of 0', and the quotient representation n = 0"/0'' must satisfy the identity ( 1 . 1 7)
for all nice 0 1 . This implies that the regular representation of GS is quasiequivalent to n and hence that the regular representation of G5 decomposes discretely. Since the same is then also true of the regular representation of each G "' v rj: S, we obtain an obvi ous contradiction . The assumption c < yields a similar contradiction and the theorem is proved. REMARK ( 1 . 1 8). Suitably modified, the proof above shows that G and G ' have the same Tamagawa number. The restriction on w is not really necessary.
c'
2. Cusp forms on GL(2). Henceforth G will denote the group GL(2). The center Z of G is z
Again Z
� p:
=
{(g �)}.
and we set G = G/Z. We also introduce the subgroups
P
=
{(� :)}.
A =
{(� �)}.
N
=
{(b i )} .
If � is a function on G(F)\G(A) we say that � is cuspidal if
S
N (F) IN (A)
�(ng) dn
=
0,
gE
G(A) .
As in § 1 , we introduce a space V(w, G) and a representation Pw · We denote by La(w , G) the subspace of cuspidal elements in V(w, G) . THEOREM (2. 1 ) . Let Pw . 0 denote the restriction of Pw to the invariant subspace La(w, G). Let � denote a Coofunction which satisfies ( 1 . 3) an d is of compact support mod Z( A). Then the operator Pw . o(�) is HilbertSchmidt. SKETCH OF PROOF. For each c > 0, recall that a Siegel domain '1J in G(A) is a set of points of the form g
_ (0I I )( 0 O)k x
tab
a
STEPHEN GELBART AND HERVE JACQUET
218
where x is in a compact subset of A, t is an idele whose finite components are trivial and whose infinite components equal some fixed number u > c > 0, a is arbitrary in Ax, lies in a compact subset of Ax, and is in the standard maximal compact subgroup K of G(A). If '!J is sufficiently large then G(A) = G(F)'!J . For a proof of this fact see [Go 3]. By abuse of language, if '!J is a Siegel domain in G(A), we call its image in N(F)\G(A) a Siegel domain in N(F)\G(A). If '!J is such a domain, we denote by V(w, '!J) the space of functions on '!J such that f(zg) = w(z)f(g) and dg < + oo . Clearly V(w, G) can be identified with a closed SN (F) Z (A) \(§' 1 subspace V of V(w, '!J) if'!J is large enough. In this case there is a bounded operator with bounded inverse A from V(w, G) onto V. Iff is in V(w, G) and
t we find that
j'(  s)
= Jr[(� ?)g] \ tl sl!Z dxt = f \ t j s1 1 2 dx t J![w(� �)(� t � x)g] d l
x
FORMS OF GL(2) FROM THE ANALYTIC POINT OF VIEW =
=
=
223
J j t j s+ l 1 2 dx t J ![(b ?)w(b I)g] dx I t j s1 /2 dxt J1[(� w I) dx ?) (b gJ S
J ![ w(b I) g, s] dx .
(Recall that w was assumed to be trivial on Fd;, ; therefore
to H(
M(s) H(s) s) M(s)\l}(g) J \l}[wng] dn /'( s) M(s)/(s). Summisupport) ng up, weisfindgiventhatby tthhee formul scalar product compact a of two Pseries FI > F2 (with fi > of F2) s (!1. (s), . ds . s (M(s)f.i (s), f2(s)) ds. s) s) s)
for t E F�. ) If we define an operator
=
(3 . 1 7)
when Re(s)
from
>
by
N (A)
!, we conclude that
=
(3. 1 8)
f2
(F�>
(3. 1 9)
=
/2(  s) )
1 2 71;{
1 + 2 1d
The integrals are taken over any vertical line with Re(s) > ! and the pairings we have written are on H( x H(  and H(  x H(s) (or simply H x H). It is worth noting that (3.20) for in and in H(s). (This is analogous to the identity (3.9) and proved just the same way.) If we think of as an operator from H to H then (3.20) simply asserts that
rp1 H(s) \1)2
(3. 2 1 )
M(s) M(s)* M(s). / (s)) Pw(g)F n,h(g)/(s)) . M(s)n.(g) n_,(g)M(s). M(s). with s purely imaginary. M(s).
Note finally that if/ is replaced b y (resp. right translate, i.e., by that (3.22)
=
f+
/(hg) then F (resp. is replaced b y its On the other hand, it is also clear
=
These facts play a key role in the next section .
4. Analytic continuation of Our goal is to use the inner product formula (3 . 1 9) to construct an intertwining operator between a subrepresentation of Pw and a continuous sum of the representations 11: 5 To this end we need to analytically continue the operator Indeed the integration in (3. 1 9) is
STEPHEN GELBART AND HERVE JACQUET
224
over a line Re( ) = x with x > t. So if we want to rewrite this formula using purely imaginary we need to know something about the poles of M( ) . A. Now it will be more convenient to replace ns and M(s) with the operators we get by performing a Mellin transform on the whole group Fx \ " rather than just Fd;,. Let r; = (fl., denote a pair of quasicharacters of with = cv . Let H(r;) be the space of functions on G(A) such that
ss Prelimminary rearks. F"\A"A fJ.V (4. 1)
q;
and
s
[(g �) J = g
v) fJ.(a)v(b) I � 1
1 12
q;(g)
(4.2) Once more we can think of the collection H(r;) as a fibrebundle over the space of pairs r;. These pairs make up a complex manifold of dimension with infinitely many connected components and our fibrebundle is trivial over any such com ponent because the character (3 g) ,_. of n is fixed there. In par ticular, H(r;) may be regarded as the subspace of functions in such that
1
fJ.(a)v(b) A(A) K L2(K)
when (3 �) E K. In any case, we denote by n� the natural representation of G(A) on H(r;). The fiber decomposes as the sum of the fibres H(r;), with r; = (fl., and fl.  1 ( ) I Is for a E Fd;, ; the representation ns decomposes correspondingly as the sum of the "principal series representations n�" ; see [Ge, p. 67] . As far as pairings are concerned, we have a natural one between H(r;) and H(7j 1 ) defined by a
v a = H(s)a
v),
(q;l > q;z) = J q;1(k)ipz(k) dk. K
(n�(g)q;l >
As before, q;2) = (q; 1 , nii t(g1 ) q;2 ) . Moreover, if we set 7j = formula (3 . 1 7) defines an operator M(r;) from H(r;) to H(7j) satisfying (4. 3)
(v,
fl.), then
M(r;)n� (g) = n�(g) M(r;)
and (4.4) Here q; 1 i s in H(r;) and q;2 is in H(� 1 ) ; cf. formulas (3.20) and (3 . 22) . In particular, if we identitfy H(r;) with and H(ij) with H (� 1 ) (keeping in mind that r; and belong to the same connected component) then (4.4) reads
7]1
(4. 5)
H(7j1)
M(r;)
*= 
M(� 1 ).
Note that (3 .21) is essentially a direct sum of identities like (4. 5). To continue analytically M(r;) we are going to decompose it as a tensor product of local intertwining operators and thereby (essentially) reduce the problem to a local one. B. Let v be a place of F and F. the corresponding
Local intertwining operators.
FORMS OF GL(2) FROM THE ANALYTIC POINT OF VIEW
225
local field . If 7Jv = (f.J.v• lJv) is a pair of quasicharacters of F� such that fl.v lJv = Wv we can form the space H(r;.) analogous to H(r;), the representation 1r�.· and the operator M(r;.) : H(r;.) + H(ij.) defined by (M(r;.) rp) (g)
(4.6)
=
JF. rp[w(6 �)gJ dx.
Since the group G. operates by right matrix multiplication on F. x F., and the stabilizer of the line F.(O, 1 ) is the group P., we may define global sections of the bundle H(r;.) by the formula (4.7)
rp(g, r;.)
I (det ) 'i'.{t• fl.vlJv � JF. cP [(O, t)g]f1.vlJ;1(t) t l d"t.
=
.
Here L(s, fl.vlJ;1) is the usual Euler factor attached to the character fl.vlJ;l, a . (x) = x I . , and cP is a SchwartzBruhat function on F. x F•. Recall that if X is a quasi character of F�. andf is SchwartzBruhat on F., the integral sF� f(t) I t l s x( t ) d"t = Z(J, x, s) converges for Re(s) � 0 ( > 0 if x is unitary) and the ratio Z(f, X• s)/L(s, x) extends to an entire function of s. (This is Tate ' s theory of the local zetafunction.) Thus the formula (4. 7) actually makes sense for all r; and indeed defines a section of our fibrebundle. Now we · want to apply the operator M(r;.) defined by (4.6) to an element of H(r;.) given by the section (4.7). After a change of variables we get fl.v I (det f) w.( (4.8) cP[(t, x)g]f1.vlJ;1 ( t ) dxt dx .  1) a � Z L(l , fl.vlJv ) Fv F�
I
JJ
Next recall the functional equation for the quotient Z(f, X• s)/L(s, x) :
Z(/, x 1 , 1  s)
____,. __, ''..,�  s,
X
L(1
1)
_ 
(
e s, X• ,, 'f' v)
Z( f, X• s) . L(s, x)
Here f denotes the usual Fourier transform taken with respect to the fixed additive character cf;., and e(s, X• cf;.) is an exponential function which also depends· on cf; •. If we define the Fourier transform of cP to be
iJ (x , y)
=
J JcP(u, v) cf;. (yu

xv) du dv
we see that (4. 8) can be written as the product of L(O, fl.vlJ; 1) L( C fl.vlJv 1 ) e (0, fl.vlJv 1 ' cf;.)
(4.9) and (4. 1 0)
Note that we pass from (4.7) to (4. 1 0) by the substitutions r; + ij, cP + iJ , and multiplication by w . ( I ) Next we write 7Jv = (f.J.v• JJ. ) , fl.v = x 1 a �2 , lJv = x 2a;s12, where x 1 and x 2 are characters. For Re s > 0, we can write M(r;.) as the product of the scalar (4.9) with an operator R(r;.) which (for each r;.) takes the element (4.7) of H(r;.) to the 
.
STEPHEN GELBART AND HERVE JACQUET
226
element (4. 10) of H(7j.). But for Re(s) >  !, every element of H(7;.) can be re presented by an integral of the form (4.7) ([JL, pp. 97 98]). An easy argument then shows that R(r;.) extends to a holomorphic operator valued function of r;. in the domain Re (s) >  ! which again takes (4.7) to (4. 1 0). Moreover, from the obvious relations (7]) = r;, (C/JA)f' = C/J, and w�(  1) = 1 , we find that R(7j.)R(r;.) = Id.
(4. 1 1)
Since (4.9) is clearly meromorphic, this also gives the analytic continuation of the operator M(r;.) in the domain Re(s) >  t . Although we do not need to, we note that R(r;.) (and hence M(r;.)) extends to a meromorphic function of all r;. (or s E It is important to note that the operator R(r;v) is "normalized" in the following sense. If r;v is unramified then H(r;v) (resp. H(7j.)) contains a unique function I then r belongs to P(F).
H(rg)
>
1 and
STEPHEN GELBART AND HERVE JACQUET
230
This lemma shows that, for a given g, the series in (5. 5) has at most one term for
c > 1 . In particular,
A c �(g) = �(g)  � N (g) X c (H(g)) if H(g) > 1, c > 1 ,
(5.7)
= �(g)  � N ( g )
if H(g) > c > 1 .
Now let 'll be a Siegel domain and Q the set of g in 'll such that H ( g ) ::;;; c. Then Q is compact (mod Z(A)) and, if c > 1 , A 1 . Then there is a finite non empty set of elements r of P(F)\G(F) such that H(rg) > c > (cf. Lemma (3. 3) again ; this finite set depends on Q but not on c). Since the resulting element rg must belong to a compact set mod P(F)N(A)Z(A), and since H is continuous, we must have H( rg) < c0• But if c > c0 we get a contradiction. Therefore (5. 8) must hold . In other words, Ac � > � uniformly on compact sets as c + oo . W e also want t o point out that A c i s a continuous hermitian operator o n V(w, G) :
1
>
(5.9) Indeed the lefthand side is (�b �2) minus the scalar product of a Pseries with �2. In particular, by (3.4) the lefthand side equals (� ! > �z)
S
P (F) N (A) Z (A) \G (A)
Similarly the righthand side is (� ! > �2) is real the desired equality follows. We also have, for c > 1 ,
� l.N (g) x c (H(g) ) �z .N (g) dg.
 f�1. N(g )952• N(g )xc (H(g)) dg ; since
Xc
(5. 1 0) i.e., Ac is an orthogonal projection in V(w, G). Indeed ( 1  Ac)� 1 is also a Pseries. So the lefthand side of (5. 1 0) is
S
N (A) P (F) Z (A) \G (A)
.
� l N ( g ) x c (H ( g ))
(J
N (F) \N (A)
)
A c �z(ng) dn dn.
But X c (H(g)) = 0 unless H(g) > c > 1 , in which case the inner integral is (by (5.7))
S
N (F)\N (A)
(�z(ng)  �z. N (ng)) dn = 0
and ( 5. 1 0) follows. (In fact (5. 1 0) is sometimes true even when �; does not belong to
V(w, G).)
C. Analytic continuation of Eisenstein series. Using the truncation operator (5. 5) we can write our Eisenstein series as
FORMS OF GL(2) FROM THE ANALYTIC POINT OF VIEW
E(h(s), g )
(5. 1 1 )
=
AcE(h(s), g) +
I:
P (F) IG (F)
23 1
EN(h (s), rg) x c (H(rg)) .
The second term on the right side is the Pseries attached to a function with support in the set {g : H(g) � c } . Thus, as noted before, for g in a Siegel domain the second term has only finitely many terms. In particular, it represents a meromorphic func tion of s whose singularities are at most those of M(s) (cf. (5.3)). On the other hand, the first term on the right side of (5. 1 1) is initially defined only for Re(s) > !. However, since it is squareintegrable (cf. (5.7) and the remarks immediately following it), it will suffice to continue it analytically as a V(w, G) valued function. Thus we need to examine the inner product (5. 1 2) By (5. 1 0) (which is true in this case) the inner product (5. 1 2) is just (E(h 1 (s 1 )), Ac £(h 2(s2) )) , which, since Ac E is a difference of Pseries, we compute (using (3.4)) to be
Now recall that EN is given by (5.3) with cp(g, s)
for t E Ft,,
a
=
h(g, s) and
E FO(A), and k E K. So by Iwasawa's decomposition, we compute
(Ac E(hl > (sl)), A c E(h z (.i'z)))
(5. 1 3)
=
{(hl> h2)c• 1 +sz  (M(s 1 )h 1 , M (s2)h2)c <s 1 +szl }
1
sl + Sz
Here Re(s1) > Re(s2) > t. Note that the right side of (5. 1 3) is a meromorphic function of (sl> s 2) which seems to have a singularity on the line s 1 + s2 0. However, by (3.21) and (4.20), the expression in the first bracket vanishes precisely along this line. Similarly the ex pression in the second bracket vanishes when s 1  s2 0. Thus we conclude that (5. 1 3) is meromorphic in s l> s 2 with singularities at most those of M(s 1 ) and M(s2) . In particular, for s s 1  s 2 # 0, =
=
=
=
(Ac£(h 1 (s)), Ac£(h2(  s)))
=
(5. 1 4)
2 (h l > h2) log c + ( M(  s)M' (s)hl> h 2)
+ { (hl> M (  s)h 2)c 2c  (M( s )hl> h2) c 2• }
is .
Now suppose that for h1 h2 h, the righthand side of(5. l 3) is analytic in the polydisc I s 1  so I < R, I Sz  s0 < R, while E(h(s)) is holomorphic in some small disc centered at s0• Then the double Taylor series of the lefthand side converges in this polydisc. Since =
=
I
STEPHEN GELBART AND HERVE JACQUET
232
I ;; AcE(h(s)) /t = ( ;;" AcE(h(s)), :s: AcE(h(s)) ) = os�;s� (AcE(h(s1)), AcE(h(sz))) 11.1=52=•
it is easy to conclude that the Taylor series of AcE(h(s)) converges in the disc I s  so l < R, i.e., AcE(h(s)) is analytic in thi s disc. Thus we conclude A cE(h(s)) (and hence E(h(s), is meromorphic in Re(s) � 0 with singularities at most those of M(s). Note that in (5. 1 3) and (5. 1 4) we needed to identify H(s) with H to define the scalar products and the derivative M'(s) of M(s). Summing up, we know that if cp(s) is any meromorphic section of our bundle H(s) then E(cp(s), is defined as a meromorphic function of s (at least for Re(s) � 0). Moreover,
g))
g)
(5.15)
E(M(s)cp(s),
g) =
E(cp(s),
g).
Indeed since cp(s) belongs to H(s) and M(s)cp(s) belongs to H(  s), the function EN(M(s) cp(s), is equal to
g +
g) g) =
=
M(s)cp(s) + M(  s)M(s)cp(s) cp(s) + M(s)cp(s) = EN(cp(s), Thus EN(M(s)cp(s), i.e., the difference between the two sides of EN(cp(s), (5. 1 5) is a cuspidal function. Therefore, since any Pseries (or its analytic continua tion) is orthogonal to all cuspidal functions, this difference vanishes as claimed . In general, say for a group whose derived group has Frank one, the same facts can be proved. The only difference is that the operator M(s) may have a finite number of poles and these poles are not necessarily known. Nevertheless, the op erator M(s) still similarly controls the analytic behavior of the Eisenstein series (cf. [Ar 1] and [La 1]) . D. The kernel Pw(cp) L�ont· Suppose F1 (g) is a Pseries attached to.fi (of com pact support mod Z(A)N(A)P(F)). Our immediate goal is to prove that, for all h e H,
g),
g) .
of in F E(h(iy), g) \(g) dg SF1 (iy)) = 2!_ s (5.16) SF1(iy) = a1(iy) (4.y.23). E(h(iy), g) F oo i ) g) dsF} 2(g) dg (Fh Fz = J  {� 2 7r: l. J"x+ioo 12 7r: l J"+ioo ds S =� g) z(g) dg =x> F j (5.17) (Fh = 217r: s:oo dy s E( l (iy), g) z(g) dg X�w (Fh y} M(iy(5.15) )/1(iy) y  y.(5.17) a1(iy) (4. 23), /1(i(5.17) (h,
G (F) Z (A) \G (A)
for almost every Here is defined by analytic continuation and as in Note first that if F2 is the Pseries attached to/2 we get from (5.2) that 1
_
E( jl (s),
G (F) \G (A)
.
Fz}
reads
_
G (F) \G (A)
E ( j l (s),
t. Shifting the integration to the imaginary axis we then get
for Re(s)
But
x ioo
+ c
x) (x. Fz).
says that the integral in is unchanged if we replace as in and then change to Thus, with formula
by
FORMS OF GL(2) FROM THE ANALYTIC POINT OF VIEW
On the other hand, computing the inner product as in
233
(4.21), we also have
� J:oo (a1(iy), az(iy)) dy + c x� (Fh x )(x. Fz) . Thus we conclude that oo (5 . 18) J oo (a(iy), SF(iy)) dy 1 J {J E(a(iy) , g) F(g) dg} dy (FI > Fz)
=
..
=
_
2

_
_
Gmww
for all a in 2 . This formula is then also true for any squareintegrable function a(y) with values in H since both sides remain unchanged when a(iy) is replaced by t (a(iy) + M(  iy) a(  iy)) . Thus we may take a(iy) = c( y)h( iy) with c a scalar function and h a "constant section" to conclude from that indeed
(5.18)
(5.16)
holds. Now extend the map S: L�ont(w, G) jo 2 to S: V(w, G) jo 2 by setting it equal to on Lij and r;p(w, G). Then S * S is the orthogonal projection of L2(w , G) onto L�ont· Since S is an intertwining operator, S * Sp.,(cp) S * S = S * n:(cp) S for any function cp of compact support. Therefore, if F1 and F2 are in V(w, G) then
coo
0
( S * Sp "'(cp) S * SF1o F2)
(5 . 19)
= =
( S * n:(cp) SF1o F2)
=
(n:(cp) SF1 o SFz)
! s:oo (n:;y(cp)SF1(iy), SF2(iy)) dy.
Before applying the identity (5.16) we make the following observation. Although (5.16) was proved only for F1 in the orthocomplement of Lij(w, G) it is actually true all F1 V(w , G). Indeed if F is in Lij( w, G) then SF 0 by definition. On the
for in
=
other hand, since every Eisenstein series is orthogonal to any cuspidal function (in the domain of convergence at first but for all s by analytic continuation), the right is also identically zero. side of to get If { tlla } is an orthonormal basis for H we can finally apply to
(5 . 16)
(5.16) (5 . 19)
=
111: J:oo dy � SFl(g) E (n:� (cp)tlla(iy), g) dg J Fz(h) E(tlla(iy), h) dh ·
with g, h in G (F)\G (A ) . Interchanging the integrations and summations then yields ( S * Sp., (cp) S * SF1 o F2)
(5.20)
= JJ Fl(g)Fz(h) dg dh {� 4� J:oo E(tlla(iy), h) E (n,�(cp) tll2(iy), g) dy} .
So if we let Pcont denote the projection S * S onto L�ont we find that the kernel of
234
STEPHEN GELBART AND HERVE JACQUET
Pcon t p., (+
""
vE
S.
STEPHEN GELBART AND HERVE JACQUET
246
We shall give two proofs of this result, both somewhat different from the one in
[JL]. First we shall take Theorem (8. 1) for granted and indicate how it together with
the trace formula implies Theorem (8.3). Then we shall sketch an alternate proof that essentially implies (rather than depends on) Theorem (8. 1). Both proofs use orbital integral techniques (cf. [Sa], [Sh], and [La 2]). B. Let w and w ' be differential forms invariant and of maximal degree on G and G ' respectively. We assume that w and w ' are related to one another as in [JL, pp. 475 and 503]. Then for each rf; S, W v = ± w� and I W v I = I w � 1 . For E S let u s say that I E C�(Gv, w;; 1) and rp E C:'(Gv, w;; 1 ) have if
Matching orbital integrals (and consequences).
v v orbital integrals (8.4)
f
Zv\G v
for "any function" Theorem (8. I ), then (8. 5)
det )
g(g)h(tr g, g dg h on F
x
P.
=
J,
, rp( ) ( tr( ), ( )
Zv\G v
matching
g h g v g) dg'
In particular, if nv and
n
� are related as in
On the other hand, tr n.(f) = tr n�(rp)
(8.6)
if nv = x a det and n� = x a ).J. Condition (8.4) can also be expressed in terms of orbital integrals as follows : (i) The hyperbolic (regular) orbital integrals of I vanish, i.e., (8.7) (ii) Let L c M(2, Fv) and L' c
D
be isomorphic quadratic extensions ; then
if there is an isomorphsim Lv > L� taking in U  px to t ' E L'x  P . (Here we select in any way a Haar measure on F:\L� and transport it to F;\L�x via the isomor phism L" > L� ; the quotient spaces are then given the quotient measures.) A consequence of the last few identities is
t
e
(8.8)
l(e) = rp( ).
(This follows, for instance, from the Plancherel formula.) Note too that if nv is a representation of G" which is neither squareintegrable nor finite dimensional then tr n v( f) = 0.
(8.9)
The following lemma results from the characterization of orbital integrals in
§4 of [La 2] .
matching orbital inGitevgenrals. in LEMMA (8 . 1 0).
rp
C�(G�, w1)
I
there are (many) in
C�(Gv, w 1 )
with
REMARK (8. 1 1). Fix a representation n� of G� and let nv be the corresponding
FORMS OF
GL(2)
FROM THE ANALYTIC POINT OF VIEW
247
representation of G. given by (8. 1). There is easily seen to be a rp in C�(G �, w;;1 ) such that tr 7C�(rp) i= but tr a'(rp) = if a' is any irreducible representation of G� (with central quasicharacter w.) not equivalent to 1r:�. Thus if f has matching orbital integrals, tr 1r:.(f) i= but tr a ) = whenever a is not equivalent to 7Cv (or finite dimensional). Now suppose f is a function on G(A) such that (zg ) = wl(z)f(g). Moreover, suppose as before that = n. where f. E C;;"(G., w;;1) and for almost all is the function defined by
0,
0 {
f f(g) f.(g.) 0,
v
fv(g.)
=
w;;1(z)
if g.
=
0
f
z.k., k . e K., z. e
0
otherwise.
v¢
f.
f.
z. ,
Suppose rp is a function on G'(A) satisfying similar conditions, rp • ..., via the isomorphism G. :::::: G� for S, and rp. and have matching orbital integrals for V E S.
PROPOSITION (8. 1 2). With rp
andfas above,
tr p�(rp)
(8. 1 3)
=
tr pw, oU)
f.
+
tr Pw, sp(f) .
PROOF. By ( 1 . 1 1) the lefthand side is
vol ( Z(A)G'(F)\G'(A))rp(e) (8 . 1 4) the sum extending over a set X' of representatives for the classes of quadratic extension L' of F embedded in Moreover
J
L' • (A) \G' (A)
D.l rp(g .;g) dg
=
nJ v
' ' Lv' IG v
t
rp.(g;; .;g.) dg
•.
Here we have selected for each v a Haar measure on F;\L'; such that (units F�\L�) = 1 for almost all v ; to Ax\£x(A) we have given the product measure. On the GL(2) side, note that S has at least two places vi> v2 and for these places f.,, fv2 have vanishing hyperbolic orbital integrals. Thus by (7.21), the righthand side of (8 . 1 3) is vol ( Z(A)G(F)\G(A)) f(e) (8. 1 5)
+
1 I; v ol "' Lx\Lx(A)) (A 2 L
I;
<E P \£";< * 1
J
L' (A) \G (A)
f(g1.;g) d
g
where L runs through a set of representatives for the classes of quadratic extensions of F in M(2, F). The measures are selected as above, and
Note that if v e S and L. is split then the corresponding local integral is by (8. 7) (i). Thus we need only sum over the set X of L which do split at any v e S, that is, which embed in But for every L e X there is an L' in X' and an isomorphism
D.
not
0
STEPHEN GELBART AND HERVE JACQUET
248
L + L'. So if we assume that the Haar measures of F�\L� and F;\L�x correspond to one another we have
for all v. Indeed for v E S this is clear, and for v ¢: S it follows from the fact that we may modify the isomorphism G. + G� so as to make it compatible with L� + L';'. We conclude from (8. 7) and our choice of for v ¢: S that in (8. 1 4) and (8. 1 5) the series on L and L' are equal. Since we have used the Tamagawa measures on G and G' we also know that
q>.,f.
vol(Z(A)G ' (F)\G'(A))
=
vol(Z(A)G(F)\G(A)).
Therefore, taking (8 .8) into account, (8. 1 2) follows. C. Let P�. o be the representation of G'(A) in the orthocomplement of the space spanned by the functions X o with = w. From (8.6) and our choice of for v ¢: S we get
Proofs of the main result. fv, q>. J f(g)x(det g) dg Z (A) \G (A)
=
Thus (8. 1 2) implies that
tr Pw. oU)
(8. 1 6)
J
=
Z (A) \G' (A)
v x2 q>(g)x (v(g)) dg.
tr P�. o(q>) .
FIRST PROOF OF THEOREM (8.2). For each place v E S fix an irreducible repre sentation of G� and let be the corresponding representation of G• . Choose = 1 , and tr 7r�(q>.) = 0 for 77: � inequivalent to E C�(G�, w;; 1 ) such that tr Thus identity (8 . 1 6) reads
({>o�.v O"�
O"v o�(q>.) n(1r') 1r'(q>) n(1r) 1r(f) 1rn(1r) n(1r'))1r' 1T:v O"v 1T:v) 1r 1r vE1r'S 7r�(q>.) vES JT:�(f.) . n(7r') vn$S 1T:�(q>.) n(JT:) vn$S 1rv(fv) O"v 1T:v O"v 1r' 1r, n(1r') n(1r) . [JL] n(1r) 1r'. nv nv 1rv(fv) 1r ®1rv ®1r�) I;
tr
=
I;
tr
where the sum on the righthand (resp. lefthand) side is over all irreducible repre sentations of G(A) (resp. of G'(A)) such that � (resp. 1r� � for all v in S, and (resp. is the multiplicity of in Pw. o (resp. P�. 0) . Moreover, if and satisfy these conditions, then, because S has even cardinality, we get Thus = TI I = TI
I;
tr
=
I;
(77:�
tr
�
for v E S,
�
for
vE
S),
and Theorem (8 .2) follows from fundamental principles of functional analysis (applied to the isomorphic groups G5 and G' S). Moreover, if corresponds to then = So since we already know from that is at most 1, we have also proved multiplicity one for SECOND PROOF. We rewrite (8. 1 6) as (8 . 1 7)
I;
tr
=
I;
tr 1T: �(q>.),
where = (resp. 1r ' = runs through all irreducible subrepresentations of Pw. o (resp. P�. 0). What we shall do now is manipulate (8 . 1 7) until it equates the
FORMS OF GL(2) FROM THE ANALYTIC POINT OF VIEW
249 one (8. 1 7) an arbiITtr!varyandplace wIToutsareide as and presentatio(8.18). n ofFiGw.x Iff= in (8.12)an irtreduci hen ble unitary """(f) right side to the trace of just representation. (Since we are summing over sub representations rather than classes, two representations on the right side of may, a priori, be equivalent.) LEMMA
S
IT tr ,.�( w
v:t:w
v;t:w
0. (8.17) (f ) f 0 But
=
can be rewritten as
1: tr ""w w IT tr ""v(fv) =
1: tr ,.�( E
1: IT tr
=
vES
(,. c Pw.O> ""w
S,
1: vES IT tr ,.�(
0 for every a e (/)b. Then {Q} is a set of representatives in Q of the left cosets of Q modulo the Weyl group of M. By the Bruhat decomposition,
S
NI (Q) \Nt (A)
= I;
tE {Ol
J
· exp (
E(nx, A) dn (/),
I; 1 N1 (Ql 1Nt (A) ("Ew/ P (Q) w, nNo (Q) INo(Q) l
(f>(w1 J..mx)
v nx))) dn, Hp(w A pp, ( 1 w1N1 wl1 s 1 tWt"1Pwt No\No Wt"1Nwt Nl\Nl . (w.N(Q)w 1 N(Q)\w.N(A)w:;1\N(A)) (f>(w 1 nx) (A pp, Hp(w:;1nx))) dn. J +
n M is the unipotent radical of a standard parabolic subgroup of M. If the group is not M itself the term corresponding to above is 0, since (/) is cuspi dal. The group is M itself if and only if s = maps a onto n1. ln this case n is isomorphic to n Therefore the above formula equals
I;
vol
sED (o, OJ)
:; n
· w,N(A) w; 1 nN(A) \N(A)
:;
exp(
+
The volume is one by our choice of measure. The lemma therefore follows. Combining the lemma with the Fourier inversion formula, one obtains COROLLARY.
ifJ
D
and P1 are associated, and that and (h are as in the discussion precediSuppose ng Lemmathat P Then 7.
EISENSTEIN SERIES AND TRACE FORMULA
S where
G (Q) IG (A)
=
(/J (/J (x) 1(x) dx
1 . )dim (2 71:1
A
s
1:
ilo+ia s r= O (a, a1J
A0 is any point in pp a+. 0 +
259
(M(s, A)W(A), Wl(  sil)) dA,
16
The proofs of Lemmas are based on rather routine and familiar estimates. This is the point at which the serious portion of the proof of the Main Theorem should begin. There are two stages. The first stage is to complete the analytic con tinuation and functional equations for a vector in £�. cu sp· This is nicely described in [2], so we shall skip it altogether. The second stage, done in Chapter 7 of [1], is the spectral decomposition of Li(G(Q)\ G(A)) . From this all the remaining asser tions of the Main Theorem follow. We shall try to give an intuitive description of the argument. The argument in Chapter 7 of [1] is based on an intricate Fix X = (BI'x , r X ' 81' x · For J'l'P , , fl' , we are induction on the dimension of for x x assuming that and are meromorphic on In the process of proving this, one shows that the singularities of these two functions are hyperplanes of the form r = (a, = a and that only finitely many + = 3 (a, e ) > 0, a r meet Let L�,.:( G(Q)\G(A » be the closed subspace of Lr( G(Q)\G(A)) generated by functions where comes from a function as above, which vanishes on the finite set of singular hyperplanes which meet + If comes from 81' x• we can apply Cauchy's theorem to the formula for for
W
W). E(x, W, A)W a+ ia {A{AE ac:ac: ¢(x), ¢ W1(A 1), P1 E
M(s,(A/Z),A)W P E WE ac. P E A)R Af1, f1 E C, E EWp}.1,'p}, W(A),a+ ia. ¢1(x) (/J (x)(/J1 (x) dx. S ( )dim S ( M(s, A)W(A), W1(sA)) dA, 2 s.;J, sA ia . ( _1_.)dim n(A) 1 2nl S.. ( M(s, tA)W(tA), W 1(tA)) dA (1.)dim n(A)_1 2nz s. M(rt 1 , tA)W(tA), W 1(rA)) dA ( 1 )dim n(A) 1 s ( M(t, A)lW(tA), M(r, A) 1W 1 (rA)) dA 'll:l G (Q) IG (A)
The result is
1
A
.
71: 1
since 
=
on
1:
.
• a s E D (a, a1)
Changing variables in the integral and sum, we obtain
A
1:
1:
1:
PzE i!l'x t E Q (az, a)
•
where
1:
1:
1:
Pz rE D (a2, a1)
·
=
az
A
=
sE O (a , a 1)
r
t az
A
t E O (az, a)
(
1: 1: 1: Pz
r
t
.
t az
2 60
JAMES ARTHUR
(I)
Fr, Pi A)
=
I;
r J (rA),
and Fp2 is defined similarly. Define i&x. x to be the subspace of the space i9x (defined in the statement of the Main Theorem) consisting of those collections {Fp2 : P2 E &>x} such that Fp2 takes values in .Yt'h x ·. We have just exhibited an isometric isomorphism from a dense subspace of L9� x to a dense .subspace of L� · x( G(Q)\G(A )) . Suppose that {Fp2} is a collection of functions in L9 . x each of x x which is smooth and compactly supported. Let h(x) be the corresponding function in L� · x( G(Q)\G(A) ) defined by the above isomorphism. We would like to prove x that h(x) equals h'(x)
=
I;
d ( ) im ASlaz. E(x, Fp2(A), A ) dA . 1
Pz E &x
n (A 2 )  1 271: 1
.
If (/J1(x) is as above, the same argument as that of the corollary to Lemma 7 shows that Az I I; n{A z)  1 _ h ' (x) /jJ J (x) dx 271: 1 G (Q) \G A)
S
=
(
( )dim .
p2
S E
· iaz r I{YA)) dA.
Since the projection of /jJ1(x) onto L�x· x( G(Q)\G(A) ) corresponds to the collection { Fl, P2} defined by (1 ) this equals Az . ( Fp2 I: n (A z) 1 _ 1 (A), Fl, Pz(A)) dA 271: 1 1az P2 ,
=
J
( .)dim s
G (Q) \G (A)
h(x)(/J1(x) dx .
We have shown that h(x) = h ' (x). This completes the first stage of Langlands' induction. To begin the second stage, Langlands lets Q be the projection of Li( G(Q)\G(A) ) onto the orthogonal complement of L�.X• x( G(Q)\G(A) ) . Then for any (/J(x) and (/J1(x) corresponding to 1/>(A) and I/>1(A1), (Qrp, ¢1) equals
( )dim A(J I; (M(s, A)IP(A), 1/>1(  sA)) dA r  s. I; (M(s, A)IP(A) , 1/>1(  sA)) dA) . A
_
I
Ao+ia sEQ (a, OJ)
71: 1
10
s
Choose a path in a+ from A 0 to 0 which meets any singular hyperplane t of s E Q ( a, a1)} in at most one point Z(t). Any such t is of the form X( t) + t�, where tv is a real vector subspace of a of codimension one, and X(t) is a vector in a orthogonal to tv . The point Z( t ) belongs to X(t) + tv. By the residue theorem (Q(/J, /jJ1) equals
{M(s, A) :
(
I 2 1. _ 71:
) Al I; J dim
r
Z (r) +irv
I;
s E Q (a, al)
Res r (M(s, A) IP (A), 1/>1(  sA)) dA .
The obvious tactic at this point is to repeat the first stage of the induction with A 0 replaced by Z(t), 0 replaced by X(t), and E(x, 1/>, A) by Res rE(x, 1/>, A).
26 1
EISENSTEIN SERIES AND TRACE FORMULA
Suppose that rv {H E a : (a, H) 0} for a simple root a E f/)p. Then rv aR , for R a parabolic subgroup of G containing P. If (/) E .Yt'P.x• define ER(x, (/), A) P (Q)I;\R (Q)rf)(ox) exp( ( A + pp, H(ox) ) ) . =
=
=
=
This is essentially a cuspidal Eisenstein series on the group MR( A) . It converges for suitable and can be meromorphically continued . It is clear that
A E ac
I; R (Q) \G (Q)
ER(ox, (/), A) whenever the righthand side converges. Suppose that AEt, and A AV E r�, Re AV E PR + a]i. Then for any small positive c, RestE(x, (/), A) 271:1 1. J2"0 E (x, (/), A + te2"iOX(r)) d() 21 . J2"0 ER (ox, rf), A + ce2"iOX(r)) do) I; (R (Q) \G (Q) I; rpv(o x) exp( ( Av + PR • HR(ox) ) ) , R(Q) \G (Q) E(x, (/), A)
=
=
X(r) +
AV,
=
=
7r:l
where
rpv(y) 2�i J:"ER (y, (/), (1 =
+ t)X _(r)e2"iO) dO,
the residue at X( r) of an Eisenstein series in one variable. One shows that the func tion MR(Q) \ MR(A), is in the discrete spectrum. Thus
m > rpv(my), m E
RestE(x, (/), A) E(x, rpv, AV), =
rpv t , A),
the Eisenstein series over R(Q)\G(Q) associated to a vector in .Ye}.,.Yt'R, cusp · Its analytic continuation is immediate. Let be the finite dimensional subspace of By examining Res M(s one obtains the £' consisting of all those vectors operators
rpv,
R
.Yt'R,x
Their analytic continuation then comes without much difficulty. for s Let f?/J ' be the class of parabolic subgroups associated to R. In carrying out the second stage of the induction one defines subspaces L'fp.' x(G(Q)\G(A)) c L i(G(Q)\G(A) ) and i'fp., x c i'fp., and as above, obtains an isom orphism between them. In the process, one proves the functional equations in (a) of the Main The orem for vectors The pattern is clear. For R now standard parabolic subgroup and flJ irP, x and associated class, one eventually obtains a definition of spaces L'fp, x(G(Q)\G(A)) . By definition, is unless R contains an element of and the other two spaces are unless an element of flJ contains an element of flJ If flJ is the associated class of R , there corresponds a stage of the induction in which and part (b) for one proves p art (a) of the Main Theorem for vectors in the spaces LrP, x and L'fp, x(G(Q)\G(A)) . Finally, one shows that
E Q(aR , aQ).
rpv E .Yt'R. x·
{0}
any .Yt'R.x {0}
Li (G(Q)\G(A))
.Yt'R .x•
rpv .Yt'R,x
=
E8 tP
L'fp x(G(Q)\G(A)) . •
any
f/Jx,
x·
262
JAMES ARTHUR
Notice that if R and
f!lJ
are fixed, i9 = E9 i9' X•
J't'R = ffiJ't'R, ...Y • )!
and
)!
L�(G( Q)\G(A) ) = ffiL�' x(G( Q\G(A) ) . )!
The last decomposition together with the Main Theorem yields V(G ( Q ) \G( A) ) = ffiL�' x(G( Q ) \G(A )) .
(2)
9, )!
This completes our description of the proof of the Main Theorem. It is perhaps a little too glib. For one thing, we have not explained why it suffices to consider only those t above such that tv = {A e a : (a, A) = 0} . Moreover, we neglected to mention a number of serious complications that arise in higher stages of the induction. Some of them are described in Appendix III of [l]. We shall only remark that most of the complications exist because eventually one has to study points X(t) and Z(t) which lie the chamber a + , where the behavior of the functions A) is a total mystery.
outside
M(s,
PART II. THE TRACE FORMULA
In this section we shall describe a trace formula for G. We have not yet been able to prove as explicit a formula as we would like for general G. We shall give a more explicit formula for GL3 in the next section. In the past most results have been for groups of rank one. The main references are
Automorphicforms on Sur Ia formule des traces de Selberg, The Selberg trace formula for groups of Frank one,
4. H. Jacquet and R. Langlands, Berlin, 1 970. 5. M. Duflo and J. Labesse, Ecole Norm. Sup. (4) 4 (197 1 ), 1 93284. 6. J. Arthur, (2) 100 (1 974), 326385.
GL(2), SpringerVerlag, Ann. Sci.
Ann. of Math.
We shall also quote from 7. J. Arthur, 32
The characters of discrete series as orbital integrals,
Invent. Math.
(1 976), 20526 1 .
Let R be the regular representation of G(A) on V(Z(R)O · G(Q)\G(A)) . If � e i0, recall that Re is the twisted representation on V(Z(R)O · G(Q)\G(A) ) given by R e(x) = R(x) exp( (�, Hc(x)) ) . We are really interested in the regular representation of G(A) on £2( G(Q)\G(A)) ; but this representation is a direct integral over � e i0 of the representations Re , so it is good enough to study these latter ones. The decomposition (2), quoted in Part I, is equivalent to V(Z(R )O · G(Q)\G( A)) = ffi L�. x(Z(R)O · G(Q)\G(A )) . 9, )!
Suppose that f e C�( G(A) ) K. Then this last decomposition is invariant under the operator Re (f). Re (/) is an integral operator with kernel K(x, y) = I; fe(x 1 ry), r oc G (Q)
EISENSTEIN SERIES AND TRACE FORMULA
where
JZ (R)O.f(zu) exp((�, Hc(zu)))dz,
263
E
u Z(R)O\G(A). For every we can expressfas a finiKte sum offunctions of theform f fiE C!"(G(A)) . "1"", W) !/(G)(Z(R)0G(Q)\G(A)).{G}. EG fe(u) =
The following result is essentially due to Duflo and Labesse. N�0
LEMMA 1 .
D
l * J2,
Let !/ ( ) be the set of X = (flJJ , in be the restriction of Re(f) to ffigoffize.9'E (G) L�. x
such that flJJ :1
Let Re. e(f)
Then
Rcusp, e (f) = Re(f)  RE. e(f)
is the restriction of Re(f) to the space of cusp forms. It is a finite sum of composi tions Rcusp, e(fl)Rcusp, e(J2) of HilbertSchmidt operators and so is of trace class. For each P and X let fJBp, x be a fixed orthonormal basis of the finite dimensional space Jft'P , x · Finally, recall that a c = a� is the orthogonal complement of 3 = ac in a . If A is in iaG, we shall write Ae for the vector A + � in ia.
2. is an integral operator with kernel KE(x, n(A)I(
LEMMA
RE(f)
y)
=
.E .E
XE.9'£ (G)
2
p
1
.
1C l
)
dim (A/Z)
·faG te�p, ;(x, lp(Ae , j)f/J, A) E(y, f/J, A)} dA.
The lemma would follow from the spectral decomposition described in the last section if we could show that the integral over A and sum over X converged and was locally bounded . We can assume that/ = fl * f 2 . If Kp, if, x, y) = a
I[Je_Eglp,
X
E(x, lp(A, f) f/J , A)E(y,
finite sum, then one easily verifies that J Kp,z(f, x, y)
J :5: Kp,z(f l * (fl) *, x, x) I I2 Kp, z((J2) *
f/J, *
A),
J2,
y,
y )1 12 .
By applying Schwartz' inequality to the sum over X • P and the integral over A, we reduce to the case that .f = l * (fl ) * and x = y. But then Re. e O , a E C/J� ( resp. p(H) > O, p E ��) } .
The following lemma gives our partition of N(A) M(Q)A(R)O\G(A). It is essentially a restatement of standard results from reduction theory. LEMMA 5. Given
P, equals !for almost all 1:
1:
{Q: PocQcP) oEQ (Q) \P (Q)
x E G(A).
FQ(ox, T)r� (H(ox)
D
We can now study the function kT(x). It equals

T)
EISENSTEIN SERIES AND TRACE FORMULA
�
(Q, P:PocQcP)
�
(  l ) di m (AIZ)
oEQ (Q) \G (Q) ·
265
FQ(ox, T)1:�(H(ox)  T)
fp (H(ox)  T)KP(ox, ox).
Now for any H E a0 we can surely write =
7:�(H) rp(H)
�
{P�o Pz:PcP tcPz)
(  l ) d im CA tiAz) 7:�(H)f2 (H)
where (J MH)
=
(J� t(H)
=
�
{Pz: Pz=>Ptl
(  l )dim CAtiAz ) 7:� (H)f 2 (H).
To study (Jh, consider the case that G = GL3 . Then a0/3 is two dimensional, spanned by simple roots a1 and a 2 • Let Q = P0, and let P 1 be the maximal para bolic subgroup such that a1 = {H E a0 : a 1 (H) = 0} . Then (Jb is the difference of the characteristic functions of the following sets :
The next lemma generalizes what is clear from the diagrams. LEMMA 6. Suppose that H is a vector in the orthogonal complement of 3 in a Q If (Jb(H) # 0, and H = H* + H * , H* E a� , H * E ab then a(H* ) > 0 for each a E f/J¢ and II H * II < c ii Hd for a constant c depending only on G. In other words H * belongs to a compact set, while H* belongs to the positive chamber in a¢. D .
We write kY(x) as
�
QcP1
�
oE Q (Q) \G (Q)
FQ (ox, T) (J�(H(ox)  T)
�
{P:QcPcPtl
(  I )di m (A IZ) KP(ox, ox).
The integral over Z(R)0 G(Q)\G(A) of the absolute value of kT(x) is bounded by the sum over Q c P 1 of the integral over Q(Q)Z(R)0\G(A) of the product of FQ(x, T)(Jb (H(x)  T) and
(1)
I � (  l )d i m (A/Z) I {P:Qcpcpl)
f
N (A)
�
f1EM (Q)
j
fe(x 1 n px) dn .
266
JAMES ARTHUR
We can assume that for a given x the first function does not vanish. Then by the last lemma, the projection of HQ(x) onto a � is large. Conjugation by x 1 tends to stretch any element M(Q) which does not lie in Q(Q). Since f is compactly supported, we can choose T so large that the only which contribute nonzero summands in ( I ) belong to QP(Q ) = Q(Q) n M(Q) = MQ (Q) NS(Q). Thus, ( 1 ) equals
f1 E
I
I;
f1
I;
11EMQ (Q) IP:QcPcP,l
which is bounded by
I;
I
I;
11EMQ (Q) IP:QcPcP,l
(  l )d i m CAIZJ
J
(  I )dim (A/ZJ
J
I;
N (A) v e N
� (Q)
I; N (A)
vE 't?o (Q)
I,
fe(x lvJ)nx) dn
l
fe(x1.uvn x) dn .
Recall that n0 is the Lie algebra of N0• Let < , ) denote the canonical bilinear form on n0. If X n0, let e(X) be the matrix in N0 obtained by adding X to the identity. Then e is an isomorphism (of varieties over Q) from n0 onto N0• Let ¢ be a nontrivial character on A /Q. Applying the Poisson summation formula to nS, we see that ( I ) is bounded by
E
I
j
J
I; I; (  l )di m CA/ZJ I; fe(xl.u e(X)x)cj;((X, 0) dX . P 11 CEn � (Q) "Q (A)
If nb(Q ) ' is the set of elements in nb(Q ) , which do not belong to any n{;(Q ) with
Q
c
P�
P 1 , this expression equals
I
1:
f:
11 CEnQ \Q) '
J
nQ (A)
fe(x 1,u e(X) x)¢ ((X, 0) dX I .
Therefore the integral of lkT (x)l is bounded by the sum over Q c 1 and MQ (Q) of the integral over (n * , n * , a , m , k) in Ni (Q)\N1 (A) X N�(Q) \ Nb( A) X Z(R) O\AQ ( R)O X AQ (R) O MQ (Q )\ MQ( A) X K of
FQ(m , T) ab(H (a)
fc I J
• CEn Ql '
 T) exp ( 
"Q CAl
P f1
2(pQ,
H(a) ) )
I
fe (k 1 mla1n ;,/ n*  ' · .ue(x) · n*n * amk)cj;( (X, O ) dX .
E
The integral over n * goes out. The integrals over k and m can be taken over com pact sets. It follows from the last lemma that the set of points { a1 n * a }, indexed by those n* and a for which the integrand is not zero, is relatively compact. Therefore there is a compact set C in Z(R) 0\ G(A) such that the integral of iF(x) l is bounded by the sum over Q c MQ (Q) and the integral over x C of
S
PI > f1 E
Z (R) 0\AQ (R) 0
· =
I;
CEnb (Q) '
J
exp(  2(pQ , H(a))) a�( H(a)  T)
jJ
I
fe(x1.ua 1 e(X)ax)cj;( <X, �)) dX da
"Q (A)
Z (R) 01AQ (R) 0
E
a� (H(a)
 T ) L;
[ J"Q (A) .fe (x 1 .ue(X)x)
C I
I
· cp((X, Ad(a)O ) dX da .
2 67
EISENSTEIN SERIES AND TRACE FORMULA
Since f is compactly supported, the sum over f.1 is finite. If Q unless of course Q = = G, when it equals 1 . If Q � Pb
Y >
J
P1
I1Q (A)
=
Ph (J� equals 0,
Y ))
fe(x1w(X)x) ¢(< X, dX,
is the Fourier transform of a SchwartzBruhat function on n¢{A), and is con tinuous in x . If H(a) = H* + H * , H* E a¢, H * E a d a , H * must remain in a com pact set. Since H* lies in the positive chamber of ab, far from the walls, Ad(a) stretches any element � in n�(Q)'. In fact as H* goes to infinity in any direction, Ad(a)� goes to infinity. Here it is crucial that � not belong to any nG (Q), Q c P � It follows that if Q � Ph the corresponding term is finite and goes to 0 exponen tially in Thus the dominant term is the only one left, that corresponding to Q = P 1 = G. It is an integral over the co mpact set We have sketched the proof of
P1 . T.
THEOREM 1 . We can choose c
\\'al/S,
S
Z (R) DG (Q) \G (A)
>
G(T).
s ch thatj01· any
0u
x dx
kT ( )
=
J
G . Then if (/J e .Yl'p, x , ATE(x, (/), A) equals the function denoted E"(x, (/), A) by Langlands in [2, §9] . This is, in fact, what led us to the definition of AT in the first place. If (/J' is another vector in .Yl'P,x , Langlands has proved the elegant formula
(PJ>, W)
S
Z (R) O·G (Q) \G (A)
= 1:
E"(x, (JJ ' , A')E"(x, (/), A) dx
I;
I;
e . The best hope seems to be to calculate residues in A and A' separately in the above formula. For GL3 the result turns out to be relatively simple. PART III. THE CASE OF GL3
In this last section we shall give the results of further calculations. They can be stated for general G but at this point they can be proved only for G GL3 • Our aim is to express the trace of Rcusp, e(f) in terms of the invariant distributions defined =
m
8. J. Arthur, On the invariant distributions associated to weighted orbital integrals, preprint.
A trace formula for Kbiinvariant functions has also been proved in 9. A. B. Venkov, On Selberg 's trace formula for SL 3 (Z), Soviet Math. Dokl . 17 ( 1 976), 683687.
First we remark that the distributions J'[(f,) and JI(fe ) are independent of our minimal parabolic subgroup so there is no further need to fix P0 . If is any Q split torus in G, let denote the set of parabolic subgroups with split component They are in bijective correspondence with the chambers in In fact, if P0 is a minimal parabolic subgroup contained in an element P of then
&i'(A)
A.
If P'
E &i'(A),
PJ>(A)
= U A1
U
Si00 (a, OJ)
w:;1P 1 w,.
and P' = w:;1 P 1 w., define
(Mp• 1p(A)(JJ)(x) = (M(s, A)(JJ ) (w,x),
PJ>(A),A.
(/J E .Yl'(P).
A
2 72
JAMES ARTHUR
Then Mr i P (A) is a map from .Yt(P) to .Yt(P'), which is independent of P0. In fact, if Re A E PP + at, (Mp , 1 p( A) q)) (x) =
(1 )
J
N (A) n N ' CA) \N' (A)
q)(nx) exp(
( t) W(X) = exp(�i tr XNX1)W(X),
illx �m
illx
(� (.i\ 1) IP(X) = / det A j m IP(XA).
and (1 .2) This representation decomposes into discrete series representations of Sp(2m, R) indexed by certain irreducible representations of 0(2m, R) (namely those which occur in the natural representation of 0(2m, R ) in L2(R2mxm) given by left matrix multiplication) ; cf. [Ge 1] and [Sa] . Howe's recently developed theory of the oscillator representation reformulates and refines Weil' s theory so as to "explain" the above results and point the way towards new ones. A key notion of the theorythe concept of a dual reductive pair formalizes that natural duality (already experimentally observed) between the symplectic and orthogonal groups. To describe this phenomenon more concretely the notion of a Schrodigner model is crucial. 2. SchrOdinger models. We adopt the notation and definitions of [Ho 1]. Thus F is a local field not of characteristic 2, V is a symplectic space defined over F,
H(V) is the Heisenberg group attached to V, X is a fixed nontrivial character of F, Px is the irreducible unitary representation of H attached to X• and Sp( V) is the isometry group of the symplectic form ( , ) on F. Recall that H( V) is a central extension of V by F. Let Sp denote the nontrivial 2fold cover of Sp( V) (whose existence and unique ness is assured by [Mo]). The oscillator representation illx is the unitary representa tion of Sp in the space of Px such that (2. 1 ) for all g E S p and h E H. This representation i s unique up t o unitary equivalence, and it is "genuine" in the sense that it does not factor through Sp( V). The cor responding multiplier representation of Sp( V) is again denoted by illx . To describe a Schrodinger model for ill x (and Px) we need the notion of a com plete polarization of V. This is a pair of subspaces ( X, Y) of V satisfying the follow ing properties : (i) X and Y are isotropic for ( , ), i.e., ( , ) is trivial on X and Y; and (ii) X EB Y = V. With respect to this polarization, the space of Px is V(X) and the action of H( V) is given by
289
DUAL REDUCTIVE PAIRS
p r_((x, 0))/(x ' ) = f(x '  x), Px (( y, O)) f(x ' ) = x( (y, x ' ) )f(x'), Px ((O , t )) f(x ' ) = x ( t ) f(x ' ),
(2.2)
x, x' E X, Y E V, x' E X, t E F, x' E X.
The corresponding Schrodinger model for wx is more difficult to describe. Thus we first restrict our attention to some distinguished subgroups of Sp( V). Let P( Y) (resp. P( X)) denote the subgroup of Sp( V) which preserves Y (resp. X), and let N( Y) denote the subgroup which acts trivially on Y. If M = M(X, Y) = P(X) n P( Y), then P( Y) is a semidirect product of M and N( Y). Using (2. 1 ) and (2.2) we find that
n
wxC )f(x) = x(  t (x, nx)) f(x),
(2.3) and
x E X,
n E N( Y),
x E X, m E M. wxCm)f(x) = Jl.(m)f(m 1 x,) In the Schrodinger model for wx then, N acts by multiplication by a quadratic
(2.4)
character, and M simply acts linearly (the factor Jl.(m) being roughly the square root of the modulus of m acting on X; cf. ( 1 . 1 )). Rather than attempt a general description of wx outside P( Y), we refer the reader to the many examples of this paper ; recall that matrices of the form [b n and [� bl generate SL(2, F). 3. The pair (Sp( V0), O(U0)). The general notion of a dual reductive pair is de scribed in [Ho 1]. The basic example we treat in this paper simply formalizes the special features of the Sp(2m, R) example just discussed. Let ( V0, < , )0) denote any symplectic vector space, and ( U0, ( , ) ) any orthogonal one. Put V = V0 ® U0, and define a form < , ) on V by '
(v
® u, v' ®
u' ) = ( v,
v ')0 (u , u') o
for v, v E V0, and u, u' E U0• Then ( V, < , ) ) is a symplectic space, and (Sp( V0) . O( U0)) is a natural dual reductive pair in Sp( V). In particular, the groups Sp( V0) and O( U0) are each other's centralizers in Sp( V). ((Sp(2m, R), 0(2m, R)) is such a pair in Sp( V), with V = R2m ® R2m = V0 ® U0. If (X0, Y0) is a complete polarization of V0, let X = X0 ® U0 and Y = Y0 ® U0. Then (X, Y) is a complete polarization of V. Moreoveraccording to §2the Schrodinger model for w" in £2 (X) is such that O(U0) acts linearly. On the other hand, if (X0• Y0) is a complete polarization of O( U0)which is possible only when U0 is splitthen the spaces X = V0 ® X0 and Y = V0 ® Y0 provide a polariza tion of V with respect to which wxC Vo) (not O( U0)) acts linearly. In either case, the problem is to describe the decomposition of wx restricted to Sp( V0) • O(U0) . Over R one has the following generalization of the results alluded to in § 1 ; cf. [Ho 2] . Let Sp0 and 00 denote the inverse images of Sp( V0) and O(U0) in Sp( V). Then wxC Sp0) and wx) = L: ,::x (FJ C!> (�) , then (} is Sp(F)invariant. In particular, the functions (4. 1 )
(} "' ( g) = O(w x ) = I: ( w ig) C!>) (�)
t;cX ( F)
are automorphic functions on SpA (slowly increasing continuous functions which are leftSp(F)invariant) . The basic goal of the global theory of the oscillator representation is to describe the automorphic forms and representations which arise from the {}distribution through functions of the form (4. 1 ) . As in the local theory, a great deal of structure is introduced by considering dual reductive pairs in SpA. If (G, G') is such a pair, it follows from the local theory that there should be a duality correspondence
29 1
DUAL REDUCTIVE PAIRS
between representations of G and G' which intertwine with wx . Moreover, this bijection should pair representations of G with repre sentations of G'. For a precise description of what to expect in general, see [Ho 1]. Rather than pursue the general theory, we refer the reader to the examples of §6.
automorphic
automorphic
5. Local examples. Throughout this section, F is a local field not of character istic 2, X is a fixed character of F, V0 is the space equipped with the skew form U0 is the orthogonal space Fn with quadratic form q, and V = V0 U0• Then Sp( V0) = SL(2, F), O( U0) = O(q), and the problem is to describe wx restricted to SL(2, F) · O( U0). Recall that we may choose a polarization in V so that wx acts in according to the formulas
F2
x1 y2  x2yt. (5 . 1 )
L2(Fn)
wx
and (5.2)
®
(6 f)J(X) = x (bq(X)) f(X),
wx
(  0I o1 )f(X) = r(q, x)f (X) .
L2(£n)
The orthogonal group acts linearly in through its natural left action in P . A. Assume that F is nonarchimedean and q is anisotropic. Then � 4, O(q) is compact, and
nThe anisotropic case. 11 Case n
) (8) 17. .E Here the sum is over in R(O(q) ) and the multiplicity of each ) is one (cf. [RS 1] ) . It remains to describe R(O(q)) , R(SL(2, F)), and the resulting duality correspondence. with O(q) = { ± 1 } , and the (i) : = 1 . In this case, = corresponding representations of SL(2, F) act on the space of even or odd func tions in L2(F). More precisely, if is the trivial representation of O(q), then ) (defined by formulas (5. 1), (5.2) restricted to the space of even functions) is the unique subrepresentation of an appropriate nonunity principal series representa tion of SL(2, F) at =  1/2; if 11 is the nontrivial representation of O(q), then ( ) (defined by (5. 1), (5.2) acting in the space of odd functions) is a supercuspidal representation of SL(2, F) which is "exceptional" in the sense explained later in § 6 ; for more details, see [Ge 2], [GPS], and [Ho 3]. Note that and lead to equivalent oscillator representations if and only if ab 1 In case F = R , the pieces of wx are squareintegrable with extreme vectors of weight 1 /2 and 3/2. (ii) : = 2. Let U0 denote a quadratic extension K of F equipped with the inner product derived from its norm form q. Then O(q) is the semidirect product of the norm 1 group Kl in K with the Galois group of K over F. In this case each in R( O(q) ) can be described in terms of a character of K I and most of the repre sentations of R(SL(2, F)) are supercuspidal. For further details, see [ST], [Cas], [G] , or [RS 1] ; in [ST] the correspondence is 2to 1 because K I (the special or thogonal group) is used in place of O(U0). This example is historically important because Shalika and Tanaka were the first to use the osciiiator representation to construct an interesting class of irreducible representations. (iii) : = 3 [RS 1]. Let H denote the unique division quaternion algebra over F, n o the subspace of pure quaternions, and q3 the restriction of the reduced (5.3)
n 11
Case n
Case n
W X l sL(2. F) ·O(q) =
1r(11
n(11 ® 11
q(x) ax2 a E P, 11
n(11
s
2 2 ax bx 2 E (P) •
11
STEPHEN GELBART
292
norm form on H to H 0 . Then every anisotropic ternary form q over F is equivalent to one of the forms aq3 with a in P/(P) 2 , and every irreducible representation 0' of O(q) belongs to R(O(q)) (SO(q) is isomorphic to (H 0 )x) . In [RS 1] it is also shown that the corresponding representations x (O') of SL(2, F) are squareintegrable, and in fact exhaust the class of all such "genuine" representations (at least when the residual characteristic of F is odd). Case (iv) : n = 4. Let U0 denote the division quaternion algebra H defined over F and equipped with the inner product derived from the reduced norm form q. Then though O(q) is not the same as Hx, each 0' in R(O(q)) corresponds naturally to an irreducible representation of Hx, all such representations of Hx thus arise, and each x (O' v) in R(SL(2, F)) is squareintegrable. Actually, x (O"v) extends in a unique way to a representation x ' (O' v) ofGL(2, F") such that
and the resulting duality correspondence 0' v +> x ' (O' ) is onto the set of classes of squareintegrable irreducible representations of GL(2, Fv) ; for further details, see [JL]. Note that Case (iv) is analogous to Case (iii) in that the x(O')'s of the local duality correspondence are characterized by their squareintegrability. Only in Case (ii) is a neat characterization of R( 0(q)) lacking. B. The noncompact (real) case. Assume that F = R and x (u) = e2triu. Let O(q) = O(k, denote the isometry group of the standard quadratic form of signature (k, on Rn , with n = k + /. In [ST] Strichartz has essentially given a decomposi tion of the natural action of O(q) in V(Rn), i.e., of wx restricted to O(q). Using results of [Re] on the tensor product of representations of SL(2, R), Howe has proved the following version of his local duality conjecture : "
I) I)
SL (2, R) · O (q)
where ds is a Borel measure on the unitary dual of SL(2 , R), O"s and 0'� are irre ducible unitary representations of O(k, I) and SL(2, R) respectively, and O's and 0'� determine each other almost everywhere with respect to s. The resulting corres pondence O's > x (O's) = 0'� is particularly interesting for the discrete series repre sentations in R(SL(2, R) ) and R( O( q)) . For complete details, see [Ho 4] and [RS 2] ; the case (k, I) = (2. 1) is described in [Ge 2]. C. Noncompact padic case. An interesting example arises when we replace the quadratic extension of example 5.A(ii) by the hyperbolic plane £2 equipped with inner product ((x, y), (x ' , y')) = xx ' yy' . In this case O(U0) splits. Thus we can find a polarization of V left fixed by Sp( V0) = SL(2, F) (take X0 = {(x, x) :x F} and Y0 = {(x,  x) : x F}). The corresponding Schrodinger model for wx then makes SL(2, F) act linearly in V(£2). More precisely, w/g)f(�) = f(rl(�)) . So since functions on £2 {0} can be identified with functions on G invariant by N  W m. the oscillator representation in this case leads to a direct integral of principal series representations of SL(2, F). The orthogonal group, however, no longer acts linearly. Indeed the operator corresponding to reflection (the element 
E

E
DUAL REDUCTIVE PAIRS
293
taking (x, y) to (y, x) ) is the Fourier transform (in £2(£2)) taken with respect to the skew form in £2 . The significance of these facts for the representation theory of SL(2, F) has already been discussed in Cartier's lectures (see also [G]). 6. Global examples. A. The anisotropic case. Case (i) : 0(1 ) . By tensoring the pieces of the oscillator
representation in V(F v) we obtain some very interesting automorphic representa tions of SL(2) and (by inducing) GL(2). The idea that nontrivial pieces of the oscillator representation could define interesting cusp forms was first communi cated to me by Howe (cf. [Ho 3]) . We shall be content to merely sketch the sub sequent development and refer the reader to [GPS] for details. If a = ®a" is an automorphic representation of 0(1) then a" is the trivial repre sentation for all v except in a finite set S whose cardinality is even. If X = IT Xv is a character of A/F, let w xv denote the oscillator representation of SL(2, Fv) in V(Fv) corresponding to Xv · Fix a character ?. = IT Av of Ax/Fx such that ?.vC  1) = 1 (resp.  1) if v ¢ S (resp. S) and let w �" denote the even (resp. odd) piece of W xv · Then extend w �v to G: = {g GL(2, Fv) : det(g) (F;) Z }
vE
E
E
by defining w�" (b gz) to be the operator 1J(x) >> J. (a) J a J  1 12 ifJ(a 1 x), and induce w �v up to GL(2, Fv) to obtain the representation w� of the double cover Gv of GL(2, Fv). This representation w� is an irreducible unitary "genuine" repre sentation of Gv which is independent of Xv fo r all v and class 1 for almost all v. The representation n(J.) = ® w� is an automorphic genuine representation of GL(2, A) which is cuspidal precisely when S # 0 . In general, when S is empty, these n(A) generate the discrete noncuspidal spectrum of GL(2, A) (cf. [GS]) ; in particular, when F = Q, n(A) generalizes the classical thetafunction 8(z) = I; �= oo exp(2nin 2z) . On the other hand, when S # 0 , n(A) generalizes such clas sical cusp forms as Dedekind's 17function. In this case, n(A) is exceptional in the sense that the Fourier expansion of any function in its representation space con tains "only one orbit of characters", generalizing the fact that the expansion of the 17function has "only square terms" : 1J(24z) = 1: ;;"=1 (3/n) exp(2nin 2z). Since the Fourier expansion of a cusp form 1t on GL(2, A) is carried out with respect to a group which is isomorphic to the product of F with the double cover f'x of P , we can say 1t is exceptional ifgiven X on Fthe Whittaker model for 7t with respect to ( x , p.) exists for only one character p. of f'x. Thus the notion of exceptionality makes sense locally too ; see [GPS] for details. Our conjecture is that the oscillator representation produces all possible exceptional representations, both locally and globally. In other words, for 0( 1) the image of the duality corre spondence a >> n(a) should be all the exceptional genuine representations of G. The classical analogue of the global part of the conjecture just described is that every cusp form of weight k/2 whose Fourier expansion involves only square terms must be of weight 1 /2 or 3/2 and (a linear combination of translates of functions) of the form
STEPHEN GELBART
294 00
I; ¢(n) n" exp(2nin 2z),
n =l
).1
= 0 or 1 .
At this conference we learned that M.F. Vigneras has proved this assertion in [V].
Case (ii) : 0(2). Let K denote a quadratic extension of the field F, X a character of F\A, and O" = @O" v a character of Kl(F)\Kl(A). Piecing together the resulting local representations n(O".) of SL(2, F.) (described in §5.A(ii)) produces automor phic cuspidal representations of SL(2, A) (provided O" is nontrivial). These automor phic forms are related to the theta series with grossencharacter constructed earlier by E. Heeke [H] and H. Maass [M] ; for more details, see [ST] and [Ge 3, Chapter 7] .
Case (iii) : 0( 4). Let D denote a division quaternion algebra over F and SD the set of places where D ramifies. Suppose (J = ®O"v belongs to R(D"(A)) . If v E SD then D� is isomorphic to the unique division quaternion algebra defined over F. and 11:'((1.) is defined as in §5. On the other hand, when v Ft SD, D� is isomorphic to GL(2, F.), and the natural duality correspondence is the identity map. Now suppose (J is actually automorphic, i.e., there is an embedding of (J into the space of automorphic forms on G'(A) = D" (A) . Then according to [JL] the repre sentation n'((J) = ®.n'((J.) of GL(2, A) will also be automorphic. Conversely, if n = ®n. belongs to R ( GL(2, A)) , i.e., n. = n'((J.) for some (Jv in R(D�) (equi valently n. is squareintegrable for each v E SD), then n will be automorphic only if (J = ®(J. is automorphic ; cf. Theorems I4.4 and I 6 . I of [JL] . These results con firm the main global conjecture of Howe's theory in the special context of division quaternion algebras. Whereas the proof in [JL] uses the trace formula, Howe ap parently has an independent proof by different, more elementary, methods. B. 0(2, I ). Let U0 denote the threedimensional space of trace zero 2 x 2 ma trices over F equipped with the inner product derived from the determinant func tion. Set V0 = (F2, ( , ) ) and V = V0 U0 • Then for each place v of F, the pair ( SL(2, F.), O( U0)) is a dual reductive pair in Sp( V). For F. = R, the duality correspondence (Jv .... n((J.) pairs discrete series repre sentations of PGL(2, R) = S0(2, R) of weight k I with discrete series repre sentations of SL(2, R) of weight k/2. For F = Q it is conjectured in [Ge 2] that the resulting global correspondence should generalize Shimura's correspondence [Shm] between classical forms of weight k/2 and k I . Evidence for this is pro vided by Howe's recent description of the local duality correspondence for class I representations (in [Ho 1]), and by Shintani and Niwa's work on the global cor respondence using integrals of thetafunctions ([Sht] and [N]). For further work on this problem, see also [GPS]. If F. is nonarchimedean, and (Jv is a discrete series representation of PGL(2, F.) of the form n'((J�), with (J� an appropriate finitedimensional representation of the orthogonal group of an anisotropic ternary form, then n((J.) should coincide with the representation n((J�) of SL(2, F.) described in § 5A, Case (iii). C. 0(2, 2) (F = Q). Let K = Q( ,.,! > 0 the discriminant, and let r: denote the Galois automorphism of K/Q. Let
®

Ll), Ll
U0 =
{x = (x3x1  xxl�)
:
x1 E
K, x3, x4 e Q
}
�
Q4
DUAL REDUCTIVE PAIRS
295
and define q : U0 > Q by q(X) = 2 det(X). Then q has signature (2, 2), and the corresponding global duality correspondence should pair together modular forms of integral weight with Hilbert modular forms (since SL(2, R) x SL(2, R) has a natural representation in S0(2, 2)) . Work on this correspondence (in the classical language of forms in the upper halfplane) has been carried out by Oda [0], Asai [A], Kudla [K], Rallis and Schiffmann [RS 3], Zagier [Z], and, in a somewhat dif ferent light (and earlier) by DoiNaganuma [DN]. Note added proof We have now proved the global "exceptionality conjecture" described at the bottom of page 293. The local conjecture has been proved by James Meister and will appear in his forthcoming Cornell Ph.D. thesis. 
in
REFERENCES [A] T. Asai, On the DoiNaganuma lifting associated to imaginary quadratic fields (to appear). [Car] P. Cartier, Representations of padic groups : A survey, these PROCEEDINGs, part 1 , pp. 1 1 1 1 55. [Cas] W. Casselman , On the representations of SL,(k) related to binary quadratic forms, Amer. J . Math. 9 4 ( 1 972), 8 1 0834. [DN] K . Doi and H . Naganuma, On the functional equation of certain Dirichlet series, Invent. Math. 9 ( 1 969), 1  1 4 ; also H . Naganuma, On the coincidence of two Dirichlet series associated with cusp forms ofHeeke's "Neben"type and Hilbert modular forms o ver a real quadratic field, J. Math. Soc. Japan 25 ( 1 973), 547554. [Ge 1] S. Gelbart, Holomorphic discrete series for the real symplectic group, Invent . Math. 19 ( 1 973), 495 8 . [Ge 2 ] Wei/'s representation and the spectrum of the metaplectic group, Lecture Notes in Math . , vol. 530, SpringerVerlag, Berlin, 1 976. [Ge 3] , Automorphic forms on adele groups, Ann . of Math. Studies, no. 83, Princeton Univ. Press, Princeton, N. J . , 1 975. [GPS] S. Gelbart and I . I. PiatetskiShapiro, Automorphic Lfunctions of halfintegral weight, Proc. Nat. Acad. Sci . U . S . A. 75 ( 1 978), 1 620 1 623 . [GS] S. Gelbart and P. J. Sally, Intertwining operators and automorphic forms for the metaplectic group, Proc. Nat. Acad. Sci . U.S.A. 72 ( 1 975), 1 4061 4 1 0 . [G] P . Gerardin, Representations des groupe S L , d'un corps local (d'apres Gelfand, Graev, et Tanaka), Seminaire Bourbaki, 20e annee, 1 967/ 1 968, n o 322, Benjamin, New York and Amsterdam, 1 969 . [GK] K. Gross and R. Kunze, Bessel functions and representation theory. II, J. Functional Analysis 25 (1 977), 1 49. [H] E . Heeke, Mathematische Werke, Vandenhoeck, Gottingen, 1 959. [Ho 1] R . Howe, &series and invariant theory, these PROCEEDINGS, part 1, pp. 27528 5 . [Ho 2 ] , Appendix to remarks o n classical invariant theory, preprint, 1 976. [Ho 3]  , Review of [GS], Zentralblatt fur Math . , Band 303, Review 220 1 0, January 1 976, p . 98. [Ho 4] , On some results of Strichartz and of Rallis and Schiffmann, preprint, 1 976. [Ho 5] , Invariant theory and duality for classical groups o ver finite fields, with applications to their singular representation theory, preprint. [JL] Herve Jacquet and Robert Langlands, Automorphic forms on GL(2), Lecture Notes in Math . , vol. 1 1 4, SpringerVerlag, Berlin, 1 970. [KV] M. Kashiwara and M. Vergne, On the SegaiShale Weil representation and harmonic polynomials, preprint, 1 976. [K] S. Kudla, Relations between automorphic forms produced by thetafunctions, Modular Functions of One Variable. VI, Lecture Notes in Math. , vol. 627, SpringerVerlag, Berlin, 1 977. [Kz] D . Kazhdan , Applications of Wei/'s representation, preprint, 1 976. [Ma] H . Maass , Uber eine neue Art von nicht analytischen automorphen Funktionen und die �,


STEPHEN GELBART
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Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung, Math. Ann . 121 (1 949), 1 4 1  1 8 3 . [Mo] Calvin Moore, Group extensions of padic linear groups, lnst. Hautes Etudes Sci. Pub! . Math. 35 (1 968), 1 5 7222. [N] S. Niwa, Modular forms of halfintegral weight and the integral of certain thetafunctions, Nagoya Math. J. 56 (1 975), 1 471 61 . [OJ T. Oda, On modular forms associated with indefinite quadratic forms of signature (2, n2), preprint. [RS 1] S . Rallis and G . Schiffmann, Representations supercuspidales du groupe metaplectique, p reprint, 1 977. [RS 2] , Discrete spectrum of the Wei/ representation, preprint, 1 975. [RS 3] , Automorphic forms constructed from the Wei/ representation, holomorphic case, preprint, 1 976. [Re] J. Repka, Tensor products of unitary representations of SL,(R), prepri nt. [Sa] M . Saito, Representations unitaires des groups symplectiques, J. Math. Soc. Japan 24 ( 1 972), 23225 1 . [ST] Joseph Shalika and S . Tanaka, On an explicit construction of a certain class of automorphic forms, Amer. J. Math. 91 (1 969), 1 0491 076. [Shz] H. Shimizu, Thetaseries and automorphic forms on GL(2), J . Math. Soc. Japan 24 (1 972), 63868 3 . [Shm] G . Shimura, On modular forms of halfintegral weight, A n n . o f Math. (2) 97 (1 973), 440481 . [Sht] T. Shintani, On construction ofholomorphic cusp forms ofhalfintegral weight, Nagoya Math. J. 58 (1 975), 831 26. [Sie] C. L. Siegel, Indefinite quadratische Formen und Funktionen Theorie. l, II, Math. Ann. 124 (1951 /52), 1 754, 364387. [St] R . Strichartz, Harmonic analysis on hyperboloids, J . Functional Analysis 12 (1 973), 341383. [V] M .F. Vigneras, Facteurs gamma et equations fonctionnelles, Modular Functions of One Variable. VI, Lecture Notes in Math . , vol. 627, SpringerVerlag, Berlin, 1 977. [We] Andre Wei!, Sur certaines groupes d'operateurs unitaires, Acta Math. lll (1 964), 1 4321 1 . [Z] D . Zagier, Modular forms associated to real quadratic fields, Invent. Math. 30 ( 1 975), 1 46. 

CORNELL UNIVERSITY
Proceedings of Symposia in Pure Mathematics Vol. 33 ( 1 979), part 1 , pp. 2973 1 4
ON A RELATION BETWEEN
St2
CUSP
FORMS AND AUTOMORPHIC FORMS ON ORTHO GONAL GROUPS
S. RALLIS This lecture is devoted to showing how, starting from the ideas of A. Weil in [22], one can construct a correspondence or lifting between SL 2 cusp forms and auto morphic forms on orthogonal groups. 1 . Siegel formula and the lifting of modular forms.
Let Rk be k dimensional be the bilinear form on = Let Q be the
x,.y,. el > .Ef=1 x,.e,. ] .Ef=1 y,.e,.. x,y,. x,.y,. a .EI.r=1x7 (a, a .E = x�.integral . exponent n L nL nL discriminant g x y y g
Euclidean space with standard basis , ek . Let [ , Rk given by [X, Yj = .E 7=1 where X = Y quadratic form on Rk given by . . .
Q(X, Y)
=
a
I;
i=l
k
 I;
i=a+l
with
. . , .;k is so that any Z basis of L, let DQ 1 2 1 CMZ + dM �b
with SL2(Z)� = {F E SL2(Z) I cr = and c71, 0 {(M, 1)) represents an element of the first column of the matrix c{(M, 1 )) { 1 , identity element of O(Q)) . Then we know that E71(z) is an eigenfunction of the center of the enveloping algebra of SL2 ; in concrete terms this means that £71 satisfies the equation :
0},
(17) with L1
(
02 02 k / 2  y2 ox2 + oy 2
)
+
I 2
k
0 + I 1 2I (b y oi v 
 a)y
0 ax·
In terms of representation theory, Siegel's formula expresses the fact that the identity representation of O(Q) corresponds (in a sense to be made precise below) to the representation of SL 2 determined by the nonanalytic Eisenstein series (£71)71• We then note that it is easy to extend this correspondence to other automorphic representations of O(Q). Namely we can do this in two ways. First we can vary the kernel function. We let P be a homogeneous polynomial on Ra ( = R span of , ea � ) of degree satisfying o(Q)(P) = where o(Q) is the differential operator Q(o/ox l > . . . , o/oxn). Then we form (Im z) t1 2 P (g · (lm z) l 1 2X) cf; (X, z, g) cj; ( X, z, g)
{eb . ..
t
*
0,
=
and as above define 1/f�.(z, g). Thus 1/f�. satisfies a functional equation similar _ to ( 1 4). The second step is then to jntegrate 8� 71 against a cusp on O(Q)/O(Qk 71• This is possible because 8� 71 is a "slowly increasing" function on •.
•.
form {3
299
CUSP FORMS AND AUTOMORPHIC FORMS
denote the Hilbert space inner H x O(Q)/C!(Q)L,�. Let 0}
n, and t are the infinitesimal generators of A , N, and k, then WsL = 2 a 2 + (n + Ad(w0) n)2. We let K ( � O(a) x O(b)) be the maximal compact subgroup of O(Q). If n is a
Then if a,  f2
+
differentiable representation of SL2 x O(Q) on a Banach space )8, then let )BJ?x K be the space of k x K finite vectors in )8. Let FQ be the space of coo vectors of n"Q in L2(Rk). An easy use of the Sobolev regularity theorem shows that if q; E FQ, then for any y belonging to u (SL 2 x O(Q)), the enveloping algebra of SL 2 x O(Q), n"Q(y)q; is a coo function on Q+ and Q_, where Q+ = {X E Rk I Q(X) > 0} and Q_ = {X E Rk I Q(X) < 0} . Then by a simple exercise in invariant theory, one can verify that the centers of the enveloping algebras of SL 2 and O(Q) collapse on each other in the representa tion n"Q · That is, nQ(3( SL 2) ) = n"Q(0(0(Q))), where 3( ) = the center of the envel oping algebra of ( ). In particular we have that n"Q(WsL,) + n"Q (Wo FQ"(A). Moreover FJ"(.A.) and F0(A) are inequivalent SL2 x O(Q) irreducible representations. (c) FJ"(s 2  2s) is SL2 x O(Q) equivalent to the tensor product L2(Whit) ,q, ® .s;�;, where L2(Whit),z_2, is the eigenspace of WsLz (with eigenvalue s 2  2s) in the space of coo vectors of the unitarily induced representation of SL2 (from the unitary character
([6 �]. 1 )
�
)
e� �lx
CUSP FORMS AND AUTOMORPHIC FORMS
301 z and i s the ei g ens p ace (wi t h ei g enval u e s of attheionLaplace Belt r ami operat o r on the space o f C"' vectors o f the st a ndard represent imeasure n L2(F+Ion• dp Here where is determined by the separationandof varidtta+blanes of inofvariant a(Q)  atazz tk  Tta with t s z s C"' ( dpI) · pf(t)pf(t;) p[, p"t C"' t·l;, t di
 2s  tkZ
+) , F+ I ·
+
k)
F+l = {X e Rk l Q(X, X) = 1 } Wl _
+
1
Wl,
O(Q)
O(Q)
1 W+, e
7I
X = · l; , I; E F+ l · REMARK 2. 1 . F0 ( 2  2s) is SL 2 x O(Q) equivalent to the tensor product 2 L (Whit);1_2, ® d;. where L2(Whit);1_2, is the representation of SL 2 on L2 (Whit),z_Zs twisted by the unique outer automorphism A of SL 2 , and d; is the  2s tkZ + k) of We, the LaplaceBeltrami eigenspace (with eigenvalue operator on the space of vectors on the standard representation of O(Q) in L2 F l > The first point in analyzing the discrete spectrum of 71:Q is simply to observe that a k eigenfunction rp (also K finite) in FQ(A) has a separation of variables property on D±, i.e. rp(X) = with functions on D+ (Q_ respectively) satisfying certain differential equations (with X = t; E r+ l , E R+). The second . point is to realize that the map rp '""+ [G + n:Q(Gl )(rp)(t;) with G E SL 2] defines an SL 2 infinitesimal intertwining map of the k finite functions in FQ(A) to the "squareintegrable" functions (L2(Whit)± space, the positive and negative Whit taker models) of the unitarily induced representation of SL 2 from the unitary character n (x) '""+ e" v'l x sgn QC
!k. Then for any rp , a K
K finite function in
x
(32)
is an absolutely convergent series. Then for any (F, r) E FL(Q) FL(Q) =
O(Q) L with
x
{( [� �J c)/ a, b, c, d E Z, ad  be = 1 , c
=
0 mod 2NL, b
=
}
O mod 2 .
we have the functional equation :
8�(GF, gr) = c(F, r) fJ�(G, g),
(33)
with c(F, r). a unitary character on FL(Q) x O(Q) L , taking values in = {z E C l = 1 } . Moreover e� is a coo function on SLz X O(Q) satisfying D * !}�(G, g) = e�Q (Dlrp (G, g) for any D in the universal enveloping algebra of SL2 x O(Q) (* denotes differentiation on the left).
S4
z4
In particular it follows from Theorem 3 . 1 that 8� is an eigenfunction of the center 0(SL 2 x O(Q)) of the universal enveloping algebra of SL2 x O(Q). That is, 2 Wsf4 * e�(G, g) = (s  2s ) e�(G, g). On the other hand, from the growth estimates (21 ) we have if r > 1 rcs1!2 + . G (34) $ M ll gl ll k• k/22 l fJL'P (G g) r(; e • LF' ® L cF+F ' ) j_ in Rk. In particular this means that PF n O(Q)L is an arithmetic subgroup of PF (relative to the Q structure Q · L on Rk). In order to study the cuspidal behavior of we must examine integrals of the following form :
{r· � IrE � E pP, maximal {g E I E
(4 1 )
with
e� (G, g), B�(G, da(v) J E compat e�(G,ibg)le with cusp form s> E F(f(s2 compat 2s)RxibKle s U) dU, dU gvr)
NFINFn rO (Q) a 1
g
E
O(Q), r
·
(u2) u3 u1 E u2 E
g(u1 u2 u3) g(u1) compatible
u3 E
E
O(Q)Q·L and da, some NF invariant measure on
NFfNF n rO(Q)a1. Here we assume that pF is NFfNF n rO(Q)a1 is compact. We know that
L. In such a case is a on O(Q) relative to the group O(Q)L if the family of above integrals vanish for all g O(Q), all r E O(Q)Q·L• and all PF with L (see [5]). and define (for On the other hand let rp tk)
(42)
(/)�( W) =
E
E
F rp( W +
with W (F + F') J. , some Euclidean measure on F. We know that the affine plane W + F (with W E (F + F')l ) is the NF orbit of the nonzero vector W in Rk. So, in particular, (/)� can be interpreted as the integral of rp over the NF orbit of (the orbit here carries an NF invariant measure which can be identified to above). Then we have SL 2 x LEMMA 4. 1 [13]. > tk. rp ..._. (/)�
dUW
Let sning mapThen the 2map 2s)KxK ontoisF(fanF(sin2fini 2s)KxK tesimalF' where intertwi of FJ'(s anded represent QF is atrestionsriocftedtheto universal envel(Inofipiningtesialmgale means relat i v e to t h e associ a t bra.) 2 Ft (s  2s) F F(f/
O(Q, (F + F')J. ) KF = K n O(Q, (F + F')j_ )
Q
REMARK 4. 1 . If QF is positive definite, then ) (when Q is indefinite).
way as
. .
(F + F')J. .
is defined in a similar
CUSP FORMS AND AUTOMORPHIC FORMS
305
FQ"is2  F!F(s2 Let g) 2 O(Q). Then and bb, and and and bb (wi th(wib th ab  a orbor a a), rp e
The most important case of Lemma 4. 1 occurs when is the case when QF has signature 1). Then 2s)
(m,
E
SL
=
2s) = {0} . Such {0} . Thus
rtJ:Q CCG, g)  l ) (tp)
in thefolrplowiFQ"(s ng cases: FQ"(s222 rprp Ft(s rp Ft(s2 gO(Q,O(Q), rpeFt(s2 polarpdecomposition g k · k e p (p malr paraboli cQ L· ofLetO(Q)Sr. icompat iblelatwititche L.in Let O(Q) · L) be the Let rp Ft(sobt2 ained byLetprowijtecth sibengartk.maxi S (relative L onto to the Q split ing of Then for any g) e L2 O(Q) CoROLLARY TO LEMMA 4. 1 . E
(a) (b) (c) (d)
E
E
E
2s)KxK 2s)KxK 2s)KxK 2s)KxK
(G,
dim F = dim F = dim F = dim F =
x
=
0
1,
=
1
=
= = 2).
1
We emphasize here that in the statement of the Corollary to Lemma 4. 1 , although i n Lemma 4. 1 the map + r[J: i s intertwining only for (F + F')l ) . The reason is that f/J:( W) = 0 for all W (F + F')j_ and all 2s)ilxK implies that
E
rtJfQ Cg)  ltp ( W )
=
rtJfQ Ck)  l (tp) (( p W) cF+F' ) l_ )
=
0.
p with (We use the of = K, E PF• and the projection of W on (F + F')l_.) Thus we can determine the cuspidal behavior of tk) , we now go back to
CUSP FORMS AND AUTOMORPHIC FORMS
307
the question of using e� as a kernel function to set up a correspondence between modular forms on the groups involved. In particular we let be as in Example 31 , and consider S� , (see (37) ) Then w e let b e a of weight tk) on satisfying = with r e LJNL · + As in § I we consider the Petersson inner product of/with 8� :
q> phic cusp form (z g) . f holomor s (s > H f(r·z) vQ(r)(c7z d7)• f(z) (51) (S�( ' g) if( ) s e�(z, g)f(z) 1 ·2 dx dy. (5, g)1)I f( ) ), e �. ( " X X , P(X)e [ J. 1, e �. 'l . ) L, , 2 ) ) =
)
I Im z
IJNL\H
The definition given in does not, at first sight, coincide with ( 1 8), the Peter sson inner product < where e�. is the o series constructed from the Schwartz function (In keeping with the notation of § we note that In particular we must determine if TJ E then we dr�p the subsc�ipt TJ from the relationship of e�.( , to 8�( , defined above. But we know that the map cf; + e�( ) is an SL X O(Q) infinitesimal intertwining map from ce�tain SL2 X O(Q) stable subspaces of V(Rk) to the space of C"" functions on SL2 x O(Q) . Thus perhaps the appropriate problem to analyze is the following : for a given K x finite function cf; and the projection ¢D of cf; onto the discrete spectrum of SL2 X O(Q) in V(Rk), what is the relation between the () series er/> and er/>D " This we now do partially in
K
LEMMA . 1
5 [16]. Let be a forKfiniallte Schwartzfunctiandon in andLet suppose that s' > Let P"j; be the projection ofFQ onto the subspace Fct(s'2 i 2s'). Then (52) ( Sf( g) I f ( )) ( S�,t cFl g) f ( )) . 5P;;, 2 FQ F(i(s' (5 1) 2s'). f'"+ (S�(f(r·z) , g) ))vQ(r) (c7z d7)• f(z) r s, vQ]z o H + vQ. f(z) Gn(z) ( 1 dr )s vQ(r) g) Gn( )) Cl ·q>�'(g), c1 g n. s> G s, vQ]o. S�( , g) i f ( n s, vQ]o} q>�' n (n q>�' q>�' s, vQ]o. g) i f( )
71: m(k((), c))(F)
=
F e ..!l s'O ckF
 71:
=
•
V(Rk)
< () �
11:
(
c
=
± 1.
fk + l .
•
REMARK . 1 . If s ' <  Hk + 1 ), then a similar statement is valid where Pt is
replaced by the projection of onto + Thus the two correspondences ( l 8) and are essentially the same. The next main problem is to characterize the image of the map .I as f varies in the space [LJNL • = {! : C l fholomorphic, f( = + for all e LJNL ' e H, and f vanishes at the cusp points of LJNL on Q U { oo} }, the holomorphic cusp forms of weight s and multiplier The first trivial observation is that if we let =
=
I; rE (.dNL) ""\.tJNL Cr Z +
e" .Jl nr C•l ,
the EisensteinPoincare series, then
= (�( , I with a nonzero constant independent of and However since tk , we know that the functions span [LJNL ' )) I fe and hence the space { ( [LJNL ' is exactly the complex linear span of the as ;:;;; 1 . From Corollary to Theorem 4.2 we know the cases when all rp�' ;:;;; I ) are cusp forms on O(Q) relative to O(Q)L. And the most general Fourier coefficient of is difficult to describe arithmetically, since is essentially very much like a Poin care series. However using the results of §4, it is possible to determine for which fe [LJNL • ek S2 S tk
Q(X, e+)•2 on {)_ ,
Q(X, X)s1
with e+ and + V 1 +  2. We are going to consider a kernel function (constructed from f{)) which is slightly =
ek1
=
CUSP FORMS AND AUTOMORPHIC FORMS
309
twisted from (51 ). We assume that the Q integral lattice L has a Q orthogonal direct sum decomposition in the form .P ED (Ztv ® Zii) with .P s;;; Rk  2 ( the R span of {e2, e3,  · · ,ek _2, ek } ) , satisfying the condition that Q(.P) £ 2 · Z and n!i' · Q(p., p.) E 2 · Z for all p. E .P *(Q), the Q dual to .P. Moreover v and span a 1 . Also assume t is an hyperbolic plane with Q(v, v) Q(ii, ii) 0 and Q(v, ii) integer so that n!i' I t and 4 1 t. Thus the exponent nL of L equals t. Then we let X be a Dirichlet character mod t and consider the formal sum =
=
Bi, x (G, g)
=
�
r mad t
ii
=
=
x(r) B&, ,.(G, g).
Then following Example (3 1 ) we define B�. x(z, g)
(62)
B&, x
=
=
(( [� �1], 1 ) , g) (Im z) l sl t2
� ,B,,x(g) l r l s 1 e" v1rz,
r�1
with z =  yfx  ,.,j  I x 2 e B, the lower halfplane, and ,8 , , x (g) =
�
�
u mod I {MELIQ (M+uv, M+uv) =r)
x(u) Q(M + u v, gl . '+ ) 52. ·
Then using the holomorphic automorphy factor � . we let
[�(g, ./  1 ek)]'2 ,B, ,x ( g1 ) with Z Then we immediately have �, .xCZ)
(63)
=
�, .i Z)
=
�
{
�
u mod l {MELIQ (M+uv, M+uv) =r) ·
=
g ./  1 ( ,.,j2ek) E T(C+).
x(u)
{  t Q(M, ; ) Q(Z, Z) + Q(M, Z) 2
+ Q (M,
./ 2 ii) +
,.,j2ur'l
Moreover �'· X is a holomorphic function on T(C+) satisfying �r. ir · Z) [�(r, Z)]•z · �'· X (Z) for all Z e T(C+ ) and all r_ e O(Q)l . {r e O(Q)! I r(v) = v mod L} . On the other hand we note that e;. X is a holomorphic modular cusp form in the z variable (z e H), i.e. =
=
B&, x< r · z, g)
=
1 j(r . z)2 1' A6, :. 1 (d7 ) x (d7) B&, x (z, g)
for all r e F0(t) where
and _
L AQ, I sl (m) 
=
{ [� �}
SL2( Z) I
c
·=
0 mod
t},
_ 1 s  k/ 2 ( (2m )k (�) m) ' m
a Dirichlet character mod t. Then we define (as in (51 ) ) the Petersson inner product ( B&, x ( , g), h1( ) ) , (i) ¢ holomorphic on where ( f(z)) with / e S21 ,1( F0( t ), ,8) {¢ : (ii) !fo (r · z) = j(r . z)21•1 ,8 (d7)¢(z) for all r e F0(t) and z e (iii) ¢ vanishes at
h1(z)
=
=
H Cl H;
H;
S. RALLIS
3 10
cusps of To(t) on Q U { CXJ } } and {3 (m) = 0.�. l s i ® ,X) (m) (  l /m)2 1 s 1 • We then define (64)
Fi Z i qJ, L, v, X· Fo(t) ) = 2). Then s all for E T(tfi'+) Fo(t)) • F0(t L, qJ, i { r, = ) Z)} ) v, FtCZ Z z L, qJ, v, i Z · X .0J( F1(r X· and all r E 0( Q)i_ v ·

k
The main problem is to give an adequate characterization of the image of this map. The difficult problem is to determine the Fourier coefficients of �r.x · As shown in [23], [11], and [13] such Fourier coefficients are highly transcendental, involving an infinite sum of Bessel functions and certain trigonometric sums, which behave like Kloosterman sums. We know that NF , ( � Rk  2) acts by translation on T(6"+). Thus there exists a lattice !l'F , in Rk 2 = (F1 + F�)j_ so that if � E !l'F , , then F1(Z +
� I qJ, L, v, X · Fo(t))
= F1 (Z I qJ, L, v , X · F0(t))
for all Z E T(tfi'+).
A simple computation shows that !l' F t = .J 2 · { � E !l' I Q (�, �) = 0 mod 2t, Q (!l', �) = 0 mod t } . Hence we have an expansion of F1(Z qJ , L, v, X • To(t)) of the form : I; a ( fJ. , f) ez" ..;IQ C". zJ , (65) F1 (Z i qJ, L, v, x, F0(t)) = 11"' (5!'FI) , (Q) n6'+ where (!l'F ) * ( Q) = {fl. E (F1 + F� ) j_ I Q(fJ., !l'F ) � Z}, the Q dual to !l' F, and
I
(66) a( fJ. , f ) = S
F1(X + .J  1 Y l qJ, L, v, x, Fo(t)) ez" ..; I Q CXUIY, "l dX.
Rk  Zf5fF1
The problem is then to determine a( fJ., f) in terms of the Fourier coefficients off at { w }. We note immediately from the Corollary to Theorem 4.2 that a(fl. , f) = 0 if fl. rt C!l'F )i Q) n tfi'+ · We begin with a heuristic discussion. First we know that the space S2s (F0(t), {3) admits a reproducing kernel function K : H x H > C, a holomorphic (antiholo morphic) function in the first (second) variable which satisfies K( n (z J ), r2 (z2 )) = J(n , ZJ ) 2 s J (r 2, Zz) 2 s {3 (dT! ) {3 (dT2 ) K(ZJ, Zz)
for all Zj , Zz E H and
n,
rz E Fo(t) .
Moreover K satisfies the formula (f(z), K(z, w)) = f(w) ( ( , ) the Petersson inner product) for all j E S2 s{F0(t), /3). Then we know for s sufficiently large, there is a convergent expansion of K(z, w) = .E n;,; I ns1 eZrr .J Inw Gn(z) where Gn is an EisensteinPoincare series belonging to S2s (F0(t), {3) . On the other hand, we know that K(z, w) = K(w, z) . Thus we have another expansion of K(z, w) = I; ns 1 Gn(w) ez" ..; Inz. n �l
The second expansion can be interpreted as the Fourier expansion of K(z, w) in z at { w } (i.e. z .... K(z, w) is an element of S2s (F0(t), {3), where the nth Fourier coefficient nsI Gn(w) is a modular form of antiholomorphic type. We, however, note the analogy of this second expansion with the Fourier ex pansion of B�(z, g) in z given in (36), where each Fourier coefficient is a modular
311
CUSP FORMS AND AUTOMORPHIC FORMS
Thus it is reasonable to ask if there is a "polarization identity" of form on C where $� of the form : �,.;;:; 1 for all is a function which satisfies by an averaging is related to n PF 1 • Also we should require that n P l ' i.e. over .E rEO(Q) L!O (Q) LnpF (qJ�1 ) * (gr) . Zagier in proved such a "polarization identity" i J the case when However such a formula can be proved generally (by Oda in for the cases k � 6, and in a very general version of the formula is shown to be valid without restriction to the case 2).
O(Q). e�(z, g) = ns1(qJ�l)* (g)G,.(z), (qJ�1)* : O(Q) > *(gr) = qJ(qJ�1 �1)(g) (qJg�e1)*O(Q) and all r e ) t � (qJ O(Q)O(Q) L dO(Q)L F ql�1(g) = [23] a = b = 2. [11] [16] b = x(z, Z)Then ewe�. x(z,havegth1)e formul [.@(g, a :  l ek)]•z (see witLeth z =sg 2k. I( vLet2ek) T(C+). I x(z, Z) = n 1 ·1,8!, x(Z)G,.(z, where G,. is the EisensteinPoincare series on the lower halfplane, given by THEOREM
6. 1
.v
(ZAGIER
IDENTITY) .
s;.
(67)
Ei_
>
v'
(6  2)) �
E
=
x).
{nEZ in:;;;  1 )
fi,
(68)
with (note here that e�. x(r·z, g) = s6, x(r)(c7z d7)5 e�. x(z, g) for Fo(t)) and {Q(�, Z) .V2Ut  ]} •z} x(Z) = { ,8,., x {� e 2 x = n,L e (singular horocyclic Q(�, �)O(Q) a(p,f) rE
+
(69)
�
� x(v)
,13:,
v  .
+
{o E SI'I Q (o, o) =n) ;jEZ
v mod t
REMARK 6. 1 . The function ,13:, is obtained from by restriction of the sum mation in (63) to precisely those lattice points of the form + tjv I � with and j Z} . This latter set lattice points) is stable under n P Fl' Then using Theorem 6. 1 , it is possible to determine (see (66)). The main idea is to apply the Lipschitz identity to the inner sum of the righthand side of (69), i.e.
(
)sz
l z r•z1 w m Lets,.;;:;>1 a1(n)Letfe be the Fouriwitehr expansi a Diriocnhletof charact e r t. Then letf( z ) = fatAssume (2Fl) (Q) and fl.¢ 2/(.V2·t). Then a(p,f) = * let (2F)iQ) with fl. = (m/(v'2·t))·� with �. a t h e ot h er hand, On primitive element of the lat ice 2 (m, a positive integer). Then � /....1
m EZ
1
+
(2 7C vI  ) S  (s 2  I) ! _
r =+oo
�
e2" ,J1 wr
r=1
With W E H.
Then we deduce
COROLLARY TO THEOREM 6. 1 .
{ oo} .
mod fl.
E
2k.
.E
n c+ fl.
E
S2.(F0(t), u)
0',
e 2" ,;1 "'
n c+
0.
S. RALLIS
3 12
a(p., f) c ( 11 t)t•z iGx ,
=
a1 ( m: I with X11 the Dirichlet character given by X11(x) a(x)( 1 /x)2• ,(x) and G(x11, t), the Gauss sum given by e t) (here ci is a nonzero constant depending only on s). 1( L, v, X11, 5. (610)
. :E x" () 11 02(A) is surjective from .9"(K�) to .9"( Oz (k) \ 02( A ) ) . PROOF. Let (xb x2) be a point in K2 such that
e: g
>
x1
and x2 span K over k. The map
(gxb gx2)
of 02(A) into K1 is a smooth injection. It also has closed image. This can be seen i n various ways, but perhaps most easily b y noting that the image o f 02(k) consists of rational points and is therefore discrete, and 02(k)\02(A) is compact. Since the image of e is closed , the pullback e* : C�(K1) > Coo( Oz (A))
is surjective onto C�( 02(A )) . Let/ E C� (K1) vanish at all points of K 2 not in the image of e. Then by Witt's Theorem [J] it is easy to compute that (2.8)
01(g) =
L: e*(f)(hg).
k 02 (k)
Since it is well known that all f in Coo(02(k)\ 02(A)) are of the form given on the righthand side of (2.8), the proposition follows. REMARK. The same result with the same proof holds for an arbitrary irreducible type I dual reductive pair (G, G') (see when the space with form attached to G has an isotropic subspace of dimension at least equal to the dimension of the formed space attached to G'.
[H])
'Actually it can be shown that 01 will transform under Sp.(A) according to an irreducible repre sentation a' determined by a , but this finer result is unnecessary for the present purpose and would unduly lengthen the paper.
COUNTEREXAMPLE TO GENERALIZED RAMANUJAN CONJECTURE
319
I f K. i s a field, i.e., i f K does not split a t then will b e anisotropic and will be compact. Therefore SI'(K�) will break up into a discrete direct sum of spaces transforming under (k.) according to its (one or two dimensional) irreducible representations. Asmuth has established the following facts [A]. (See also [H].)
v, {3.
02
PROPOSITION 2. If K
02(k.)
sum decomposition does not split over at v, then there is an orthogonal direct over ch actspmodule. saceby theTherestmappi ricitsioninrreduci ogf btole underalthl eirjoireduci nandt tacthbusleiohasnrepresent ofthe02form (k.at)ioandns of 0as2(k.0()whi. Each 2 2(k.) into the (irreducible admissible deunifitnesarizanablein)jectirepresent on of athetionsrepresent a t i o ns o f 0 ofl on S02, (the siigsnumthe nontri vialat!diion)mtensi onalis super repre whi c h i s t r i v i a represent h en sentati o n of 0 2 cuspidal. 02 J,f' 02(k.) ( (r t r1 �t1 ) ) f t l rt l 1 f' (O, J f(xl > x2) dx1 dx2. l r 4 t 2 1 ., {3. 02 k
SI'(K;)
1: SI'(K�, O")
�
a
SI'(K;, O" )
Sp4(k.)
SI'(K�, O") )
O"
®
0"
1
x
w.
0" +
Sp4
Sp4(k.). If O"
O"
0"1
'
If O" is any representation of other than the signum, the corresponding O"' is not even tempered [KZ]. We will show this is true when O" ' is the trivial representa tion, which will suffice for the present purpose. The point is that there are functions e SI'(K;) which are invariant under and which are positive and do not vanish at zero. We compute the matrix coefficient
(2.9)
0 0 0 0 J, w. 0 0 0 0 0
(/')
Since has compact support, we see that for r and sufficiently large, (2.9) becomes
0)
±
Kv
Since the afunction of Sp4 is we see the matrix coefficients (2.9) decay too slowly at oo for O" ' to be tempered. If K. is not a field, then is a hyperbolic plane whose isotropic lines are the two places of K lying above v. The joint action of and Sp4 on SI'(K�) can be analyzed in a manner directly analogous to Proposition 2, but the work is slightly longer since the decomposition is continuous rather than discrete. Since we cannot refer to the literature, we prove only what we need. We consider only nonArchimedean places. Let C and C' be the subgroups of integer matrices in (k and Sp4(k.). Let SI'(K;)C' be the C'fixed vectors in SI'(K;).
There is a linear isomorphism ( 02 The map is an 02(k.) intertwining map.
02 .)
PROPOSITION 3.
a
a : CC:
(k.) / C)
_..,
SI'(K�)c' .
PROOF. By performing a partial Fourier transform ( see [RS] for the analogy in
R. HOWE AND I. I. PIATETSKISHAPIRO
320
the (02 , Sl2) pair) we find w': is equivalent to the representation of Sp4(k.) on .9'(k�) induced by the natural linear action of Splk.) on k�. In this realization of w';', the action p';' of S02(k.) becomes the (scalar) dilations of kt normalized for unitarity. To get 0 2 , you add Fourier transform on kt with respect to the symplec tic form
of which Sp4(k.) is the isometry group. It is easy to check that the orbits of C' in kt are the sets 11:k((J�  11:0t) , together with 0. From this one sees that .9'(k�)c' has a basis composed of the functions (h. where f/Jk is the characteristic function of 11:k8t. All the f/Jk are invariant by dilations of x: by elements of � Assuming the Fourier transform 1\ on k� to be normal ized properly, as we may, it will be seen that (2. 10) In particular ¢0 is invariant under lV� and under A, that is, under C. Define a map by
pf¢
a (/) = .( )( 0) for
f
e
C{ 0 2(k.)/C) .
Since the dilations by powers of 11: are a set of coset representatives for p.(02(k.))fp.(C), it is easy to see that a is a linear isomorphism. It is also trans parently an 02 (k.) intertwining map, so the proposition follows. We will say an irreducible admissible representation of Sp4(k.) is if it contains a C'fixed vector. Similar terminology applies to 02(k.).
C'spherical an (irreduci ble admissibinletertwi ) C'nspheri cal represent a t.9'io(nK�)of to the spLetaceSuppose Y11of• be thThen eis kernel of all i n g operat o rs from the joint actionwhereof is an irreducible Cspheri on cal (K�)/ isonirreduci .9'representati b l e , and of t h e form of ationsTheofcorrespondence from the asettioonsfallof Cspherical represent into the set of iC's anspinherijecticalonrepresent a
PROPOSITION 4. Sp4(k.). Ya'
'
Sp4(k.)
' a .
a
0 2(k.).
02 (k.)
®
' a , a
+
a
'
a
02 (k.)
x
Sp4(k.)
Qa '
=
Sp4(k.).
PROOF. Since C' is compact, the natural projection map .9'(K�)C ' + Q�; is sur jective. The space Q�: will be invariant by 02(k.), and since there is precisely one C' fixed vector in a ' [C), Q... will be irreducible under 02 x Sp4 if and only if Q�; is irreducible under 02 • According to Proposition 3, there is a surjective 02 inter twining map
a' : C';"(0 2/C) + Q�: . Therefore the irreducible 0 2 representations in Q�: will correspond to the Cfixed vectors in Q�;. So to show irreducibility of Q .. under 02 x Sp4, it will suffice to show that Q�' xc is onedimensional. To this end, consider the action of the Heeke algebras [C] C';"(02 //C) and C';"(Sp4//C') on .9'(kt)C x C', as described in Proposi
COUNTEREXAMPLE TO GENERALIZED RAMANUJAN CONJECTURE
32 1
tion 3. (We will pass freely between .?(k!) and .?(K�).) If the ¢k are as in (2. 10), then it is easy to see that the set of functions qZk¢k + qZk ¢ k where q = l 1rl;;1 form a basis for .?(k�)Cxc' . Let T1 be the Heeke operator on 0 2 defined by T1 (f)(x) = qZ /(1rx) + q2J( 7r  1 x) for f e .?(k�)cxc' . Then T1 generates the Heeke algebra C';'(02 //C). Let Ti. and T� be the Heeke operators on Sp4 , each of total mass 1 and supported on the C' double cosets with respective coset representatives
[� ! � g] [� � � g ]· and
1
0 0 0
1
1
0 0 0
1t'  1
These operators generate C';'(Sp4 //C'). Direct computation yields the equations T1 (¢o) = q z f/J  1 + q2¢ � o (2. 1 1 )
Ti( ¢o) =
1 ((q  1 ) ¢  1 + (q4  2q + 1 ) ¢0 + q4(q  1 ) ¢1 ), q( q 4 _ I )
T2(¢o) =
1 ((q2  1 ) ¢ 1 + (q4  2 q2 + 1 ) ¢o + q4(q2  1) ¢1 ) · q2(q4 _ 1)
From Proposition 3, we know that .?(k�)cxc' is isomorphic as C':'(02ffC) module to the regular action of C':'(0 2//C) on itself. In particular ¢ 0 is a cyclic vector for .?(k�)cxc ' under the action of T1 . Since T1 and the Tj commute with each other we have, for any fin .?(k�)cxC' , (2. 1 2)
q(q4  1 ) Ti_(f) = ( q4  2 q + 1)/ + qZ(q  1 ) T1 (f), q2 (q4  1 ) T2(f) = (q4  2qZ + 1 ) / + q2(q2  1 ) T1 (f).
Now return to Q�,xc' . It will be the quotient of .?(K;)cxc' . Therefore if ip0 is the image of ¢0 in Q�,xc' , then ifio is a cyclic vector for T1 acting on Q�,xc' . On the other hand, since Q11, is a sum of copies of (}' 1 as Sp4 module, we see ifi o will be an eigen vector for Ti and T?_. Hence by (2. 1 1), we see ip0 will be an eigenvector for Tr . so Q�,xc ' is onedimensional as desired. Hence Q11, � (}' ® (}' 1 • Furthermore (2. 1 2) shows that the eigenvalue of T1 acting on ip0 determines those of T{ and T2 , and vice versa. Since these eigenvalues determine in turn (}' and (}' 1 [C], we see also that 1 (}' + (}' is injective, so the proposition is proved. We note that, from the realization of w';: on .?(k�). it is obvious that no C' spherical quotient of w';: can be tempered, because all such quotients will admit a distribution which is invariant by the isotropy group in Sp4 of a point in k�, and existence of such a distribution precludes temperedness. We may now proceed as indicated above. Select/ in .?(K1) such that 81 trans forms under 0 2(A) according to an irreducible representation whose local com ponent is the signum representation at at least one place where K does not split over Such representations obviously exist. Then under SplA), we know from Propositions 2 and 4 that 81 will transform according to an irreducible representa
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tion 17;, determined by 17 v• of Sp4(kv) for all places v where K does not split over k, or where K does split and 81 i s fixed by c;. Together, these account for almost all places. Since 81 transforms by a supercuspidal representation at at least one place, it will be a cusp form. Therefore the (£2 closure of) the SplA) translates of 81 will decompose into a direct sum of cuspidal automorphic representations. At all the places mentioned above, the local components of these automorphic representa tions will have to be the 17; determined by the 17v· Therefore these automorphic representations will be nontempered at almost all places. This completes the con struction. To conclude, we note that if k Q and K is an imaginary quadratic extension, then some of the forms we construct will be ordinary Siegel modular forms of weight Also, since automorphic forms on 02 obviously fail to satisfy strong m ultiplicity one, the corresponding forms on Sp4 will also contradict strong multiplicity one. =
I.
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