Automorphic forms and Shimura varieties of PGSp (2)

Automoiphic Forms and Shirnura Varieties PQ( of 2) This page intentionally left blank Automorlhk FONTIS and Shimur...

Author: Yuval Z Flicker

39 downloads 886 Views 13MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

Automoiphic Forms and

Shirnura Varieties PQ( of

2)

This page intentionally left blank

Automorlhk FONTIS and Shimura Varieties PGsp( I) of

hby

Yuval

Z Flicker

The Ohio State University, USA

1;World Scientific NEW JERSEY * L O N D O N * SINGAPORE * BElJlNG

*

SHANGHAI

*

HONG KONG

-

TAIPEI

*

CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd.

5 Toh Tuck Link, Singapore 596224 USA ofice: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK ofice: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library.

AUTOMORPHIC FORMS AND SHIMURA VARIETIES OF PGSp(2) Copyright 02005 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereoj may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-256-403-9

Printed in Singapore.

PREFACE This volume concerns two main topics of interest in the theory of automorphic representations, both are by now classical. The first concerns the question of classification of the automorphic representations of a group, connected and reductive over a number field F . We consider here the classical example of the projective symplectic group PGSp(2) of similitudes. It is related to Siege1 modular forms in the analytic language. We reduce this question to that for the projective general linear group PGL(4) by means of the theory of liftings with respect to the dual group homomorphism Sp(2, C ) ~f SL(4, C). To describe this classification we introduce the notion of packets and quasi-packets of representations - admissible and automorphic - of PGSp(2). The lifting implies a rigidity theorem for packets and multiplicity one theorem for the discrete spectrum of PGSp(2). The classification uses the theory of endoscopy, and twisted endoscopy. This leads to a notion of stable and unstable packets of automorphic forms. The stable ones are those which do not come from a proper endoscopic group. This first topic was developed in part to access the second topic of these notes, which is the decomposition of the &ale cohomology with compact supports of the Shimura variety associated with PGSp(2), over an algebraic closure F , with coefficients in a local system. This is a Hecke-Galois bi-module, and its decomposition into irreducibles associates to each geometric (cohomological components at infinity) automorphic representation (we show they all appear in the cohomology) a Galois representation. They are related at almost all places as the Hecke eigenvalues are the Frobenius eigenvalues, up to a shift. In the stable case we obtain Galois representations of dimension 4[F:Q1. In the unstable case the dimension is half that, since endoscopy shows up. The statement, and the definition of stability, is based on the classification and lifting results of the first, main, part. The description of the Zeta function of the Shimura variety, also with coefficients in the local system, follows formally from the decomposition of the cohomology. The third part - which is written for non-experts in representation theory - consists of a brief introduction to the Principle of F’unctoriality in the theory of automorphic forms. It puts the first two parts in perspective. Parts 1 and 2 are examples of the general - mainly conjectural - theory described in this last part. Part 3 can be read independently of parts 1 and 2. It can be consulted as needed. It contains many of the definitions V

vi

Preface

used in parts 1 and 2, but is not a prerequisite to them. For this reason this Background part is put at the back and not at the fore. Regrettably, it does not discuss the trace formula. But this would require another book. Part 3 is based on a graduate course at Ohio State in Autumn 2003.

Yuval Flicker Jerusalem, Tevet 5765

ACKNOWLEDGMENT It is my pleasure to acknowledge conversations with J . Bernstein, M. Borovoi, G. Harder, D. Kazhdan, J.G.M. Mars, S. Rallis, G. Savin, R. Schmidt, J.-P. Serre, R. Weissauer, Yuan Yi, E.-W. Zink, Th. Zink, on aspects of part 1 of this work. P.-S. Chan suggested many improvements to an earlier draft, and to part 3. For useful comments I thank audiences in HU Berlin, HU Jerusalem, Princeton, Tel-Aviv, MPI Bonn, MSC Beijing, ICM’s Weihai, Saarbrucken, Koln, Lausanne, where segments of this part 1 were discussed since the completion of a first draft in 2001. This work has been announced in particular a t MPI-Bonn Arbeitstagung 2003, ERA-AMS and Proc. Japan Acad. [F6], in 2004. I wish t o thank M. Borovoi for encouraging me to carry out part 2 after I communicated to him the results of part 1, R. Pink, D. Kazhdan, and A. Panchishkin, for inviting me to ETH Zurich, and the Hebrew University in Dec. 2003, and Grenoble in Jan. 2004, and the directors, esp. G. Harder and G. Faltings, for inviting me to MPI, Bonn, in summer 2004, where I talked on part 2 of this work. My indebtedness to the publications of R. Langlands and R. Kottwitz is transparent. I benefited also from talking with P. Deligne, N. Katz, G. Shimura, Y. Tschinkel. The interest of the students who took my Autumn 2003 course at Ohio State prompted me to write-up part 3. The editorial advice of Sim Chee Khian and TeXpertise of Ed Overman are appreciated. Support of the Humboldt Stiftung at Universitat Mannheim, Universitat Bielefeld, Universitat zu Koln, and the Humboldt Universitat Berlin, of the National University of Singapore, of the Max-Planck-Institut fur Mathematik a t Bonn, and of a Lady Davis Visiting Professorship a t the Hebrew University, is gratefully acknowledged.

vii

This page intentionally left blank

CONTENTS PREFACE ACKNOWLEDGMENT

V

vii

PART 1. LIFTING AUTOMORPHIC FORMS OF PGSp(2) I.

TO PGL(4)

1

PRELIMINARIES 1. Introduction 2. Statement of Results 3. Conjectural Compatibility 4. Conjectural Rigidity

3 3 4 28 30

11. BASIC FACTS 1. Norm Maps 2. Induced Representations 3. Satake Isomorphism 4. Induced Representations of PGSp(2, F ) 5 . Twisted Conjugacy Classes

35 35 39 46 47 50

111. TRACE FORMULAE 1. Twisted Trace Formula: Geometric Side 2. Twisted Trace Formula: Analytic Side 3. Trace Formula of H: Spectral Side 4. Trace Formula Identity

60 60 66 73 78

IV. LIFTING FROM SO(4) TO PGL(4) 1. From SO(4) to PGL(4) 2. Symmetric Square 3. Induced Case 4. Cuspidal Case

86 86 94 97 101

V.

LIFTING FROM PGSp(2) TO PGL(4) 1. Characters on the Symplectic Group 2. Reducibility 3. Transfer of Distributions 4. Orthogonality Relations ix

113 113 119 122 129

Contents

X

5. 6. 7. 8. 9. 10. 11.

Character Relations Fine Character Relations Generic Representations of PGSp(2) Local Lifting from PGSp(2) Local Packets Global Packets Representations of PGSp(2, R)

VI. FUNDAMENTAL LEMMA 1. Case of SL(2) 2. Case of GSp(2)

133 140 148 152 160 161 174 182 182 184

PART 2. ZETA FUNCTIONS OF SHIMURA VARIETIES OF PGSp(2) 205 I. PRELIMINARIES 207 1. Introduction 207 2. Statement of Results 209 3. The Zeta Function 218 4. The Shimura Variety 222 5. Decomposition of Cohomology 223 6. Galois Representations 225 11. AUTOMORPHIC REPRESENTATIONS 227 1. Stabilization and the Test Function 227 2. Automorphic Representations of PGSp(2) 229 3. Local Packets 231 4. Multiplicities 234 234 5. Spectral Side of the Stable Trace Formula 6. Proper Endoscopic Group 236 111. LOCAL TERMS 1. Representations of the Dual Group 2. Local Terms at p 3. The Eigenvalues at p 4. Terms at p for the Endoscopic Group

238 238 240 245 247

N. REAL REPRESENTATIONS

248 248 249

1. 2.

Representations of SL(2,R) Cohomological Representations

Contents

3.

Nontempered Representations

4. The Cohomological L(vsgn, ~ - ~ / ’ 7 r 2 k ) 5. 6. 7.

V.

The Cohomological ~ 5 ( { d / ~ 7 r 2 k + l ,a v-l/'. EXAMPLE. In the case of GSp(2, F ) and P(1,1),the representation denoted I ~ ( p 1p2) , normalizedly induced from the character P = udiag(a, b, X / b , X / a ) H Cl1(ab/X)ILa(a/b) = pL1p2(a)(CL1/CL2)(b)C111(X)

is the same as p1p2 x p1/p2 xlpL1. Its central character is trivial, namely it is a representation of H(F) = PGSp(2,F). If J2 = 1 then I ~ ( J p 1 , J p z=) PlP2 x

cLlIcL2

UPl.

4. Induced Representations of PGSp(2,F)

49

4.1 LEMMA.(i) The centraZcharacterof7rlx...x7rTxa isw,w,, ... w,,. if (nl < n; here wTi are the central characters of 7ri (w, ofa,a being a representation of GSp(n- In[,F ) ) . It is a2w,, . . . w,, zf In1 = n. (ii) For a character p we have p(7r1 x . . . x T, x a ) = 7r1 x . . . x 7rr >a pa. I n particular p ( M~a ) = T x po. (iii) W e have 7r x o = ir x w,a Recall that two parabolic subgroups of a reductive group G over F are called associate if their Levi subgroups are conjugate. This is an equivalence relation. An irreducible representation 7r of G = G ( F ) , F a padic field, is supported in an associate class if there is a parabolic subgroup P in this class such that 7r is a composition factor of a representation of G induced from an irreducible cuspidal representation of the Levi factor M of P extended trivially to the unipotent radical U of P. In our case an irreducible representation 7r of H = PGSp(2,F) is supported in P(1,1), P(11,P p , or it is cuspidal. An unramified representation is supported on P(l,l). It is a subquotient T H ( ~ I , ~ ofL ~a )fully induced I ~ ( p 1p2) , = p1p2 x p l / p 2 M p I 1 , where the pi are unramified characters of F X . An irreducible representation T is called essentially tempered if v% is tempered for some real number e , where (v"n)(g) = v(detg)%(g). The following is the Langlands classification for GSp(n, F ) . 4.2 PROPOSITION. Each representation ve17r1 x . . . x uer7rr x u, where el 2 . . . 2 e, > 0 , 7ri are irreducible square integrable representations of GL(ni, F ) , and a is an irreducible essentially tempered representation of GSp(n - In\,F ) , has a unique irreducible quotient: L(ve17rl,. . . ,ver7rr,a ) . Each irreducible representation of GSp(n, F ) is of this form. 0

With these notations we shall use the results stated in [ST]. These concern the reducibility of the induced representations, and description of their properties. In particular [ST], Lemma 3.1 asserts that for characters X I , x 2 , a of F X the representation x1 x x2 >a a is irreducible if and only if xi # v f l and x1 # v f 1 x ; l . In case of reducibility the composition series are described in [ST], together with their properties. The list is recorded in chapter V, section 2, 2.1-2.3 below. Moreover, we shall use [ST], Theorem 4.4, which classifies the irreducible unitarizable representations of GSp(2, F ) supported in minimal parabolic subgroups. It shows that

II. Basic Facts

50

4.3 LEMMA. The representation L ( v x v tarizable.

>a

v-') = 7 r ~ ( v1), is not uni-

0

5 . Twisted Conjugacy Classes The geometric part of the trace formula is expressed in terms of stable conjugacy classes, whose definition we now recall. We shall need only strongly regular semisimple (we abbreviate this to "regular") elements t in H = H(F), those whose centralizer &(t) in H is a maximal F-torus TH. The elements t , t' of H are conjugate if there is g in H with t' equal to Int(g-')t (= g - l t g ) . Such t , t' in H are stably conjugate if there is g in H(= H(F)) with t' = Int(g-')t. Then gu = ga(g)-' lies in TH for every CT in the Galois group r = Gal(F/F), and g H {a H g b } defines an isomorphism from the set of conjugacy classes within the stable conjugacy class o f t to the pointed set D ( T H / F ) = ker[H1(F,TH) 4 H1(F,H)]. Using the commutative diagram with exact rows

H 1 ( F ,TE)

+

H 1 ( F ,Hsc)

1 --$ D ( T H / F ) + H 1 ( F , T ~ )+

H1(F,H),

-

1

I

where Hsc5 Hder H is the simply connected covering group of the derived (commutator [H,H])group Hderof H, and noting that for a padic field F one has H1(F,HSC) = {l},one concludes that D ( T H / F ) = Im[H'(F, Tsc)-+ H 1 ( F ,T)] for such F , in particular it is a group. Here T S C - r-'(Te'), = Hde'nTH. Indeed, if {a H gu} is in D ( T H / F ) , thus gu = ga(g)-', write g = g l z , using H = HderZH(ZH denotes the center of H), with z in ZH and g1 in Hder.Then gu = gIUzu,and zu = za(z)-' is a coboundary, as ZH C TH,and H'(F,TE) surjects on D ( T H / F ) . It is convenient to compute H'(F,TH) using the Tate-Nakayama isomorphism which identifies this group with

TF

H - ' ( F , X,(TH)) = { X E X,(TH);NX = O } / ( X - TX;T E Gal(L/F)). Here L is a sufficiently large Galois extension of the local field F which splits TH,N denotes the norm from L to F , and X,(TH) is the lattice Hom(Gm 1 T H ) *

5. Twisted Conjugacy Classes

51

In our case of H = PSp(2) = PGSp(2), Hsc is Sp(2), and H'(F,H) = {0), hence D ( T H / F )= H 1 ( F ,TH)is a group. Denote by N the normalizer Norm(TT,,H) of T& in H, and let W = N / T & be the Weyl group of TT, in H . Signify by H1(F,W ) the group of continuous homomorphisms 6 : r 4 W , where r acts trivially on W . 5.1 LEMMA.The set of stable conjugacy classes of F-tori in H injects naturally in the image of ker[H1(F, N ) 4H1(F,H)]in H 1 ( F ,W ) . When 0 H is quasi-split this map is an isomorphism. This is proven in Section 1.B of [F5] where it is used to list the (stable) conjugacy classes in GSp(2). Our case of H = PSp(2) is similar but simpler. The Weyl group W is the dihedral group D4,generated by the reflections s1 = (12)(34) and s2 = (23). Its other elements are l,(l2)(34)(23) = (3421) (taking 1 to 2, 2 to 4, 4 to 3, 3 to l), (23)(12)(34) = (2431), (23)(3421) = (42)(31), (3421)' = (32)(41), (23)(23)(41) = (41). Our list of F-tori follows that of loc. cit. The list of F-tori TH is parametrized by the subgroup of W . If THsplits over the Galois extension E of F then H 1 ( F , T ~=) H1(Gal(E/F), T & ( E ) )where T & ( E )= {t = diag(a, b, X/b, X / a ) mod ZH; a, b, X E E X }and Gal(E/F) acts via p : -+ W . If p ( ~ = ) Int(g,) then I' acts on TT, by a*(t) = g, . a t . gil, and at = ( a a ,a b , a X / o b , aX/aa) mod ZH. The split torus corresponds to the subgroup (1) of W , its stable conjugacy class consists of a single class. There are nonelliptic tori T H ,with trivial H1(F,TH),corresponding to p(r) being ((23)), ((14)), ((12)(34)), ((13)(24)). The elliptic tori are: (I) p ( r ) = ((14)(23)),TH N {diag(a,b,X/b = ab,X/a = a a ) ; a,b E E X , X E N E I F E ' ) , [E:F]=2. To compute D(TH/F) we take the quotient of X*(TZ) = ((z, y, -y, -z);z,y E Z) (note that the generator a of Gal(E/F) maps (z,y, z - y, z - z) to ( z - z, z - y, y, z) and the norm N E / F = N is the sum of the two) by the span ( X - a X = (z, y, z - y, z 2) - ( z - z, z - y, y, z) = (22 - z , 2y - z , z - 2y, z - 22)) (X ranges over X,(TH));it is 2/2. (11) p ( r ) = ((14)(23),(12)(34),(13)(24)),then TH splits over an extension E = E1E2, biquadratic over F , Gal(E/F) = 2 / 2 x Z/2 is generated Ez = E("'), El = E ( T ) , by a and T whose fixed fields are EB= say p ( a ) = (14)(23), p(7) = (12)(34). Then H-l is the quotient of ((x,y, -y, -2)) by ((2z - z , 2y - z , z - 2y, z - 2z)), namely 2/2.

II. Basic Facts

52

(111) p ( r ) = ((14),(23)), again E = E1E2, Gal(E/F) = 2 / 2 x 2 / 2 is generated by a and T with E3 = E ( " ) ,E2 = E("'), El = E ( 7 ) ,and p ( ~ =) (23), TO) = (14),and H - l is {0}, being the quotient of ((2,y, -y, -x)) by ((2x - *, 070, z - 2x1, (0,2Y- .z, z - 2Y,0)) = ((2,0,0, -Z), (0, y, -y, 0)). (IV)When p(r) contains an element of order four, H-' is {0}, as explained in [F5], I.B, (IV). Next we describe the (stable) 0-conjugacy classes of a strongly ®ular element t in G. We fix a &invariant F-torus T*, to wit: the diagonal subgroup. The stable 8-conjugacy class of t in G intersects T* ([KS], Lemma 3.2.A). Hence there is h E G ( = = G ( F ) )and t* E T* such that t = hM1t*8(h).The centralizers are related by Z ~ ( t 6= ) h-'ZG(t*O)h, and ZG(t*8) = T*'. The centralizer of &(to) in G is an F-torus Tt: it is ZG (ZG ( t o ) )

= h-lT*h = Tt. The torus

Tt is Ot = Int(t) o &invariant:

We have zG(t6) = TP: if tl E Z ~ ( t 8=) h-lT*'h C h-'T*h = Tt then . t e . tl = te, thus e t ( t l )= te(tl)t-l = tl. The 8-conjugacy classes within the stable 8-conjugacy class of t can be classified as follows. If tl = g-'tB(g) and t are stably &conjugate in G then gu = ga(g)-l E Z G ( t 6 ) = T:t. The set

t;l

D ( F ,0 , t ) = ker[H'(F, TP) + H 1 ( F ,G ) ] parametrizes, via ( t l , t )H {a H gu}, the 0-conjugacy classes within the stable 8-conjugacy class oft. The Galois action on Ti: = a(h-lt*e(h))= h-I

. h a ( h ) - l . a ( t * ) e@(h)h-l)e(h), .

induces the Galois action a* on T*, given by a*(t*)= ha(h)-' . a ( t * ). O(a(h)h-'),and H1(F,Tp) = H 1 ( F ,T*').


53

The norm map N : T* TL is defined to be the composition of the projection T* TZ = T*/(l - O)T*and the isomorphism TZ TL. If the norm Nt* o f t * E T* is defined over F then for each 0 E r there is l E T* such that a*(t*) = lt*O(l)-'. Then ---f

---f

h-lt*o(h) = t = a(t)=

at *.~ ( ~= ha(h)-let*e(e-la(h)), )

hence t* = h,l

*

t* 19(h,l)-', *

h, = ha(h)-',

and h,l E ZG(t*O) = T*', so that h, E T*. Moreover, (1 - 8)(h,) = t*a*(t*)-', hence (h,,t*) lies in the subset

and it parametrizes the 8-conjugacy classes of strongly ®ular elements which have the same norm. We put Vt = (1- &)Ti. While not necessary in our case, recall that the first hypercohomology f group H1(G,A-+B) of the short complex A +f B of G-modules placed in degrees 0 and 1 is the group of 1-hypercocycles, quotient by the subgroup of 1-hypercoboundaries. A 1-hypercocycle is a pair ( a , @with ) a being a 1-cocycle of G in A and p E B such that f ( a ) = a@;ap is the 1-cocycle 0 H @-'a(@) of G in B. A 1-hypercoboundaryis a pair (aa,f(@)), a E A. This hypercohomology group fits in an exact sequence

We need only the case where A = Tt, B = Vt = (1- &)Ti,f = 1 - O t , G = Gal(F/F). The exact sequence 1 4 TP 4 Tt ' 3 Vt + 1 induces the exact sequence H o ( F ,Tt) = TF = Tt + H o ( F ,Vt) = V,

Hence H1(F,T!t) = H1(F,Tt3 V t ) . If t is a strongly ®ular element in G, then Tt = zG(zG(te)') is a maximal torus in G . Denote by TF the inverse image of Tt under the natural homomorphism 7r : Gsc -+ GderL, G. Note that G = n(GSC)Z(G).

II. Basic Facts

54

If tl = gP1t8(g) E G is stably &conjugate to t E G then g = r ( g 1 ) t for some g1 E Gsc and t E Z(G). Then a ( g l ) g c l lies in TF, and (1-Ot)r(c(gl)g;')

= a(b)6-l

where 6 = O ( Z ) Z - ~ = ( 1 - O t ) ( t - ' )

-

E

V,;

(1-8tb

a(g1)gT1,b) defines an element inv(t,tl) of H1(F,TY

Vt). This element parametrizes the 8-conjugacy classes under Gsc within the ( ~ 7H

stable 8-conjugacy class of t. The image in H 1 ( F ,Tt ' 3 V,) under the map [TT ---f V,] 4 [T, --+ V,] induced by r : T:' 4 Tt is denoted by inv'(t, t l ) . The set of 8-conjugacy classes within the stable 8-conjugacy class of t , D ( F ,0, t )

is the image under

of (1-8t)"

hence a subset of the abelian group

In our case of G = PGL(4), the pointed set H1(F,G) is trivial, hence

D ( F , O , t )= H1(F,T2)= H 1 ( F , T t 1 2 V t ) . Since H1(F,T)is trivial for every maximal torus T, we have that H1(F,TtZ V , ) is & / ( l h h h We list the stable 0-conjugacy classes of strongly ®ular elements t in G = PGL(4) as in [F5]. Thus we describe the F-tori T, as ZG(t8) = T> and T = Tt = Z G ( Z G ( t 0 ) ) . The conjugacy classes of F-tori T are determined by the homomorphisms p : r + W = W(T*0,G8) = W ( T * , G ) 8 . We list only the &elliptic, or &anisotropic (T8 does not contain a split torus) as the other tori can be dealt with using parabolic induction. (I) p ( F ) = ((14)(23)),[ E : FI = 2, T" = { ( a ,6, ab, a a ) ;a , 6 E E X } / Z ;

V = { ( a , b, b, a ) } / Z .


55

Hence V = { ( a ,b, b, a ) = ( z a a ,zab, zab, z a a ) ; z , a , b E E X } .Then a / a a = b/ab, or a / b = a ( a / b ) , and ( a ,b, b, a ) = (1, b / a , b/a, 1) with b/a in F X . Finally ( 1 - O)T*= { ( a a a ,bob, bab, aaa);a, b E E X } / Z , V / ( l - O)T*= F X / N E X= 2 / 2 . (11) p ( r ) = ( p ( ~ =) ( 1 4 ) , p ( ~ )= (23)). The splitting field of T is E = E1E2, where El = E ( f i ) = E ( T ) l

are the quadratic extensions of F in E. Then

(since ( a ,b, b, a ) = (TO,rb, T b , T U ) = (aa,ab, ab, a a ) mod Z implies a / b E FX), (1 - Q)T*= { ( a a a ,k b , b ~ baaa); , a E ET , b E E ; } / Z , hence

(111) p(r) = ( p ( ~ = ) (12)(34),p(a) = (14)(23)), the splitting field of T is E = E1E2, a biquadratic extension of F , Gal(E/F) = ( l , a , ~ , a ~ ) , El = E(') = F ( f i ) , E3 = E(") = F ( & i ) , Ez = E("') = F ( m ) are the quadratic subextensions, and so T* = { ( a ,T U , TOU, a a ) ;a E E X } / Z , and (1- O)T*= {(u~u,~(u~u),~(u~u),u~u);u E EX}/Z. Now V consists of ( a , b, b, a ) which equal (aa,ab, ab, aa) mod 2. Thus a / b = a ( a / b ) lies in E" = E3, and also ( a ,b, b , a ) = ( ~ b , ~ u , ~ u , ~ b ) m o Hence b/a = T ( u / ~ )and , so f = u / m , u E hE:. Then

( a ,b, b, a ) = ( b u / ~ ub,,b, b u / ~ u=) (u, T U , T U , u) Hence V / ( l - O)T* is E:/NE/E3EX = 2/2. (IV) If p ( r ) contains p ( a ) = (3421), T is isomorphic to the multiplication group E X of an extension E = F ( a ) = E 3 ( f i ) of F of degree

II. Basic Facts

56

4, where E3 = F(&) is a quadratic extension of F ( A E F - F 2 , D = a , B a E E3). The Galois closure E / F of F ( a ) / F is E = F ( G ) when F ( f i ) / F is cyclic, and E = F ( a ,C ) when F ( a ) / F is not Galois. Here C2 = -1 and Gal(fi/F) is the dihedral group Dq. In either case T* = { ( a ,a a , a3a,a2a); u E E X } / Z ,

+

(1 - 8)T* = { ( u a 2 a , ~ ( a a 2 ~ ) , a ( ~ a 2 a ) l a a E 2 aE)X ; a} / Z ,

and V consists of ( a ,b,b, a ) with ( a b , a a , aa, a b ) = ( a ,b, b, a ) z , thus a / a b = b / a a , or a / b = a b / u a so a / b = a 2 ( a / b )lies in E3, and a / b = u/aufor some u E E t . Thus ( a ,b, b, a ) = (bu/au,b, b, bu/au) (u, cm, cm, u), u E E:, and V/(1 - 8)T* is E:/NEIE3EX = 2/2. We recall some results of [F5] concerning representatives of (stable) 8twisted regular conjugacy classes. These are listed according to the four types of &elliptic classes: I, 11, 111, IV. A set of representatives for the 8-conjugacy classes within a stable semi simple 8-conjugacy class of type I in GL(4, F ) which splits over a quadratic extension E = F(@) of F , D E F - F 2 , is parametrized by (r,s) E F X / N E I F E Xx F X / N ~ , ~ E ([F5], X p. 16). Representatives for the 8regular (thus t 8 ( t ) is regular) stable 8-conjugacy classes of type (I) in GL(4,F ) which split over E can be found in a torus T = T ( F ) , T = h-lT*h, T* denoting the diagonal subgroup in G , h = 8 ( h ) , and

Here a = a1 + a 2 a r b = bl + b2@ E E X, and t is regular if a / a a and blab are distinct and not equal to f l . Note that here T* = T * ( F )where the Galois action is that obtained from the Galois action on T . A set of representatives for the 8-conjugacy classes within a stable 8conjugacy class can be chosen in T . Indeed, if t = h-lt'h and tl = h-ltTh in T are stably &conjugate, then there is g = h - l p h with tl = gt8(g)-11 thus t; = pt*O(p)-l and t;O(t;) = pt*O(t*)p-l. Since t is 8regular, p lies in the 8-normalizer of T * ( F )in G ( F ) . Since the group W e ( T *G , ) = Ne(T*G , ) / T * ,quotient by T * ( F )of the 8-normalizer of


57

T*(F)in G ( F ) ,is represented by the group W e ( T * , G )= N ' ( T * , G ) / T * , quotient by T* of the 8-normalizer of T* in GI we may modify p by an , is replace tl by a &conjugate element, and element of W e ( T * , G )that assume that p lies in T*(F). In this case pO(p)-l = diag(u,u',au',au) (since t , tl lie in T * ) ,with u = au, u' = uu' in F X . Such t , tl are &conjugate if g E G I thus g E TI so p = diag(v,v',av',uv) E T* and pO(p)-l = diag(vav, d a d , w'ou', uau). Hence a set of representatives for the 0-conjugacy classes within the stable 8-conjugacy class of the ®ular t in T is given by t ' diag(r,s,s,r), where r, s E F X / N E / F E X .Clearly in PGL(4, F ) the &classes within a stable class are parametrized only by r, or equivalently only by s. A set of representatives for the 0-conjugacy classes within a stable semi simple 8-conjugacy class of type I1 in GL(4, F ) which splits over the bi, quadratic extension E = E1E2 of F with Galois group ( a , ~ )where El = F ( 0 ) = E', E2 = F ( m ) = E"', E3 = F ( a ) = E" are quadratic extensions of F , thus A , D E F - F 2 , is parametrized by rEFX/N~l/~E s ET F, ' / N E ~ / F E ; ([F5],p. 16). It is given by

(

alr 0

0

0 b'ls z b~ zbAl sD s azr 0

0

""') 0

= h-'t*

h . diag(r, s, s,r),

t* = diag(a, b, ~ baa). ,

alr

+

+

Here a = a1 a 2 0 E E:, b = bl b z m E E;, B(h) = h. In PGL(4, F ) the 0-classes within a stable class are parametrized only by r, or equivalently only by s. A set of representatives for the 8-conjugacy classes within a stable semi simple 0-conjugacy class of type I11 in GL(4, F ) which splits over the biquadratic extension E = ElE2 of F with Galois group ( a , r ) ,where El = F ( 0 ) = E', E2 = F ( m ) = E"', E3 = F(&) = E" are quadratic extensions of F , thus A , D E F - F 2 , is parametrized by r(= r1 r 2 a ) E E : / N E / E ~ E([F5], ~ p. 16). Representatives for the stable regular 8-conjugacy classes can be taken in the torus T = h-lT*h, consisting of

+

t=

(z 'f)

= h-lt*h,

t* = diag(a, ~ aa ,~ ano), ,

where h = B(h) is described in [F5],p. 16. This t is ®ular when a / u a , T ( c ~ / c are ~ ) distinct and # f l . Here

II. Basic Facts

58

+

+

+

Further a = a b a E E X, a = a1 a 2 a E E: , b = bl b 2 a E E:, aa = a - b a , ra = r a + r b a . Representatives for all 8-conjugacy classes within the stable 8-conjugacy class of t can be taken in T . In fact if t' = gtQ(g)-' lies in T and g = h-lph, p E T*(F),then p 8 ( p ) - l = diag(u, ru,uru, au)has u = uu, thus u E E l . If g E T , thus p E T " , then p = diag(v, r v , urv, a v )

and

pO(p)-l = diag(vav,~ v u r vr ,v a r v , vav),

with vav E N E / E ~ EWe ~ . conclude that a set of representatives for the 8-conjugacy classes within the stable 8-conjugacy class of t is given by t . diag(r, r ) , as T ranges over E: / N E / EE~X . Representatives for the stable regular 8-conjugacy classes of type (IV) can be taken in the torus T = h t l T * h , consisting of

t=(i

t* = diag(a, ua, a3alu2a),

= h-lt*h,

where h = 8(h) is described in [F5],p. 18. Here a ranges over a quadratic extension E = F ( a ) = E3(&) of a quadratic extension E3 = F ( 0 ) of F . Thus A E F - F 2 , D = d l d2& lies in E3 - Ei where di E F . The normal closure E' of E over F is E if E/F is cyclic with Galois group 2/4, or a quadratic extension of El generated by a fourth root of unity a pT1 with p! = 1 = pi are irreducible (by [ST]), hence the operators M are scalars which in fact are equal to 1. Next we consider the .AM for Levi subgroups other than H and Ao, and the w in the Weyl group which map A M to itself with fixed points (0) only. These are A:+/ = {0}, .A$& = {0}, A '.?; = (0) and A%, = 0

(0). The reflection sa+O acts on M, by mapping diag(A,XA*), A* = wtA-'w, to (XA*,A). The representation 7r2@p1from which 1~~(7r2,p) = 7 r ~x p is induced, takes diag(A,XA*) to p(X)7r2(A). Since 1~,(7r2,p)is a representation of PGSp(2), we have p2w = 1, where w is the central character of 7r. The representation 7r2@ptakes (XA*,A) to p(X)n2(XA*) = p(X)w(X>w-'(detA)7r2(A). Then we have sa+p(7r2 €3 p ) = 7r2 @ p when w = 1, thus 7r2 is a representation of PGL(2). Since [WO] = 8, [ W p ]= 2, det(1 - s)ldM, = 2 and Mb, contributes a term equal to that contributed by M a , the contribution to the trace formula from the Wd(dM) with d = ( 0 ) and M = M a and M', is (7r2 is a representation of PGL(2, A))

76

III. Truce Formulae

The representations IP, ( ~ 2 p, ) = 7r2 x p are irreducible (by [ST]),hence the operators M are constants, in fact equal 1. The only element of WOwhich maps AMp = W(2a p)" to itself with ( 0 ) as its only fixed point is s ( P + ~ ~=) (14). It takes t = diag(a, A , det A / a ) to diag(det A / a ,A , u ) . The representation p 8 7 ~ 2of GL(1, A) x GL(2, A) from which the representation Ipo( p ,7 r 2 ) = ~ X T of Z PGSp(2, A) is induced takes the value p(a)7rz(A) at t , and p(det A/u)7r2(A)at sza+pt. Since it is a representation of the projective group we have pw = 1, where w denotes the central character of 7r. From p @ 7r2 11 sza+p(p @ 7 r 2 ) we conclude that p2 = 1 and 7r2 11 W T ~ ,w = p is 1 or has order 2. We have = 8, [W,""] = 2, and the contribution det(1 - s z a + p ) l A ~ , = 2, [WO] from M&is the same as that from Mp, hence the contribution to the trace formula of H from Mp and Mb and the unique element in W d ( A M p ) when A = (0) is

+

The representations Ipo( p ,7 r z ) = p x 7r2 are irreducible when p # 1, in which case the operator M is a constant, equal to 1. When p = 1, the representation 1 x 7r2 is a product of local representations 1 >a 7r2,, which are irreducible unless ~ 2 is , square integrable or one dimensional. The operator M can be written as a product m 8, R, of a scalar valued function m and local normalized operators R, (they map the K,-fixed , itself). When 1 x 7 ~ 2 , is irreducible, vector in an unramified 1 x ~ 2 to R, acts trivially. When 7r2, is square integrable 1 >a 7r2, decomposes as a direct sum of two tempered constituents, .rr;t), and xi,, and R, acts on one constituent trivially, and by multiplication by -1 on the other. When 7r2, is J l 2 , J 2 = 1, 1x 1 1 2 has two irreducible (nontempered) constituents: L ( Y ,1 x v - l l 2 J ) and L ( v ' /sp2, ~ v - ~ / ~ Jand ) , R, acts on the first trivially and by multiplication with -1 on the second. The scalar m is 1. Similarly we describe the spectral discrete contributions to the trace formula of the endoscopic group C O= PGL(2) x PGL(2) of H = PGSp(2). The terms corresponding to the parabolic group COitself is as usual a sum over the discrete spectrum representations T I , 7r2 of PGL(2, A):

3. Truce Formula of H: Spectral Side

77

The proper parabolic subgroups are Mp = A0 x PGL(2), M, = PGL(2) x Ao, Mo = A0 x Ao, where A0 denotes here the diagonal subgroup in PGL(2). Thus Mo consists of t = diag(a, 1)' x diag(b, 1)*. The roots are a ( t ) = a,p(t) = b. They can be viewed as CY = ( l , O ) , p = (0, l),in the lattice X*(Mo)= Z x Z in dg = R x R. The coroots a' = 2 a / ( a , a ) = (2,0), p" = (0,2) lie in the lattice X,(Mo) = Z x Z in d o = R x R. Since X*(Mp) = X,(Ao x (0)) = Z x (0) and X,(M,) = ( 0 ) x Z,we have A, = d M , = RP' and d p = d M M B = Ra'. The Weyl group WO= W ( M 0 )is generated by the commuting reflections s, and sp, where s,(t) = diag(l,a)* x diag(b, 1)*and s p ( t ) = diag(a, 1)*x diag(l,b)*. Identifying X*(Mo)with a lattice in RxR these reflections become s,(z, y) = (-x,y), s p ( x , y) = (x,-y). The other nontrivial element in Wo is sasp = -1. For d = (0) we have Wd(&) = {s,sp). Since 1- s,sp = 2 and d imd o = 2, det(1 - s,sp)ldo = 4. Further, W{o}(d,) = { s p } and W{O}(dp) = {sa}, (l-s,)(O, y) = (0,2y) hence det(1-s,))dp = 2 and det(1-sp)ld, = 2. The representation p1 @ p2 of Mo, taking !I to p l ( a ) p 2 ( b ) , is equal pz), whose value at t is p;'(a)p;l(b), precisely when the to s,sp(pl 18 characters pi are of order at most 2. The representation p1 @ 7r2 of Mp is equal t o s,(p1 I8 7r2) precisely when p: = 1. We obtain

Note that the representations which occur in these three sums are wellknown to be irreducible, from the theory of GL(2). Hence the operators M are scalars, equal 1. Similar analysis applies to the &twisted endoscopic group

C = [GL(2) x GL(2)]'/Gm, whose group of F-points consists of (91,g2), gi in GL(2, F ) , det g1 with (g1,gz) ( z g l , a g z ) , z E F X .A character (Pl, P i ; P 2 1 4 ) mod(p, p - 7 ,

PlP;p2P; = 1,

= det 9 2 ,

III. Pace Formulae

78

of the diagonal subgroup Mo of

t = diag(a1, a2) x diag(b1, b2) mod(z, z ) , a1a2 = b1b2, invariant under s,sp

for all

a1a2

satisfies

= b l b 2 , thus

which replaces the requirement p: = pi = 1 in the case of C O . As for a representation (PI, p i ) x r2 of the Levi subgroup Mp, thus y1pkw, = 1, if it is s,-fixed then its value at diag(a,b) x g, a b = detg, which is pl(a)pi(b)r2(g), is equal to its value at diag(b,a) x g, which is pl(b)pi(a)7r2(g). Here a b = detg, so we conclude that = 1 for all a , g , so P; = Pl. Since all of the representations which contribute to the spectral sides of the trace formulae of H, C , COassociated to proper parabolic subgroups are induced and are irreducible, except in the cases of 1x r 2 , the intertwining operator M ( s , O ) in each case where the representation is irreducible is a scalar which comes outside the trace. Hence our assumption on the components of the test function f, hence also on the matching functions f H ,fc,fc,,implies the vanishing of the contributions from the properly induced representations to the spectral sides of the trace formulae of HI

g(%)

c, co. 4. Trace Formula Identity We now review the trace formula identity for a test function f = @fv on G(A) = PGL(4,A), and matching functions fH = @ f on ~ H(A) ~ = PGSp(2,A), fc, on Co(A) = PGL(2,A) x PGL(2,A), fc on

C(A) = [GL(2,A) x GL(2, A)]’/Ax where the prime indicates (g1,92) = ((glv),(g2v)) with det glv = det ~2~ in F,” for all v, and fc, = @fc,,v on CE(A) = A; x . :A The $-elliptic

4. Trace Formula Identity

79

O-semisimple strongly ®ular part of the geometric side of the &twisted trace formula, Te(f,G, O ) , is

The assumption that at the place uo of F the components f v o , f ~ , ,. .~. , vanish on the (0)- singular set of G(Fv,),H ( F v o ) ,. . . , and the assumptions that the components of f , fH, . . . at v = 211, 212, q have (8.) orbital integrals which vanish on the strongly-(&) regular non-(8-) elliptic sets, imply that the geometric sides of the (&twisted) trace formulae are equal to the ( 0 - ) elliptic parts. The geometric sides are equal to the spectral sides - these are the trace formulae for each of the groups under consideration. The spectral side Tsp(f,G , 0 ) of the 0-trace formula for G and f will be equal to the (weighted) sum of the spectral sides of the trace formulae:

Here is a summarized expression of the form of the spectral side of the &twisted trace formula for f on G(A) = PGL(4,A): Tsp(f, G ,0)

=I

1

1

1

+ T1(3,1) + 5 1 ( 2 , 2 ) -k zI(2,1,1)+ I l ,

where 71

.rr ranges

over the (equivalence classes of) discrete spectrum representations of G(A) which are &invariant. Note that each of these 7r occurs with multiplicity 1 in the discrete spectrum of L2(G(F)\G(A)). Further, .rr

where T ranges over the discrete spectrum representations of PGL(3, A) which satisfy T 11 T$: here O(g) = J-ltg-' J and J is (&+j), and x is any quadratic character of A X / F Xor 1.

111. Trace Formulae

80

Furthermore,

1(2,2)

is the sum of Ii2,2),

where 1 I [ ~ ,=~2)

C tr M ( J ,

f

e)

7 ,T ) I P ( ~ ,(~ ~ ) ~ x7 ;

7

+

c

tr~(~,71,7z)1P(~,~)(71,x 7 20). ;f

712 7 2

Here we put b ( z ) = p ( z / F ) for a character p of AL/E1; fi is a character of A G / A X E X ;T~ and 7 2 (and 7) are discrete spectrum representations of PGL(2, A) and the sum over 7 1 # 7 2 is over the unordered pairs; the sum over b1 # b2 is over the unordered pairs too. Note that for representations of PGL(2) we have i 2: 7. Next 1 ( 2 , ~ , 1 is ) the sum of

and

where

Here x is a quadratic character of A X / F X ,71 is a discrete spectrum representation of PGL(2,A), and X E signifies the character # 1 on A X / F X which is trivial on N E I F A ~ .

4. Trace Formula

Identity

81

Finally,

The twisted trace formula for f on G(A) is equal to a sum of trace formulae listed below. First we have T,,(fH,H), which is

The fourth contribution here involves a properly induced representation Ip, (7r2, p ) , which is irreducible. Consequently the intertwining operator M(s,+p,O) is a scalar which can be taken outside the trace. Then tr Ipu( ~ 2p; , f ~ is) a product of local terms, and those local terms at v = vl, v2, 213 are zero by the assumption that we made, that the orbital integrals of fHvi vanish at the regular nonelliptic orbits. Similar observation applies to the third term in the spectral side of the trace formula of H and f H ( I P ~ ( ~ p, T #Z 1= ) ,p 2 ) , as well as to the contributions from Po that we did not write out here: they vanish for our test function fH. Only the first two terms remain under our local assumption. From this we subtract of the spectral side Tsp(fc,,,C O )of the trace

III. Trace Formulae

a2

formula of C Oand fc,:

The representations

IpB(p1, 7 r z ) = I(pl) x r2,

are properly induced, where I ( p ) denotes the representation of PGL(2, A) induced from the character

(:

a!l)

H

p(u) of the proper parabolic sub-

group. Since the group in question is PGL(2) they are irreducible, hence the operator M ( s ,0) is a scalar, can be taken in front of the trace (in fact it is equal to l),and the tr I ~ ( Tfco) ; are products of local factors, those indexed by 11 = 111, 112, 113 are zero by our assumption on the vanishing of the orbital integrals of the components f ~ , , on ~ ,the regular elliptic set, hence the only contribution is the first:

To this we add !j of the spectral side of the stabilized trace formula for C and fc; it is stabilized by subtracting C E T ~ ( f c E To) explain . this trace formula, recall that a representation of C(A) is an equivalence class of representations 7r1 x 7rz of GL(2, A) x GL(2, A) with w,,w~, = 1 under the equivalence relation r1 x r~ N xr1 x x-lrz for any character x of A X / F X .Thus the terms

1

4 t r M ( s , , O ) ~ P ~ ( P l , p l , rf2); ,

sum over the discrete spectrum representations 7rz of GL(2, A) and characters p1 of A X / F X ,which appear in the trace formula for GL(2) x GL(2),

4.

Trace Formula Identity

a3

would contribute to the trace formula of C precisely one term: I(p1,p1,7r2) would make a representation of C(A) precisely when p?w,, = 1, and I ( p 1 , p2, ~ 2 N) 1(1,1,pL17r2) for any p1 as a representation of C(A). For any representation 7rz of PGL(2, A) (thus the central character w,, is trivial), 1 ( 1 , 1 , 7 r 2 ) is irreducible. Hence the intertwining operator M is a scalar, which can be evaluated to be equal to one by standard arguments. Similarly, the terms

contribute just

1 4

- tr IP, ( r z ,

which is in fact equal to

a tr I p P

1,1;fc),

(1,1,7r2;

fc).Thus we have

The first sum ranges over all discrete spectrum representations

of C(A). The second sum ranges over all discrete spectrum representations 7r2 of PGL(2,A). As the group C = [GL(2)xGL(2)]'/Gm is a proper subgroup of [GL(2)x GL(2)]/Gm,the induced representations Ipo might be reducible (I have not checked this). Recall that a representation I ( p ) of SL(2, F ) , induced from a character p(x) =

(t a ! l )

H

p(a) is reducible precisely when p has order 2 (or

1x1*'), in which case I ( p ) , p2 = 1 #

p , is the direct sum of two

tempered constituents. To derive lifting consequences from this identity of trace formulae or rather their spectral sides - we use a usual argument of "generalized linear independence of characters", which is based on the "fundamental

a4

III. Trace Formulae

lemma". Almost all components of a representation 7r = @7rV of G ( A )are unramified and for any spherical function f V on G(F,) we have tr 7rV(f V x 8 ) = f,"(t(7rV) x 8 ) where f," is the Satake transform of f V and t(7rV)x 8 is the semisimple conjugacy class in 2 x 8 parametrizing the unramified representation xu. A standard argument (see, e.g., [FK2]) shows that the spherical functions provide a sufficiently large family to separate the classes t(7rV)x 8. The trace identity takes then the form where we fix a finite set V of places of F including the archimedean places, and an irreducible unramified representation 7rV of G(F,) at each place u outside V , and then all sums range only over the 7r (or I ( T whose )) component at u is 7rVl while the sums of representations of the groups H(A), C(A),Co(A),. . . range over the representations T H = @ T H ~ ,7rc = @7rcV, 7rc0 = etc., whose components at u outside V are unramified and satisfy

Note that by multiplicity one and rigidity theorem for discrete spectrum representations for PGL(4,A), there exists at most one nonzero term in all the sums in T,,(f, G ,0). However, fixing t(7rV)at all v $! V does not fix the t(7rcu)and at this stage it is not even clear that the number of T H , 7rcl 7rc0which appear in the trace formulae is finite. The terms themselves in the sum are replaced by a finite product of local terms over the places u at V , taking the forms

nu

The intertwining operator M ( s ,T ) is of the form m(s,T ) R ( s ,T ~ )where , m(s,T ) is a normalizing global scalar valued function of the inducing representation T on the Levi subgroup, and the R(s,T,) are local normalized intertwining operators, normalized by the property that they map the normalized (nonzero) KV-fixedvector in the unramified representation to such a vector. We shall view the identity of spectral sides of trace formulae stated above for matching test functions as stated for a choice of a finite set V, unramified representations 7rV at each u outside V , and matching test

4. Trace

Formula Identity

85

functions fu, f H u l . . . at the places u in V , where the terms in the sum a.re such finite products over u in V. For the statement of the fundamental lemma in our context and its proof we refer to [F5]. The existence of matching functions follows by a general argument of Waldspurger [W3] from the fundamental lemma. The statement (“generalized fundamental lemma”) that corresponding (via the dual groups homomorphisms) spherical functions have matching orbital integrals] follows from the fundamental lemma (which deals only with unit elements in the Hecke algebras of spherical functions] namely with those functions which are supported and are constant on the standard maximal compact subgroups) by a well-known local-global argument I which uses the trace formula. We do not elaborate on this here, but simply use the ( “generalizedll) fundamental lemma and the existence of matching functions.

IV. LIFTING FROM SO(4) TO PGL(4) 1. From SO(4) to PGL(4) We begin with the study of the lifting X1, and employ the trace identity Tsp(f,

1 G, 0) = T s p ( f H , H) - qTsp(fco, CO)

with data of a term in T s p ( f ~C). , We choose the components at w outside V to be those of the trivial representation l c = 1 2 x 1 2 of C(A). The parameters tc(l2,,

x 1 ~= ~[diag(q;I2, ) q;1/2) x diag(q;l2, q;1’2)]/{H}

of its local components are mapped by XI to t = diag(q,, 1,1,q;’), thus to the class of 1 3 , 1 ( 1 3 , l ) , the unramified irreducible representation of PGL(4, F,,) normalizedly induced from the trivial representation of the standard parabolic subgroup of type ( 3 , l ) . Consequently the only nonzero contribution to Tsp(f,G, 0) is to ;1(3,1). Had there been a nonzero contribution to T s p ( f ~ , Halmost ), all of its local components would have the parameters t , associated to r p G S p ( 2 ) ( v , 1) = L ( v x v M v-’), which is not unitarizable by [ST], Theorem 4.4. However, all components of an automorphic representation of PGSp(2, A) are unitarizable, hence there is no contribution to T s p ( f H). ~, Similarly there is no contribution to TSp(fco,Co),since had there been a contribution its local components would have to be 7r2(v, v-’) x 7 r 2 ( 1 , 1 ) at almost all places, but the irreducible 7r2(v, v-’) is not unitarizable. There ) , a contribution from a pair is no contribution to any of the T E ( ~ Esince p1, pa of characters of C,(A) = : A x: A corresponds to a (cuspidal) representation 7rz(p1)x 7 r 2 ( p 2 ) of C(A). But we fixed the parameters of the trivial representation l c = 1 2 x 1 2 of C(A) = [PGL(2,A) x PGL(2, A)]’/Ax. 86

1. From SO(4) to PGL(4)

87

Moreover, since a discrete spectrum representation of PGL(2, A) with a trivial component is necessarily trivial, the only contribution to Tsp(fc,C) is from l c . We conclude that the trace formula reduces in our case to (10

Since each of the representations I C ,is~elliptic - its character is not zero on the regular elliptic set in C, = C(Fv)- we can choose three of the functions fv so that fc,unot only has orbital integrals which vanish on the regular nonelliptic set but moreover be supported on the regular elliptic set of C,, and tr l ~ fc,v) , ~#(0. The equality (1) then implies that , ,~ u )for( all matching tr14v(13,1;fu x 0) is a nonzero multiple of tr l ~ f c

fv,

fc,u.

1.1 PROPOSITION. For every place v of F , and for all matching functions fv and fc,uwe have

PROOF.Name the place v of the proposition VO. We apply the displayed identity with a set V containing at least 3 places, but not the place wo, and use f v such that fcv is supported on the regular elliptic set for 3 places v in V. We then apply the displayed identity with the set V U {wo}, and with the same functions f v , fcvfor v E V . Of course we use f v l fcv with tr l ~ , ~ ( f#c0,for ~ )all w in V. Taking the quotient, the proposition 0 follows. Let us derive a character relation for the 0-elliptic 0-regular elements t from the equality of the proposition, using the Weyl integration formula. 1.2 PROPOSITION. W e have the character identity

where ’t denotes the element stably 0-conjugate but not 0-conjugate to t , and r ranges over F / N E / F E in case I, E: / N E / E 3E X in case 111, and

IV. Lifting from SO(4j to PGL(4)

88

denotes the nontrivial character of this group. Here case 111, and 0 in cases I1 and IV.

K

L

is 2 in case I, 1 in

PROOF.In local notations,

is equal to

for test functions f and fc with matching orbital integrals. Matching means that for &elliptic ®ular t of type I or 111, the ( K , 0)-orbital integral o f f on GI denoted FF(t) = F f ( t ) - Ff(t’) (here t’ is an element stably 8-conjugate but not &conjugate to the 8regular t ; K indicates the nontrivial character on the group of 0-conjugacy classes within the stable 8-conjugacy class) is equal to the stable orbital integral of fc on C at the norm N t o f t , denoted

where (Nt)’ denotes an element stably conjugate but not conjugate t o N t . Implicitly we use the fact that the norm map N is onto. It is defined for elliptic elements only in types I and 111, as recalled in chapter 11, section 5 . The notation Ff(t) and F f c ( t ) for the (0-) orbital integral multiplied by the A-factor was introduced in Definition 11.2.4. To determine the group of conjugacy classes within the stable class of TC = N T in C where T is of type I or III, we compute H-‘(F,X,(Tc)). It is the quotient of the lattice { X E X * ( T c ) ;N X = 0)

by

(X

- TX; T E

Gal(F/F)) = {(z, -x; y, -y); x,y E Z},


89

namely 2/2. Indeed, in case I the Galois group is Gal(E/F) = (a),with

In case I11 the Galois group is Gal(E/F) = (a,T ) with

We choose the function fc to be supported only on the regular elliptic set, and stable, namely such that Ffc(Nt) and Ffc((Nt)') be equal. We choose the function f on G to be supported on the ®ular &elliptic set and unstable, thus F f ( t ) = -Ff(t') if t, t' are stably @conjugate but not &conjugate. Thus we choose f,fc related on elements of type I by

and

and similarly in case 111. Taking f , fc with Ff(t) supported on a small neighborhood of a @-regularto, the proposition and the Weyl integration formulae imply - since the characters are locally constant functions on the ®ular set - the character identity

where tr denotes the element stably @-conjugatebut not &conjugate to t , and r ranges over F X / N E I F E Xin case I, E2/NEIE3EX in case 111, and K. denotes the nontrivial character of this group.

IV. Lifting from SO(4) to PGL(4)

90

Since the 8-conjugacy classes of type I1 and IV are not related by the norm map to conjugacy classes in C ,whatever the choice of f is on these classes, the integral

is zero, hence xR(tO) vanishes on the 6'-regular 8-conjugacy classes of type I1 or IV. The torus It remains to compute the numbers [We(T)]and [W(Tc)]. Tc consists of elements

(( :i ':), (:i 'if),

cf

-

Dci = d: - D d i m o d F X 2

Its normalizer (modulo centralizer) in C ( F )is generated by (diag(i, -i), I ) ,

( I ,diag(i, -i)),

where i E Fxwith i 2 = -1. Hence [W(Tc)] is 4. The 6'-normalizer modulo the ¢ralizer of the torus T is generated by w = and diag(i, 1,1,-2) in case I. Hence [We(T)] is

(y

z),

( y i),

8, and [ W e ( T ) ] / [ W ( T= c )2.] In case I11 the 6'-normalizer modulo the 6'centralizer is 2 / 2 x 2 / 2 , generated by the matrices diag(-1, -1,1,1) and 0 diag(-l,1, -1, l),hence [We(T)]/[W(Tc)]= 1.

REMARKS. (1)The computation of the twisted character x ~ ( ~ , ~ ) (ist reached by purely local means in the paper [FZ] with D. Zinoviev. (2) The propositions remain true when the local field F,, is archimedean. Indeed, we choose the global field F to be Q or an imaginary quadratic extension thereof, and apply the global identity (1) once with a set V consisting of 3 nonarchimedean places (and f,, fc, supported on the (0-) elliptic set for Y E V ) ,and once with V U {YO}. In the real case, where F,, = R, the only &elliptic elements are of type I, and we obtain the character relation A(tO)xI(l,l)(trO)= 2 ~ ( r ) A ~ ( N t ) x ~ ~ ( KN:tRx/R: ), z{&l}. In the complex case there are no &elliptic elements, and all $-regular elements are &conjugate to elements in the diagonal torus T . For t =


91

diag(a, b, c, d ) and N t = (diag(ab, cd), diag(ac, b d ) ) , noting that W e ( T )= 0 4 has cardinality 8, and W ( T c ) is 212 x 212, generated by ( w , I ) and

( I ,w), w =

( -;),

when F,, = @,

Ibl,P I 1 ) x

7r

of cardinality 4, we have

lc, or F is any local field, 7rc = = Xl(7rC) = I ( P l P 2 , P l / P 2 , P Z l P l , 1/PlP2)

= I ( 1 3 , l ) and

I ( P 2 , PT1)

and

7rc

1

are induced.

1.3 DEFINITION. The admissible representation nc = 7r1 x

7rz

of

C = [GL(2,F ) x GL(2, F ) ] ' / F X IZfts to the admissible representation 7r of G = PGL(4, F ) , F local, and we write 7r = X1(nc), if for all matching functions f, fc we have t r n ( f x 0)

= trnc(fc).

Equivalently we have the character relations xr(tO) = 0 for ®ular elements without norm in C (type I1 and IV for &elliptic elements, as well as non-&elliptic elements of type (2), (3) of [F5], p. 15 and p. 9, where T* = {d'iag(a,b, a a , ab);a , b E E X } ) ,and

for ®ular t in G with norm in C , thus of type I and I11 for &elliptic t , for split t and for t of type (1) and (1') of [F5], p. 15 (and p. 9). Type (1) has T* = {diag(a,aa,b , a b ) ; a , b E E X } ,type (1') has T* = {diag(a,b, ab, n u ) ; a , b E E X } ,[ E : F ] = 2. The norms are (diag(aaa, bob), diag(ab, aaab)) and

(diag(ab, aaab),diag(aaa, b a b ) ) ,

in cases (1) and (1'), and stable 8-conjugacy coincides with 8-conjugacy in cases (l),(1') and the split elements. Thus rc = 1 and r = 1 in these cases. In the case of split t , W e ( T )= Dq has 8 elements while W c ( N T )= 2 / 2 x 2 / 2 has 4. Fort of type ( l ) , W e ( T *consists ) of 1, (12)(34), (13)(24), (14)(23) ( W e ( T )is generated by diag(1, -1,1, -1) and antidiag (IJ)), and

92


W c ( N T ) = 2 / 2 x %/2 too, generated by (

( y -:)

,I) and (diag(-1, l),

diag(-1,l)). In type (1’) W e ( T )is generated by d i a g ( - l , - l , l , l ) diag(w, w), and W c ( N T )by (diag(-1, l ) ,diag(-1,l)) and ( I ,

( y -:)

and

).

Then in cases (1) and (1’)we have [ W e ( T ) ] / [ W c ( N= T )1, ] and = 2 for split T or T of type I. In type 111, W e ( T )is generated by diag(1, -1,1, -1) and diag(-I, I) (which act on diag(a, T ( Y ,O T Q , acr) in T’ as (43)(21) and (32)(41)), and W c ( N T )by (diag(i, -i), I) and ( I ,diag(i, -i)), hence the type III. quotient [ w ’ ( T ) ] / [ w ~ (isN1Tin) ] Given a representation 7rc = 7r1 x 7r2 of C = [GL(2,F) x GL(2, F)]’/FX - thus the central characters w1,wz of 7rl17r2 satisfy w1w2 = 1 - and characters X I , x2 of FXwith xfxi = 1, F local, we write ~ 1 x x1 ~ 2 x 2 for the representation (g1,g2) H (w(g1) @ m(g2))xl(gl)x2(g2);note that xl(gl)x2(g2) = X I X ~ ( S I )= xlx2(g2) since detgl = detg2. The character relation implies 1.4 PROPOSITION. If 7r1 x 7r2 lifls to a representation G = PGL(4, F), then x17r1 x ~ 2 x 2lifts to ~ 1 x 2 7 ~ .

7r

of the group

PROOF.The characters XI, x2 depend only on the determinant. As the norm map is N(diag(a, b, c, d ) ) = (diag(ab, cd), diag(ac, b d ) ) , we have

Denote by sp, or St2 the special (= Steinberg) square integrable subrepresentation of the induced representation 1(v1l2,v-l12) of PGL(2, F ) , and by St, the Steinberg square integrable subrepresentation of the induced representation I(v,1,v-’) of PGL(3, F ) . Put also sp2(x) = x @ sp2 for a character x of FX/FX2. Since ~ I ( , , I / Z , ~ - = I/Z xspz ) xlZ vanishes on the regular elliptic set of PGL(2, F), for a function h on the regular elliptic set of PGL(2, F ) we have tr sp2(h) = - t r 12(h). Hence for a function fc on C,supported on the regular elliptic set of C , we have

+


93

= - tr(12 x SPZ)(.fC) = tT(SP2 x SPZ)(fC).

Let 7r2 denote an irreducible unitarizable representation of PGL(2, F ) . Let J(v1/'7r2, v - ~ / ' T z ) be the unique ("Langlands") quotient of the induced representation I = I ( ~ ~ / ~ 7 r 2 , v - ~ /of~ 7PGL(4,F). r2) It is unramified if 7 ~ 2is, in fact it is the unique unramified constituent of I if I is unramified. 1.5 PROPOSITION. The representations 7rc,vo= 7rwo x l,, and l,, x 7rwo of C,, lift via XI to J(v:L27rwo, vw~"7rw0)for e v e y square integrable representation 7rvo of PGL(2, Fvo).

PROOF. We choose a number field F whose completion at a place and 3 other nonarchimedean places: u1, vz, 213. Fix a cuspidal representation 7rvl of PGL(2,Fw,), and let 7rv2 and 7rw3 be the special representations sp, at 212 and 213. Using the trace formula for PGL(2, A) one constructs a cuspidal representation 7~ whose components at v1,v2,2r3 are our rWi, which is unramified outside V = ( q , v 2 , v 3 ,cm}, and a cuspidal representation IT'whose components at u in V' = {VO, 211, 212, 213, cm} are our T,, which is unramified outside V'. We use the trace identity with the sets V (resp. V ' ) , such that 7r x 1 2 and 1 2 x n (resp. IT' x 1 2 and 1 2 x d)are the only contributions to the trace formula of C(A). We choose test functions f (resp. f') such that their components at 212, 213 are supported on the ®ular elliptic set, and such that the stable &orbital integral of fu, and fv, are zero. This guarantees that in the trace identity there are no contributions from H . Now in the trace identity, for v outside V we fix the class

uo is our nonarchimedean field F,,,

tc,w(l2

x

7r2v) =

[diag(9y2,4 y 2 ) x diag(P&, P f , ' ) I l { ~ " } l

where 7 ~ 2 , is the unramified component of I ( P z , , P ~ ~ and ) , P;, = p z v ( r , ) . This class is mapped by XI to

t, =

1/2

4v

0-1

Pzu

-1/2& 7 %

7

4,'

which is the parameter of J ( v ~ / 2 7 rvi1/27r~v). ~w, Thus the unique contribution to the trace formula of G = PGL(4, F ) is the discrete spectrum noncuspidal representation

J ( v ' / ~ Tv, - ' / ' ~ ) .

94

IV. Lifting from SO(4) t o PGL(4)

Since the trace formula for C appears in our identity with coefficient $, and both 1 x IT and IT x 1 make equal contribution, we conclude the equalities

where the products range over V, and over V'. Since we can choose f for which the terms on the right are nonzero, dividing the equality for V' by that for V, and noting that fv, is arbitrary, the proposition follows. 0

2. Symmetric Square The diagonal case of the lifting A1 : ITC = IT^ x IT^ H IT, that is when irl = IT^, coincides with the symmetric square lifting from SL(2) to PGL(3) established in [F3]. More precisely we shall use here the results of [F3] to relate terms in our identity of trace formulae, and in particular obtain the (new) character relation A ~ ( I Tx~ el) = 1(3,1)(Sym2IT^, 1) for admissible representations IT^. The global and local results of [F3] are considerably stronger than what we need here. Not only that we work in [F3]with arbitrary cuspidal representations T I , and put no local restrictions (that 3 local components rlvof IT^^ be elliptic) as here, but more significantly, [F3] lifts representations of SL(2) - rather than of GL(2). Consequently [F3] proves in particular multiplicity one theorem for discrete spectrum representations of SL(2,A) as well as the rigidity theorem for packets of such representations, as well as it characterizes all representations of PGL(3, A) which are invariant under transpose-inverse as lifts from SL(2, A) (or PGL(2,A)). For our purposes here we simply observe that the restriction of a representation of GL(2,A) (resp. of GL(2,F), F local) to SL(2,A) (resp. SL(2,F)) defines a packet of representations on SL(2, A) (resp. SL(2, F ) ) . At almost all places of a number field, the unramified components of 2 IT = @IT, satisfy XI(IT, x ir,) = I(s,l)(Sym IT^, l ) ,where if IT^ = I(uvlbv) then S Y ~ ~ ( b,) U ~ =, ( a v / b v ,1,b,/a,). Here is a summary of the symmetric square case in our context of PGL(4).

2. Symmetric Square

95

2 . 1 PROPOSITION. (1) For each cuspidal representation 7r2 of GL(2, A) there exists an autornorphic representation 7r = Sym2(7r2) of PGL(3,A) which is invariant under the transpose-inverse involution 83 such that A l ( r 2 x 5 2 ) = I(3,1)(Sym2(X2),l). (2) If 7r2 is of the form 7r2(p), related to a character p : A i / E X -+ C x where E I F is a quadratic extension of number fields, then Sym2(n2) is I(2,1)(7r2(p/p),~E), where F ( x ) = p ( ? f ) ,x H ?f denotes the action of the nontrivial automorphism of E / F and XE is the quadratic character of AX/FXtrivial o n the norm subgroup N E , F A i . (3) If 7r2 is cuspidal but not of the f o r m 7 r 2 ( p ) , then Sym2(7r2) is cuspidal. (4) If sym2(7r2) = Sym2(7ri) then 7rb = x7r2 for some character x of AX/FX. ( 5 ) Each &-invariant cuspidal7r3 is of the form Sym2(7r2). The analogous results hold locally. For each admissible irreducible representation ~ 2 ,of GL(2, F,) there is an irreducible representation 7r3, = Sym2(7r2,) of PGL(3, F,), invariant under the transpose-inverse involution 83, such that the character relation A1(7rzV x jr2,) = I(3,1)(Sym2(7r2,),1) holds. If Sym2(7r;,) = Sym2(7r2,) then 7rhv = xv7r2, for some character xv of F," . Each 83-invariant cuspidal 77-3, is of the f o r m Sym2(7r2,). As sym2(sp2)= ~ t 3 we , have ~ l ( s xp sp2) ~ = 1(3,1)(St3,1).

PROOF.The global claims (1)-(5) are consequences of the results of [F3]. The new claim here is the character relation. Note that the character relation has already been proven by direct computation for 7rzV which is an induced representation, as well as for the trivial representation 7r2v = 12,. Thus we need to prove the character relation for square integrable ~ 2 , . We fix a global field F which is Q if F, = Iw or totally imaginary if F, is nonarchimedean , whose completion at a place vo is our F,, cuspidal , the ~ special ~ representation 7rzV3of GL(2, F,) representations 7 r 2 u l , ~ 2 and at the nonarchimedean places v = v l , 712, v g of F , and construct a cuspidal representation 7r2 whose components at 2ri (0 5 i 5 3) are those specified, while those outside the set V consisting of the archimedean places and zli (0 5 i 5 3), are unramified. We apply the trace formula identity with the set V and a contribution 7rc = 7r2 x i r 2 to the trace formula identity. We take the test function f,, to be supported on the 8-elliptic regular set, such that tr.rrC,,,(fC,,,) # 0 and with fHIV3= 0 (thus the stable &orbital integrals of f,, are zero).

96

IV. Lifiing from SO(4) to PGL(4)

This choice is possible by the character identity t r ( l a Wx lzw)(fcw) = trI(3,1)(13w,lw)(fw x 8 ) . Consequently we get a trace identity with no contributions from the trace formula of HI while the contribution to the 8twisted trace formula of G is only I(3,1) (Sym27r2, l),by the rigidity theorem for GL(4). Note that the coefficient of T,,(fc,C) is and so is the coefficient of the term 1(3,1) in Tsp(f , G I8). Denoting by Vf the set {vo, v1, v2, v3}, we conclude, for all matching the identity functions fwo and fc,wo,

i,

trI(3,1)(Twil)(fw x 8 ) =

m(a3a2a1,T) W€Vf

trTC,w(fC,w). W € V f

Here we wrote the intertwining operator hf(a3a2all7),T = Sym2(7r2) ,T,) where 7rc = 7r2 x j r 2 , as a product of local factors R ( c I ~ c x ~ c x ~over all places v and a global normalizing factor m(7r2) = m ( a 3 ~ 2 ~T1) , , and incorporated the local factor in the definition of the operator 0, thus l)(fw trI(3,1)(~ ~ , x 8 ) stands for tr R(a3Q2a1,Tw)I(3,1) ( T w ,

l)(fw

8).

Note that R(T,) = R ( a 3 a z a 1 , ~ ~is) normalized by the property that R(Tw)7rw(8) fixes the K,-fixed vector when 7rw = I ( 3 , 1 ) ( ~1) ~ ,is unramified. We now repeat our argument with the set Vf’= Vf- {vg} and construct a cuspidal IT; unramified outside Vf’whose components at vl , v2 v3 are as above (we are assuming that vo is nonarchimedean). Dividing the identity for Vf by the new identity for Vf’we get

T’)X ~is, independent of the global The constant ~ ( C Y ~ C X ~ TC )X / ~V, L ( Q ~ Q ~ C representations 7 , ~ ’it; depends only on the local representation 7rzw0, and will be denoted m(7rawo). It is equal to 1 for ~2~ = 1 z W ,the trivial representation, by Proposition 1.1,hence also for the special representation

SP2W. Hence m(7r2) = n m ( ~ 2 ~product ), only over the cuspidal components 7rzW of 7r2, and we replace R(T,) by rn(7rzw)R(.r,)when 7rzWis cuspidal to 0 obtain the character relation as claimed.

3. Induced Case

97

3. Induced Case We then turn to the study of the XI-lifting of 7r1 x 7r2, wlw2 = 1 (wi is the central character of ri),when 7r2 is not the contragredient 51 of T I . Note that 5l(A) = 7rl(A) where A = wtA-lw. This 771 is equivalent to wT17r1. 3 . 1 PROPOSITION. Let 7r2 be a n admissible representation of GL(2, F ) ( F is a local field) with central character w . Let 7r1 = I(p1,pi) be a n induced representation of GL(2,F) with p1p:w = 1. Then X 1 ( I 2 ( p l r 4 )x 7r2) = 7 r , where 7r = 14(p17r21pi7r2)is the representation of PGL(4,F) induced f r o m the parabolic subgroup of type (2,2) as indicated. PROOF.Since

it suffices to show that Al(I2(1,w-’) x 7r2) = I 4 ( ~ 2 , + 2 ) , as the contragredient jT2 of 7r2 is wP17r2. We then compute the &twisted character of the induced representation IT = 7r4 = I ~ ( Q 52). , Put p = 7r2 @ 772. Write Xl(p7r1

x pV17r2)= X l ( 7 r l x

7r2)’

p(diag(A, C)) for p(A, C) = 7r2(A) 8 jT2(C). Its space consists of

+ :G

+

p with

m = diag(A, C) with A , C in GL(2, F ) and n is a unipotent matrix (upper triangular, type (2’2))’ b(m)= I det(Ad(m)ILieN)I = I det(AC1)I2, and 7r acts by right translation. Note that 7r = 7r4 is &invariant. Namely there exists an intertwining operator s’ : 7r -+ 7r with s17r(g)= 7r(Og)s’. Fix s’ to be (s+)(g) = p(O)(+(Og)). Here p(8) intertwines 7r2 8 ir2 with 772 8 7r2 by p ( O ) ( [ @ = [ 8 [ and

i)

Note that s is well defined. When 7r is irreducible (this is the case unless 7rz = v1I27rL, 7rk 2 ir;), d2is a scalar by Schur’s lemma, so we can multiply s’ by a scalar to assume st2 = I, and so s‘ is unique up to a sign. It is

IV. Lifting from SO(4) t o PGL(4)

98

easy to see that our choice of s' = s here is the same as our usual choice of 7r(0), preserving Whittaker models or a K-fixed vector if 7r2 is unramified. Extend p, by p ( 0 ) = s, to a representation of [GL(2,F ) xGL(2, F ) ]>a ( 6 ) . Note that trp(O)(n2(A) 8 irz(C)) = tr[nz(A) 8 1 . p ( 0 ) . (1 8 & ( C ) ) ]= t r ( ~ z ( A C8 ) l)p(0).

To compute tr p(B)(na(A)@ 1) choose an orthogonal basis vi for 7r2, and a dual basis 6i for i r 2 . (It is standard to "smooth" our argument on using test functions). Then 7r2(A) 8 1 takes vi 18ijj to 7r2(A)vi 8 ijj, and p ( 0 ) takes 7rz(A)vi8 6j to 6j187rz(A)vi. The trace t r p ( O ) ( ~ ( A8) 1) is then

C(6,87r2(A)vi,vz8

q = C(6jlUi)(7r2(A)Wi,fiJ)

ij

..

23

= C(7r2(A)wiIiji)= tr7r2(A). i

As usual,

=

(7r(0f dg)4)(h)

is

//If

(0(h)-'nmk)61/2(m)(rr2 x ir2)(0m)$(k)6-'(m)dn dm dk.

Write m = diag(A,C) as 0(m;')moml with ml = diag(I,C) and mo = diag(A', I),where A' = wtC-'wA. We have tr(n2 x ir2)(Odiag(A1C))= tr7r2(AwtC-'w). Put

MO = {diag(X, I);X E GL(2, F ) } ,

M I = {diag(I, X ) ;X

E

GL(2, F ) }

well as for the images of these groups in PGL(4, F ) . Note that 6(m)= 6(m0).Then, putting m = 0(m;')m,m1, mo = diag(A, I),we have a5

t r r ( 0 f dg) =

///

f ( 0 ( k - 1 ) n l m k ) b - 1 / 2 ( mt)r ~ z ( A ) d n l d m dk.

Change variables n1 H n, where nl = nrnO(n-')m-'. This has the Jacobian I det(1 - Ad(m0))l LieNI. Replace n by 0(n)-' and note that Ad(O(mll)moml .O) = Ad(O(myl))Ad(moO) Ad(O(m1)).

3. Induced Case We obtain

t r r ( 8 f dg) =

X=

/// K

N

99

sMoA ~ ( r n o 8tr1~2(A)Xdmo, ) where f(8(lc-ln-lrnr1)rnornlnlc)dndlc,

Mi

and A ~ ( r n 0 0= ) 6-1/2(rno)ldet(1 - Ad(rno8))l Lie(N)I.

Note that if rno = diag(A, I ) and A has eigenvalues a , b, then 6(rno) = labiz, and AM(rno8) = l(1- a ) ( l - b ) ( l - ab)/abl has the same value at A and at wtA-lur (or A-'). Writing the trace again as

rno = diag(A,I), we use the fact that the 8-normalizer of M in G is generated by M and J . Since 8(J-')

(t y ) J = (i wiw) = O(rn[')m~rn,,

rnl = diag(I,wAw),

rno = diag(tA-l, I),

and since X ~ , ( ~ A - '= ) xKZ(detA-l. A) = w-l(detA)x,,(A), we finally conclude that tr.rr(8f dg) is

On the other hand, using the 2-fold submersion

Moreg x M\G

4

Gg- reg,

(my9) H o(g-')mg,

whose Jacobian is

I det(1 - Ad(rn8))I Lie(G/M)I = 6-'(rn)l

det(1 - Ad(rn8))l LieNI2,

and noting that the 8-Weyl group W g ( M )= {g E G; B(g)-lMg = M } / M is represented by I and J , if g t-+ xT(g8) denotes the &character of IT then we have tr ~ ( 8 dg) f =

f(g)xT(g8)dg= /M ;6-'(rn)l det(1 - Ad(rn8))l LieNI2


100

On writing m = B(rnl)-lmoml this becomes

We conclude that

and that xx(gO) is supported on the set 8 ( g - ' ) m o g , m o E M0,g E G. In particular it vanishes on the &elliptic stable conjugacy classes of types I, 11, 111, IV. Now

as

Nmoe)/Ac(Nmo) = I

(1 - aI2(1- b)2 11/2

ab

if A has eigenvalues a and b, and A2 is the usual Jacobian of GL(2) : Az(A) =

.2/1-I

We rewrite our conclusion as

where N(diag(A,I))=

((

7) ,A), since

(and it is zero on the elliptic element in GL(2)). But this is precisely the statement that I(1,w-') x 7r2 XI-lifts to 7r = I(7r2,?2). 0

REMARK. The character relation implies that elliptic conjugacy classes.

xx vanishes on

the 8-

4.

Cuspidal Case

101

3.2 COROLLARY. Let F be a local field. For every cuspidal representation .rr2 of PGL(2, F ) the representations .rrc = .rr2 x sp2 and sp2 x.rr2 of C X1-lift to the subrepresentation

of the fully induced I(v1127r2,~ - ~ / ~ . r r z ) .

PROOF.To simplify the notations we write simply 0 + S

J -+ 0 = I and

+ I --+

~ - ~ / ~ 7 r 2 )Since . X1(I(v1j2,v-1/2) x n2) x 7r2) = J , and since the composition series of I(v112,v-l12) consists and spa, while the composition series of I consists of J and S, the

omitting the Xl(l2

of 1 2 claim of the corollary follows from the additivity of the character of a 0 representation: xxl+xz- xxI xxz.

+

4. Cuspidal Case It remains to XI-lift cuspidal representations of C. 4.1 PROPOSITION.Let F be a local field. Let 7rh and .rrg be (irreducible) cuspidal representations of GL(2, F ) with central characters w',w" with w'w" = 1 so that .rrc = .rrh x .rrg is a cuspidal representation of C . Then

x $') exists as a n irreducible 0-invariant representation This 7r is cuspidal unless (1) 7ry = irhx, x2 = 1, where

XI(7ri

7r

of G.

or ( 2 ) there i s a quadratic extension E of F and characters p1 and p2 of E X with p1p2JF'X = 1 such that 7rb = 7 r ~ ( p 1 )7,rg = T E ( p 2 ) , in which case Xl(rE(p1) x T E ( p 2 ) ) = I(2,2)(TE(pli&)? rE(111112)). I n particular, if .rrh and ng are monomial but not associated to the same quadratic extension, then A l ( 7 r h x 7rg) is cuspidal. When F is a global field and .rr;, ng are automorphic cuspidal representations of GL(2,A) (with w'w'' = 1) the analogous global results hold. I n particular XI (7rh x 7rg) exists as an irreducible automorphic 0-invariant

IV. Lifting f r o m SO(4) t o PGL(4)

102

representation 7r of G(A), which is cuspidal except at the indicated cases. I n this global case we require that at least at 3 places the components of 7 r 4 , 7r; be square integrable.

PROOF.We denote the local fields of the proposition by F', El. Suppose F' is nonarchimedean. Choose a totally imaginary global field F whose completion at a place vo is our F'. Fix four nonarchimedean places WI, vz, 213, 214 (f WO) of F , and cuspidal representations nLl (i = 1, 2, 3, 4) of PGL(2, F v x ) .Let V be the set of places of F consisting of w, (0 5 i 5 4) and the archimedean places. Construct cuspidal representations T I ,7r2 of GL(2,A) (with w 1 w 2 = 1) which are unramified outside V whose compcnents at 210 are 7 r a 1 7 r ; of the proposition, at 211 are sp2,vl (resp. 7rhl), and at v 2 , v4 are d2 7rhs 7rh4 (resp. sPZ,v21 spZ,v3I sPZ,v4)* Set up the trace formula identity with the set V such that 7r1 x 7r2 (and 7r2 x TI)contribute to the side of C. Take the components fc,v,( i = 1, 2, 3) to have orbital integrals equal zero outside the elliptic set. Consequently we may and do choose the matching fv, (i = 1, 2, 3) to have zero stable &orbital integrals. Hence fH,v, (i = 1, 2, 3) are zero, and there is no contribution to the trace formulae of H and Co. We need to show that there is a contribution 7r to the 0-twisted trace formula of G. If-there is then it is unique, by rigidity theorem for automorphic representations on GL(n), and it is cuspidal, since XT;,) = S(vv4 1 / 2 7rv4 I , ~ & ' / ~ 7 r L , & is not induced from any proper parabolic subgroup. 7

7

We may apply "generalized linear independence" of characters at the archimedean places of F . There the completion is the complex numbers. Hence the local components are induced and the local lifting known. All matching functions fv, fcv are at our disposal. There remain in our trace identity only products over the set Vf = {v,;O 5 i 5 4) of finite places in

V. Note that both 7r1 x 7r2 and 7r2 x 7r1 contribute to the side of C, which has coefficient while the coefficient of the cuspidal contribution to the &twisted trace formula of G is 1. Choosing the fc,+,,(O 5 i 5 4) to be pseudo coefficients of the 7 r ~ , we ~ , obtain on the side of C a sum of 1's. We conclude that the side of G is also nonzero, hence 7r exists. Next we make this choice only at the places v, (i = 1, 2, 3, 4). Observe that tr7rc,vt(fc,vt)is 1, and tr7r'v,(fv,) = 1, since for i = 1, 2, 3, 4 the ~ ;~ ~' ,/ ~ 7 rwhich ~ , ) , is the XI-lift of T C , ~ ,= component 7rvt of 7r is S ( Y : ! ~ ~ F

i,

4.

Cuspidal Case

103

x sp2,,,. In fact the character relation Xl(7rh, x sp2,,,) = S,, on the &elliptic set alone, and the orthogonality relations for twisted characters (used below), would imply that the component 7rvt is Svzor J,,. Had the component 7rvt been J,, for an odd number of places w, (1 5 z 5 4) we would get a coefficient -1, and a contradiction. We obtain, for all matching functions fvo and fc,,,,the identity (w = wo) 7rht

(10

Here 7r&,, are representations of C, not equivalent to 7 r ~ , ,= IT; x IT; of the proposition, under the equivalence relation generated by d x 7rtf E 7rtt x 7r’ and d x x 7r”x-l N 7rf x d’.The coefficient n, counts the number of equivalence classes of global automorphic cuspidal representations whose components at each w are in the equivalence class of 7rlW x 7r2, (where 7r1, 7r2 are our global representations). W e claim that all the 7r&, which appear on the right are cuspidal. For this we use the central exponents of the representations which appear in our identity. Since all of the n(7rbv) arc nonnegative real numbers, a familiar ([FKl], [F4;II])argument of linear independence of central exponents, based on a suitable choice of the functions fc,,,implies that if one of the T&,, which appears with n(r&,) > 0 has nonzero central exponents namely it is not cuspidal - then IT,must have matching &twisted central exponents. This means that 7rv is the XI-lift of some 7rk x 7ri where 7ra of 7ri are not cuspidal, since we already know to XI-lift IT; x IT; where 7r; is fully induced or special. Linear independence of characters (after replacing t r r v ( f v x 0 ) on the left by tr(7r; x 7r;)(fcV)) gives a contradiction which implies that all the 7r&, which appear on the right are cuspidal, as claimed. Note that the identity exists for each local cuspidal ITC,,. W e claim that 7r, is uniquely determined by TC,,,and that the identity defines a partition of the cuspidal representations ofC,. For this we use the orthogonality relations for characters of elliptic representations of Kazhdan [K2] in its twisted form [Fl;II].These assert the existence of pseudo coeficients: if fv is a pseudo coefficient of a 0-elliptic IT; inequivalent to IT, then tr 7rv(fv x 0 ) = 0; this is # 0 if 7rh = 7rv, and = 1 if 7rh = 7rv is cuspidal. Now let f, be a pseudo coefficient of T ; inequivalent to 7rv for

104

IV. Lafling from SO(4) to PGL(4)

which we have the identity (sum over the

T;,

with n(n&,) > 0)

where f&, is the matching function on C,. We then have that the stable , ; ,the ~ elliptic set of orbital integrals of f&,,are equal to C n ( ~ ~ , ) x on C,. Evaluating our identity (1) at f: and f&,,,we note that only finite number of terms in (1) can be nonzero. Indeed, T&,, of (1)is a component of an automorphic representation of C(A) which is unramified outside V , its archimedean components (hence their infinitesimal characters) lie in a finite set, and the ramification of the remaining components is bounded (fixed at v1, 212, 213, 214; bounded by f,, - which is biinvariant under some small compact open subgroup - at 210). Hence 0 = tr 7rw(f, x 0) is

The inner products (TC,, 7r,Kw), or (xrCv, x,;~),are nonnegative integers, hence no 7r& can equal TC,, or T&,, as claimed. Given a cuspidal 7rcvwe now denote the 7rw specified by (1) by Xy(7rcw). We claim that

where p 1 , p2 are characters of the local quadratic extension E'IF' of the proposition, with p1 # Til, pz # jZ2 and p1pZIF' = 1. As in the beginning of this proof, we choose a totally imaginary global quadratic extension E I F such that at the place WO of F the completion E,,/F,, is our E'IFI. Then we choose global characters p l , p2 of A i / E X with our local components at V O , with p1p21AX = 1, which are unramified outside a set V consisting of the archimedean places of F , vo and three finite places 211,212, 213 (# V O ) which do not split in E l where the components are taken to satisfy pivj # piWj(bar indicates the action of the nontrivial automorphism of E I F ) . The existence of p 1 , p 2 is shown on using the summation formula for A g / E X , which is the trace formula for GL(1). First we construct p 1 , and then p2 - which is known to be pT1 on A XI F X-has to be constructed on A;/ E A X .

4. Cuspidal

Case

105

Set up the trace formula identity with the set Vf = {vi;0 I iI 3 ) such that ~ ~ ( px 71 r )~ ( p 2contributes ) to the trace formula of C . We choose the components fcviof the test function to have stable orbital integrals which vanish on the regular nonelliptic set. Hence we may take fvi (i = 1,2,3) to have zero stable orbital integrals, so that we can choose f~~~to be zero, hence there is no contribution to the trace formulae of H and CO. The trace formula of G will have the (unique) contribution

Indeed, we follow the homomorphisms of the Weil group

which define ~ ~ ( p Tl E) (,p 2 ) , and their composition with Pi(Z),

Pa

A1

(put pi =

= pi@)):

This homomorphism implies that

at all places where (E/F and) , are unramified. Following the aerguments leading to (1), we obtain (1) for our C,vv= E( 1)x E( 2), where as claimed. Note that when 1= 2-11 we have E( 1 2)= E( 1/ 11) and E( 1 2) thus is

as is known already.

106

IV. Liftang from SO(4) to PGL(4)

At this stage we note that we dealt with all square integrable repre) where sentations 7rc, of C,, except pairs 7r1 x 7r2 = ~ ~ , ( px17rEZ(p2), El, E2 are two distinct quadratic extensions of the local field F', and pi are characters of EC with pi # pi and p 1 ~ 2 1 F '= ~ 1, and pi) is not monomial from Ej ({i,j} = {1,2}). In fact, in residual characteristic 2 there are also "extraordinary" representations, which are not monomial; we shall deal with these later. We claim that 7ru in (1)f o r such a T E , (p1) x T E (p2) ~ is cuspidal. Let us review the homomorphisms of the Weil group which define the product T E (PI) ~ x 7rE2(p2). Denote by E the compositum of El and E2 and by ( a , ~the ) Galois group of E/F, so that El = E', E 2 = E" are the fixed points fields of r , c respectively (thus Gal(EI/F) = (0) and Gal(E2IF) = ( T ) ) . To multiply ~ ~ , ( pand 1 ) 7rE2(P2)we view their parameters as homomorphisms of WE/,, an extension of Gal(E/F) by E X, which factorize through W E , / Fand WE,/,. For T E (p1) ~ we have:

viewedin

We simply pull Ind(p1; W E ~ /W FE , , / E ,from ) W E , / , to W E / , using the diagram

The middle vertical (surjective) arrow is the quotient by

The arrow on the left is also surjective. Its restriction to CE C WEIE,is

and

maps to

4. Cuspidal For

Case

107

~ ~ ~ ( 1 we 1 2 have: )

a(€ Gal(E/F)) T(E

H u2

( E E,X - N E / E ~ EH ~)

(

Gal(E/F), viewed in W E ~ / H F)

p2&z)

h)

.

Composing these two representations by X I , we obtain the 4-dimensional representation p of W E I F :

z (E E )

diag (111‘Pl

112 “112

I

P1‘111 ‘112 ‘“112 ,“11 1“‘P

where a p means p ( a ( z ) ) ,or z

H

,

1112 “112 “111 “‘Pl ‘ p 2 ‘“112

)

diag(p,Tp,“p,uTp)where

When p1 # “111 and 112 # %2 this 4-dimensional representation p is irreducible, hence - repeating the global construction employed twice already in this proof we conclude that (1) is obtained with 7r, cuspidal, as had 7ru been induced from a proper parabolic subgroup of G,, the representation p would have had to be reducible. This establishes the claim. After completing the study of the lifting from PGSp(2) to PGL(4) we shall conclude the same result - that X l ( 7 r l x 7 r 2 ) has to be cuspidal if 7r1 is not x i r 2 , x2 = 1, and T I , 7 ~ 2are cuspidal but not ~ ~ ( p 7l r)~,( p 2 )by , showing that there are no induced G-modules that such 7r1 x 7r2 can XI-lift to. Since X y ( ~ ~ ~ (xp ~1 ~) , ( p z = ) )7r, is cuspidal, the orthonormality relations for twisted characters of cuspidal representations on G,, and the orthonormality relations for characters on C,, imply that the identity (1) ~

108

IV. Lifting from S O ( . ) to PGL(4)

reduces to only one contribution on the right side, namely our 7rc, = 7 r ( p~ l ) ~x 7rEz ( p 2 ) ,with coefficient n, = 1, thus x 1 ( 7 r ~ (, P I ) x 7rEz ( p 2 ) ) = T,, a cuspidal representation of G,. The orthogonality relations now imply (in odd residual characteristic) that for 7rc, = 7 r ~ ( p 1x) 7rE(p2), the trace identity (1) has only the term 7rc, on the right, and it becomes

Clearly when one of pi) is induced (thus pi = Tii), we have this identity with n, = 1. We claim that n, is 1 for all 7 r ~ ( p 1x) 7rE(p2). But first let us explain the meaning of the n,. A global (cuspidal, automorphic) representation 7rc = 7 r 1 ~ 7 r 2(with at least 3 square integrable components) defines an automorphic representation 7r of G ( h )on using the trace formula identity, by the arguments used repeatedly above (we choose test functions such that the function f H is zero at one of the places where 7rc is square integrable, and such that t r 7rc,(fc,) is 1 for square integrable 7rcw). Note that both 7rc = 7r1 x 7r2 and iic = 7r2 x 7r1 contribute to the trace formula of C when 7r1 74 x i r 2 , x2 = 1, as we now assume. The trace formula identity (for a suitable finite set V ) then takes the form

On the other hand we have the local character relations

for each T , on the left, where n, = 1 unless 7rcw= 7 r ~ ( p 1x) ~ ~ ( p z Replacing then the left side by n, tr7rc,(fcw) we conclude (applying n,,pairs 7r&, linear independence of characters on C,) that there are 5& of cuspidal representations of C(A) whose local components belong to the pair {TC,, iic,}. In other words, the representations 7r& = 7r1 x 7r2 which contribute to the right side are obtained from each other on interchanging the local components of 7r1 and 7r2 at a set S of places of F which is infinite and whose complement is infinite (if T I , 7ri differ by only finitely many components and both are cuspidal then they are equal by

nwEV

nwEV

4. Cuspidal Case

109

rigidity theorem for GL(2)). If all nu are 1, we would have on the right side only the contributions tr 7rcv(fcv) and t r iicv(fcv).

n

n

Let E‘IF‘ be a quadratic extension of local fields, and T: = rE’(p1) x TEj(p2) a cuspidal representation of C ( F ’ ) , where p, # pi,p1p21F’ = 1, p1p2 # x o NE‘IF‘ for any quadratic character x of F‘. Our claim is that the associated nu is 1. For this we construct first a totally imaginary number field F whose completion at a place vo is F’, and a cuspidal representation 7rc of C ( A ) which is unramified outside the set V consisting of the archimedean places and the finite places W O , . . . , v3. The component of 7rc at vo is our 7r; and at v l , wz, v3 it is sp,,, x spzv,. Then there are nu, (a positive integer depending on 7rE) pairs 7r&, if& of cuspidal representations of C(A) whose components at v 1 , v 2 , v 3 are sp, x sp2 and at each v the components belong Now we apply the theory of basechange for GL(2) to the pair {TC,, iicW}. for a quadratic extension E of F whose completion E @ F F,, = E,, is the local quadratic extension E’ of F’ = F,, . Then TC,,, = 7rg = ~ p ( p 1 x) TE‘(p2) lifts to a fully induced representation 7r&, = I ( p ~ l ~ jxZI(p2,jZ2) ~) of C ( E w , )and , the global 7rc lifts to the cuspidal representation 7rE whose components at the places of E above v1, v2, w3 are sp2 x sp,, at vo it is 7rg,vo specified above, and it is unramified at the places outside V . Since 7rg has no components of the form 7 r ~ ( p ix) 7 r ~ ( p ; )where , pil pk are characters of M X, M a quadratic extension of El and it has at least three square integrable components, it and its companion iig are the only cuspidal representations (with the indicated components at the places of E above v1, v2, w3) which XI-lift to 7rE = X l ( 7 r z ) . Consequently, each of the nu, pairs 7r&, iib of cuspidal representations of C(A) which XI-lift to X1(7r&), basechange from F to E to the pair 7 r z l iig of cuspidal representations of C(AE), which XI-lift to 7 r E = Xl(7rg). But the fiber of the base change map BCEIF, which takes 7rc to 7 r z l consists only of 7rc and XEIFTC, where X E / F is the quadratic character of kAx/FXNE,Fkli. Consequently each pair {7r& iib} is equal to the pair { T C , iic},up to multiplication by XE/F. But this implies that n,, = 1, as asserted. In residual characteristic two there are also the LLextraordinaryll cuspidal representations, which are not associated with a character of a quadratic extension. But since the relation (1) defines a partition of the set of representations of C , and we already handled the monomial representations,

IV. Lafling f r o m SO(4) to PGL(4)

110

the orthogonality relations imply the lifting and character relation, and the proof of the proposition is complete. 0 We obtain the following rigidity theorem for representations of S0(4), that is of C(A). Note that X l ( x 7 r 1 x x - ' 7 r 2 ) = X l ( 7 r 1 x 7 r 2 ) = X l ( 7 r 2 x T I ) . 4.2 COROLLARY. Let T I ,7 r 2 , 7 r i , 7r4, be cuspidal representations of GL(2,A) with central characters w1, w2, w i , w;, satisfying wlw2 = 1, wiwb = 1. Suppose that there is a set S of places of F such that (xiw, 4,)= ( ~ I ~ X ~ , f o~ r ~all ~ in X S; and ~ ) (xiw,&,) = ( ~ 2 u ~ u l ~ f o~ r w all v outside S , for some character xu of F," (for each v). Then the pair (xi,~ 4 is) ( n i x ,7 r 2 x - l ) or ( 7 r 2 x , 7 r l x - l ) for some character x of A X / F X .

A considerably weaker result, where the notion of equivalence is generated only by 7rlw x Tr2w N irzWx 7rlw but not by 7rlw x 1 : xw7rlwx follows also on using the Jacquet-Shalika [JS] theory of L-functions, comparing the poles at s = 1 of the partial, product L-functions L V ( S , 7r;

x i r l ) L V ( S , 7r; x

*I)

= L V ( S , n-1 x i r l ) L V ( S , 7r2

x

el).

Moreover, such a proof assumes the theory of L-functions. This has a consequence purely for characters. 4.3 COROLLARY. Let E I F be a quadratic extension of number fields, and ~ 1 p2, , p i , p; characters of A g / E X such that the restriction to A XI F X of the products p1p2 and pip; is trivial. Suppose that at 3 places v of F which do not split in E we have that piw# piw ( i = 1 , 2 ) . Suppose that there is a set S of places of F , and characters xw of F," for each place v of F , such that if piware the local components of pi on E," = ( E @ FF w ) x , then PL,) = PI^ * X V 0 N , ~2~ . (xW0 N ) - l )

(dW,

for all v in S , and ( P i w ,P L ) = ( P 2 w

*

x w

. N , Plw . ( x w O W - l )

for all v outside S (where N is the norm map from E, to Fw). Then there is a character x of A X I F Xsuch that (PLPk) = (P1.XON,P2.(XON)-1)

or ( P ; , P ; ) = ( P 2 . X O N , P 1 . ( X . N ) - 1 ) .

4.

Cuspidal Case

111

PROOF.Consider the cuspidal representations ~ ~ ( p l~ )~ , ( p z Note ) . ) 7r~(,u. that they are cuspidal at least at three places, and that x 7 r ~ ( p= x o N ) . Apply the previous corollary. O 4.4 PROPOSITION. Let IT,,be a square integrable 8-invariant representation of the group PGL(4, F,,). Its 0-character is not identically zero on the 8-elliptic regular set ( b y the orthonormality relations). Suppose it is not a 0-stable function o n the 0-elliptic regular set. Then it is a Xl-lij? of a of C(F,,), and its &character square integrable representation 7rzuO x is a $-unstable function.

PROOF.Let F be a totally imaginary global field such that FUz= F,, (i = 0 , 1 , 2 , 3 ) . We use a test function f = @ fv such that f,, (i = 1, 2, 3) is a pseudo-coefficient of a &invariant cuspidal representation rut= X , ( T E ; ( ~ ~ ) X T E ; ( ~ Z ) )E, i , Ei are quadratic extensions of FUzand plp21F,X = 1, and f,, is a pseudo coefficient of 7ruo. At all finite v # v, (0 5 i 5 3) we take f, to be spherical, such that for 6 # 1 which corresponds to the endoscopic group C and with f" = @f,, v finite, the K-&orbital integral @:(fa) is not zero at some ®ular elliptic y in G ( F ) ; this simply requires taking the support of the f, 2 0 for v # v, (0 5 i 5 3) to be large enough. Since the 8-stable orbital integrals of fv, (1 5 i 5 3) are 0, the &elliptic regular part of the &trace formula consists entirely of K-&orbital integrals, by a standard stabilization argument. As G ( F ) is discrete in G ( A ) ,for every f m = @,fV, v archimedean, f = f m f " is compactly supported, we can choose fa to have small enough support around y E G ( F )with a:( f") # 0 in the ®ular set of G(F,), to guarantee that a:( f ) # 0 for a single &stable ®ular conjugacy class y in G ( F ) ,which is necessarily &elliptic. Hence the geometric part of the 0-trace formula reduces to the single term @(: f ) , which is nonzero, hence the geometric part is nonzero, and so is the spectral side. The choice of the pseudo coefficients fv, implies that in the spectral side we have a &invariant cuspidal representation 7r of G ( A )with the cuspidal (i = 1 , 2 , 3 ) and the square integrable component 7ruo of components the proposition (note that 7r is cuspidal since it has cuspidal components at w, (i = 1 , 2 , 3 ) , hence it is generic). The components of 7r at any other finite place are spherical. Since the @-stableorbital integrals of f,, (i = 1 , 2 , 3 ) are zero, we may take f H v , and so f H to be zero. Hence there is no contribution to the spectral form of the trace formula identity from the

112


trace formulae of H and CO. Using generalized linear independence of characters we get the form of the trace formula identity with only our 7r as the single term on the spectral side of G, while the only contributions to the other side - 7rc - depend only on fc. Any unramified component of 7rc XI-lifts to the corresponding component of 7 r , and similar statement holds for the archimedean places. Using the pseudo-coefficients fwi at the places wi ( i = 1 , 2 , 3 ) we see that T C , ~= , TE;(pl) x TE;(p2). We are left with an identity of t r 7 r w o ( f v o x 0) with a sum rn(7rc)t r 7 r ~( f, ~ ~, ~~for , ) all matching f V o , f ~ , from ~ , which we conclude as usual using the character relations that the 7 r ~ are , square ~ ~ integrable, finite in number, and in fact consist of a single square-integrable 7rZVO x 7rkwo which X1-lifts to 7rwo. This has already been treated by our 0 complete description of the XI-lifting.

REMARK. The central character of a monomial

T ] T E (is ~)

If 7r~(p1)x 7 r ~ ( p zdefines ) a representation of C then the product of the central characters is 1, thus p1p2\FX= 1. Hence 7 r ~ ( p 1 i i ~7r~(p1p2) ), have central characters x . p17i21FX= x = x . p1p21FX. Thus

will not be in the image of X - see Proposition V.5 below: it is not I(7r1, TZ), 7r1, 7r2 on PGL(2).

V. LIFTING FROM PGSp(2) TO PGL(4) 1. Characters on the Symplectic Group Next we proceed with preliminaries on the lifting of representations of H = PGSp(2) to those on G = PGL(4). Recall that the norm map N : G + H is defined on the diagonal tori N : T* -+ TL by N(diag(a, b, c , d ) ) = diag(ab, ac, db, dc), and on the Levi factors on the other two proper parabolic subgroups by N(diag(A,B ) )= diag(det A , EBEA,det B ) where

E

=

(i -:)

,

and N(diag(a,A,d)) = diag(aA,deAe). The dual, lifting, map of representations takes the induced-from-the-Bore1 representation

where H = PGSp(2, F ) , G = PGL(4, F ) , F a local field. Lifting is defined by means of the character relation. Before continuing, let us verify 1.1 LEMMA. The Jacobians satisfy AG(t0) = A,(Nt).

PROOF.We take t = diag(cu, p, y, 6), a6 = By, and compute

Aw(t) = I det(Ad(t)(Lie N)(-’”I det(1 - Ad(t))(Lie NI, where N denotes the upper triangular unipotent subgroup in H . The Lie algebra Lie N consists of X E LieH = { X = - J - l X J } of the form

113

V. Lijling from PGSp(2) to PGL(4)

114

and the effect of 1 - Ad(t) is

Thus

gives AG(h0) when A H ( t ) is evaluated at t = N h .

0

We are now ready to extend the basic lifting result from the minimal parabolic to the other two proper parabolic subgroups of H .

1.2 PROPOSITION. We have that w, >d ir = wR1 XI 7r (A-) lifts to 7r4 = I G ( T ,i r ) , where w, is the central character of the representation 7r = 7r2 of GL(2, F ) = GSp(1, F ) , and f r = w;ln is the contragredient o J n . For the representation 7r2 oJPGL(2, F ) we have that p17r2 >dpT1(A-) l$ts to 7r4 = 1G(p1, 7r2, p ; ' ) , and 12(p1, p ; ' ) x 7r2 Xo-li.fts from CO to p17r2 x p;' on H .

PROOF.Recall that at mo = diag(A,I ) , A E GL(2, F ) , the value of the character xT4(m00), where 7r4 = I4(n,ir),has been computed to be

Since N(diag(A, I ) ) is diag(X, A , l),X = det A, we have to compute the character xw-l x, a t diag(X, A, 1). A general element m of the Levi M H of type (1,2,1) in H has the form m = diag(a,A,X/a), a E F X .If N = NH is the corresponding upper triangular unipotent subgroup then

6 ~ ( m=)Idet(Ad(m)ILieN)I = la2/detAI2 (using the X of the proof of Lemma 1.1 with u = 0). The usual argument, using the measure decomposition dg = S&'(m)dndmdk, shows that

( 7 r H ( f d g ) d ) ( h )is

S,JMJK

f (h-

=

'721mk)c5Z2( m )(w, ' M

7 r )( m ) $(k )

6,' (m)d n d m d k .

1. Characters on the Symplectic Group

115

Hence

of variables has I det(1 - Ad(m))l LieNI as

The change n1 = nmn-lm-' Jacobian. Hence with

AM,(^) = d&112(m)ldet(1- Ad(m))lLieNI =

a - b a-c u-d b c d

--

(we denoted the eigenvalues of A by b and c), the trace is

Now the Weyl group of M H in H (normalizer/MH) is represented by J g has the effect of mapping m = 1 and J . Changing variables g diag(u, A, A/.) to m' = diag(A/u, A, a). We have xW,l,T ( m )= x,,(a-lA) and U x ~ , I ~ ~=(x,(-A) ~ ' ) = w,(det(a-lA))-lX,(a-lA).

x

The trace becomes

Hence x,;l,,(diag(X,

A , 1))= (1+ wil(det A))Xr(A)/AM(diadk A , I)),

where AM,(diag(ab, A , 1))= 1-

U-1 b-1 .U b . (ab - 111

(where a, b are the eigenvalues of A), and we recover XT4(moO). We are done by Lemma 1.1: A ~ ( t 0 = ) AH(N~).

To show that X(p17r2 >a p;') = 1G(p117r2rp11), we first compute the &character of 7r4 = I~(p1,7r2,p;'). Note that 4 E 7r4 takes nmk to

V. Lifting from PGSp(2) to PGL(4)

116

S ~ 2 ( m ) p l ( u / d ) 7 r 2 ( A ) ~where ( k ) , m = diag(a,A,d) is in the standard Levi M of G of type (1,2,1). The measure decomposition contributes a factor SM(m)-', SO we have

The Jacobian of n1 H n, 721 = nmO(n-')rn-' is I det(1- Ad(m8))l Lie NI. Putting A ~ ( m 0= ) Sh''2(m)I det(1- Ad(m0))l Lie" we get

The 8-Weyl group W e ( M )of M in G (O-normalizer/M) is represented by {I,J } . Hence the trace is

+

and the character is [ p l (a/d) p1 (d/a)]xa2(A)/AM(mO). This we compare with the character of ITH = pl7r2 >a pT1, the representation of H normalizedly induced from the representation A

*

( 0 &€A€)

=

(

A

*

0 AwtA-Iw)

det A)7r2(A)

of the standard parabolic subgroup of H whose Levi M H is of type (2,2). As usual we have that t r r ~ (dg) f

The Weyl group of M H in H (normalizer / M H ) is represented by {I,J } . Writing (A1for det A , and X for JMH,H f (h-'rnh)dh,

I . Characters on the Symplectic Group

117

I - L (~ X / ( A I ) T ~ ( X ( A ( - ~, L J A ~ )

and we obtain

The character of

ITH

= p17r2 >a pT1 is then

1 -[Pl(X-' det A) f CLl(X/detA)]xnz(A)/AM~(m). 2

We need to compare the characters at an element g = diag(a,A,d) whose norm is h = N g = diag(aA,deA&).Note that on this h, the character from which T H is induced takes the value

( "," g A t )

PI ( a l d b 2 (4 = PI ( a l 4 7 r 2 (A) 7

the last equality since 7r2 has trivial central character. As A,(gQ) = AH(Ng) our character identity be complete once we show 1 . 3 LEMMA. We have Aw(g0) = AM,(Ng).

PROOF.As these factors depend only on the eigenvalues of A, we may take t = diag(a, b, c, d ) , and N t = diag(a,p, y,6) = diag(ab, ac, db, dc). Then Q,,,,,,,(t0) is the product of 6M(t0)-', where

with

I

det(1 - Ad(t0))l Lie N(1,~,111 = 1(1 -

:)

(1 -

namely

A(l,Z,l)(W= Similarly

(ab - cd)'(ac - bd)'(u - d), a3b2 c2d3

a>

(1 -

1

%),


118

I I =

a2b2acd2b2cd

as ( a ,0, y,6) = (ab,ac, db, dc). We conclude that A(l,2,1)(tO) =Ac,2)(Nt). This completes the proof of the lemma, hence also of the proposition. 0 Let x denote a character (multiplicative function) of F X / F X 2It.defines one dimensional representation X H of H by h H x(X(h)). If h = Ng (on diagonal matrices, if g = diag(a, b, c, d ) then h = diag(ab, ac, db, d c ) ) then X(h) = det g. Hence 1.4 LEMMA.The one dimensional representation X H , or x.l ~of ,H , A-lifts to the one dimensional representation x : g ++ X(det g) of G. The trivial representation of H lifts to the trivial representation of G. 0 We conclude 1.5 COROLLARY.The Steinberg representation of H A-lijh to the Steinberg representation of G.

PROOF. We use Lemma 3.5 of [ST],which asserts the following decomposition result: v2 x v >a v-3/2a is equal to

in the Grothendieck group (Z-module generated by the irreducible representations) of H . Here o2 = 1 to have trivial central character, and as usual v(x) = 1x1. The terms on the right decompose into irreducibles (on the right of the following four equations, which in fact define the square integrable Steinberg representation of H = PGSp(2, F ) ) : a) b) c) d)

+

~

~ x vp3l2a / ~ = 0 . l1G S p ( 2~ ) ~ ( vv-la ~ , sp2), v2 N v - ' c . S P = ~ 0 . StGSp(2)+ L ( v 2 ,v - l a . SP,), v2 v - l a ' 1 2 = 0 ' l G S p ( 2 ) L(v3l2sp2, v - 3 / 2 a ) , y 3 l 2S P > ~ a ~ - ~=/CT~' a StGSp(a)+L(v3l2S P ~~, - ~ / ~ a ) .

+

We can apply the A-lifting to a), as the lifts of two of its term is known:

2. Reducibility

119

Next we apply the X-lifting to b) and note that aI(u-lsp2 x u s p , ) , the left side, is known to be of length two, consisting of the Steinberg representation a.St4, and an irreducible which lies in the composition series of ~ I ( u - ~1 2/, u3l2) ~ , (which is also of length two, the other irreducible being a . 1 4 ) . Hence the common irreducible is X(L(u2,u - l a . sp2)),and the X-lift of a S t G S p ( 2 ) is aSt4. An alternative proof is obtained on X-lifting c) to get ul'(.U-112

X V12)

X(L(v3I2Sp2, V - 3 / 2 a ) ) ,

=a14

the last irreducible is the one common with G I ( V - ~spa, / ~ ,u3 / 2 ) , which is the X-lift of the left side of d); the latter has o Stq as the other irreducible in its composition series, hence the X-lift of the a S t ~ s ~ (on 2 ) the right of d) has to be a St4.

2. Reducibility It will be useful to record the results of [ST], Lemmas 3.3, 3.7, 3.4, 3.9, 3.6, 3.8, on reducibility of induced representations of H . This we do next. Note that the case of v2 x v x u-3/2a is discussed in the proof of Corollary 1.5 above. 2.1 PROPOSITION. ( u ) The representation x1 x x 2 x a of H , where characters of F X ,is reducible precisely when XI,x 2 , ~ 1 x or 2 x1/x2 equals u or u-l (its central character is ~ 1 x 2 ~ 7 ~ ) . ( b ) If x $! { [ u * ' l 2 , uf3l2} for any character [ with = 1, then * sp, x a and x . 1 2 x a are irreducible and

X I , ~ 2u ,are

c2

u1/2x

x u-1/2x x a =

If x # 1, vfl, u f 2 then

xxa

x

'

12

x u

x

+ x .sp, x u .

x x a.1 2 are irreducible and x x u x u-1/2a = x x a .sp, +x x a.1 2 . '

sp2 and

For any character a we have that irreducible and

u >a

u-'l2a. sp2 and u

>a

u - 1 / 2 a .1 2 are

120

V. Lifting f r o m PGSp(2) to PGL(4)

c2,

x u-ll2a contains a unique essentially square (c) I f [ # 1 = UJ x integrable subrepresentation denoted 6([u1/2sp,, u - ' / ~ u ) .Since

we have

as well as

the 4 representations o n the right of the last two lines are irreducible. ( d ) The representations 1 x a . sp, and u1l2sp, x u - 1 / 2 a (resp. ~ ' / ~ x1 u-ll2a) have a unique irreducible subquotient in common; it is essentially tempered, denoted by 7(v1I2sp,, ~ - l / ~ (resp. a ) 7 - ( ~ ~ / ~ U1 -2~ , / ~ ' T ) ) .These two 7's are inequivalent, and we have

= 1 x u x u-1/2a = 1 x a .sp,

fl x

0.12,

as well as the following decomposition into irreducibles:

Note that the 4 x 4 matrix representing the last four equations is not invertible, hence the irreducibles on the right cannot be expressed as linear combinations of the representations on the left.

2

2. Reducibility

Note that the X-lift of v J x J

XI

121

v-1/2a is a I ( ~ l V/ ~~ ,/ ~v J- ,~ / ~v-ll2) J,

+ a I ( v ’ / 2 , J 1 2 ,v - q = aI(sp2 xJsp2) + a I ( l 2 x Jsp2) + a q s p , x J 1 2 ) + a I ( 1 2 x J 1 2 ) . = a l ( v ’ / 2 , [ s p 2 , v-112)

It is invariant under multiplication by 6 ( J 2 = 1). To determine the liftings of the constituents of v J x J XI v-ll2a we shall use the trace formula identity. We shall also state the results of [Sh2], Proposition 8.4, [Sh3], Theorem 6.1, as recorded in [ST], Propositions 4.6-4.9, on reducibility of representations of H supported on the proper maximal parabolics P(z) of type (2,2) and P(l)of type (1,2,1). 2 .2 PROPOSITION. (a) Let 7r2 be a cuspidal representation of PGL(2, F ) and a a character of F X. Then v1/27r2XI v-1/2a has a unique irreducible subrepresentation, which is square integrable. Inequivalent ( ~ 2a, ) define inequivalent square integrables, and each square integrable representation of H supported in P(2) is so obtained (with w,,u2 = 1). (b) All irreducible tempered non square integrable representations of H supported in P(2) are of the f o r m 7r2 XI a where 7r2 is cuspidal unitarizable and a is a unitary character (with w,,a2 = 1). The only relation is 7r2 xu = 772 XI W,,zJ.

( c ) The unitarizable nontempered irreducible representations of H supported in P(2)are L(v07r2, a ) , 0 < ,B 5 $, a a unitary character of F X ,7r2 a cuspidal representation of PGL(2, F ) . 0

2.3 PROPOSITION. (a) Let 7r2 be a cuspidal unitarizable representation of GL(2, F ) such that 7r2J = 7r2 for a character 5 # 1 = J 2 of F X . Then vJ >a v - 1 / 2 7 r 2 has a unique subrepresentation, which is square integrable. Inequivalent (7r2, I ) define inequivalent square integrables. All irreducible square integrable representations of H supported in P(l) are so obtained, with Jw,, = 1. (b) All temi.’ered irreducible non square integrable representations of H supported in P(1) are either of the form x x 7r2, 7r2 cuspidal unitarizable representation of GL(2,F) and x # 1, as well as xw,, = 1 (the only equivalence relation o n this set is x XI 7r2 e x-’ XI x 7 r 2 ) , or one of the two inequivalent constituents of 1 XI 7r2.


122

(c) The irreducible unitarizable representations of H supported on P(ll which are not tempered are L(vP 0 for the 7rk with which we started, which occurs in the trace identity for t r 7r1 x 1r2, and which is square integrable or elliptic tempered constituent of 1 XI 7 r 2 , square integrable 7r2. Clearly X X ( N t ) = X X ( t x 0) is a stable class function on H , hence perpendicular to the unstable function x, that is

Now the f i are nonnegative and fi(7rk) > 0 for the 7r; for which m"(7rk)# cl 0. Hence m" takes both positive and negative values. In particular we see that a , the number of irreducible TH with ~ " ( T H#) 0, is at least two, hence a = 2 when 7r1 and IT^ are inequivalent.

6.3 LEMMA.Suppose that 7r1 and 7r2 are (irreducible) cuspidal (resp. square integrable) inequivalent representations of PGL(2, F ) . Then there are (irreducible) cuspidal (resp. square integrable) representations 7r& = 7r$(7r1 x 7r2) and 7r; = 7r;(7r1 x 7r2) such that for all matching functions f H , fc, we have

6. Fine Character Relations

143

a

PROOF. We have a = 2 and see that lm"I = 1.

(Clm"1)2 5 4.

As the m" are integers we

We now return to the identity (l),and evaluate it at fHv, (i = 1,2,3) which are matrix coefficients of nGv,(7rlv, x 7rzv,). This choice determines fcov, and fv, (or rather their orbital integrals), and our identity becomes

+(2m(n,)

+ 1)trnG(fH) + 2 ~ m ( ~ H ) t r ~ H ( f H ) . TH

Here we deleted the subscript uo to simplify the notations, as usual. The are inequivalent (pairwise and to T ; ) , hence c # 0. Since

XH

is a linear combination with positive coefficients of some tr7rH(fH), we conclude that c = 1 (in fact there is so far a possibility that c = but this will be ruled out later). Once again, the TH which occur with m ( 7 r ~#) 0 are cuspidal and finite in number.

4,

6.4 LEMMA.Suppose that n1 and 7r2 are (irreducible) cuspidal (resp. square integrable) inequivalent representations of PGL(2, F ) . Then m ( n z ) = m(n&).W e write m(n1 x 7 r 2 ) for the joint value.

PROOF.The twisted character X ( N t ) = ~ ~ ( ~x 0) ~ of, I(7r1, ~ ~7r2)) is( a t stable function, while x+ - x-, X* = x f ,is unstable. Hence their inner RH product is zero:

Let us discuss the case of a square integrable

7r1

= 7r2 on PGL(2, F )

6.5 LEMMA.If 7r1 = n2 are square integrable, they satisfy the conclusion of Proposition 5.

PROOF.The representation 1x7r2 is reducible (see V.2.l(d), V.2.3(b)). It is the direct sum of its two irreducible constituents, 7r$ and njj, which are tempered. The induced representation 1x 7r2 of H A-lifts to the induced


144

representation fH

I ( 7 ~ 2 , 7 r 2 ) of

G by V.1.2, narnely we have, for matching f ,

7

For the other identity of Proposition 5 we denote our representation by 7raW0,choose a totally imaginary number field F whose completion at wo is our local field, F,, , and construct two cuspidal representations, 7r1 and 7r2, of PGL(2,A), which have the same cuspidal components at three places w1, w2, w3 (f w 3 ) , which are unramified outside the set V = { v 0 , ~ 1 , w 2 , ~ 3 } , such that 7rlv0 is unramified while the component at wo of 7 ~ 2is our square integrable 7rzvO. We use the trace formulae identity, and the set V, such that the only contributions are those associated with I(7r2,7r2). These contributions are precisely those associated with 1(7r2,7r2), 1 x 7r2 on H(A) and 7r2 x 7r2 on Co(A). Note that at the three places w l , 212, w3 we work with f,, whose twisted orbital integrals vanish outside the O-elliptic set, while I(7rz,,, 7rzWi) is not O-elliptic. Hence the contribution from I(7r~,7r2)to the trace formula identity vanishes for our test functions. Now 1 x 7r2 enters the trace formula of H(A) as

where R(7rzv,) is the normalized intertwining operator on 1 x 7rav,, while 7r2 x 7rz enters the trace formula of Co(A) as n 0 < % 5 3tr(7r2,, x 7r2,%)( fc,,,). In the identity of trace formulae, the trace forkula of Co(A) enters with (see e.g. first formula in Chapter IV). We conclude that coefficient

-+

Repeating the same argument with 7r1 instead of 7r2 we get the same identity but where the product ranges over 1 5 i 5 3 instead. In both cases fc,,, (1 5 i 5 3) can be any functions supported on the elliptic set of C,,. Taking the quotient we conclude that


145

for all matching functions fc,,,, fHvo. The normalized intertwining operator R(7rzu0)has order 2 but it is not a scalar on the reducible 1 >a 7rzv0. It is 1 on one of the two constituents, which we now name 7 r i v o , and -1 on 0 the other, which we name IT;,^, as required. We can now continue the discussion of the case of square integrable 7r1 # 7r2. We claim that

6.6 LEMMA.The (finite)sum over T H ( # 7 r g ) in our identity (for all matching f , f H , where the m are nonnegative)

is empty.

PROOF.To show this we introduce the class functions on the elliptic set of H

x1 = (247F.$) + 1)x,; + (2m(x,) + l ) X X i and XH

Also write xy(,,,,,, for the class function on the regular set of H whose value at the stable conjugacy class Ng is x ~ ( ~ ~x ,0). ~~)(g Our first claim is that x1 (and xo) is stable. It suffices to show that (xl,d ~ ( x ~7 r Li ) ) ~is 0 for all square integrable 7ri x 7rh on CO.By (3) and since m+ = m- this holds when 7ri x 7r; is equivalent to 7r1 x 7 ~ 2(or 7r2 x TI). When 7ri x 7r; is inequivalent to 7r1 x 1r2 or 7r2 x 7r1, the twisted orthogonality relations for twisted characters imply that ( x ? (, T~z ),, xl(T; e , X h ) ) ~is zero. Since the coefficientsm are nonnegative, if T H E { 7 r 2 ( 7 r 1 x 7r2)}U{7r;(7r1 x ~ 2 ) then ) it is perpendicular to d H ( 7 r i x T ; ) , and the claim follows. xo - (xl Next we claim that xo is zero. If not, x = (x' + x o , x l ). ~ x 0 , x o ). x1 ~ is a nonzero stable function on the elliptic set of H . (Note that (xo,x l ) =~0 ) . Choose f:, on G,, such that @ ( tf:,, x 0) = X ( N t ) on the O-elliptic set of G,, and it is zero outside the O-elliptic set. As usual fix a totally imaginary field F and create a cuspidal &invariant representation

+


146

which is unramified outside 210, 211, v2, v3, has the component StWiat wi (i = 1, 2, 3), and tr7r,,(fLo x 19) # 0. Since 7r is cuspidal as usual by generalized linear independence of characters we get the local identity 7r

for all matching f,,, fH,vo. The local representation 7r = 7rwo is perpendicular to I ( 7 r l 1 7 r ~since ) (x,xo x l ) =~ 0, and xo x1 = x;(,,,,,,. Since x1 xo is perpendicular to the &twisted character x; of any 8invariant representation ll inequivalent to I(7r1,7r2), x is also perpendicular t o all x i , hence trII(fL, x 6) = 0 for all &invariant representations ll, contradicting the construction of 7rwo with tr7rW,(f~,x 19) # 0. Hence x = 0, which implies that xo = 0, namely that for 7r1 # 7r2 we have

+

+

+

Since the character on the left is stable, it is perpendicular to the unstable character on the left of (3). So the right sides of (3) and (4) are orthogonal, hence m(7r;) = m(7r;).

6.7 LEMMA.The integer m = m(7ri)= m(7r;) is 0 . We show at the end of section 10 that precisely one out of 7rL, 7r; is generic. Our proof of the vanishing of m(7ri)= m(7r;) is global. It is based on the theory of generic representations. This latter theory implies that given automorphic cuspidal (generic) representations 7r1 and 7r2 of PGL(2, A) there exists a generic cuspidal representation T H of PGSp(2, A) which is a Xo-lift of 7r1 x 7r2, namely XO(7r1, x ~ 2 , ) = TH, at almost all places Y of F , where 7rl, 7r2 and T H are unramified and the local lifting A0 is defined formally by the dual group homomorphism XO : 6'0-+H . Moreover, in Corollary 7.2 below we prove that ITHoccurs in the discrete spectrum of PGSp(2, A) with multiplicity one. To use this, beginning with our local square integrable representations 7riwo and 7rhw0, we construct a totally imaginary field F with FWi= F,, at three places q , w2, 213 and global cuspidal representations 7r1 and 7r2


147

of PGL(2,A), which are unramified outside wi (0 5 i 5 3), with cuspidal components 7rlV3 and XZ,, , and 7rjvi N r;,, (i = 0, 1,2;j = 1,2). We set up the identity (2), which in view of (3) and (4) takes the form

TH

21

where Y ranges over the finite set {wi; 0 5 i 5 3). Corollary 7.2 below asserts that m ( 7 r ~is) 1 for at least one T H = C&TH~ (product over all places Y of F ) . Hence the corresponding number ni(2mvl 1) f 1 is 2 m ( n ~= ) 2. Since mvo= m,, = mvn,and 33 f 1 > 2, mVO is zero. The 0 proposition follows.

+

REMARK. Our proof is global. It resembles (but is strictly different from) the second attempt at a proof of multiplicity one theorem for the discrete spectrum of U(3) in [F4;II], Proposition 3.5, p. 48, which is also based on the theory of generic representations. However, the proof of [F4;II],p. 48, is not complete. Indeed, the claim in Proposition 2.4(i) in reference [GP] to [F4;II], that “L& has multiplicity I”, is interpreted in [F4;II] as asserting that generic representations of U(3) occur in the discrete spectrum with multiplicity one. But it should be interpreted as asserting that irreducible 7r in L& have multiplicity one only in the subspace Li,l of the discrete spectrum. This claim does not exclude the possibility of having a cuspidal 7r‘ perpendicular and equivalent to 7r c L;,l. Multiplicity one for the generic spectrum would follow via this global argument from the statement that a (locally generic) representation equivalent to a globally generic one is globally generic (multiplicity one implies this statement too). In our case of PGSp(2) this follows from [KRS], [GRS], [Shl]. A proof for U(3) still needs to be written down. The usage of the theory of generic representations in the proof above is not natural. A purely local proof of multiplicity one theorem for the discrete spectrum of U(3) based only on character relations is proposed in [F4;II],Proof of Proposition 3.5, p. 47. It is based on Rodier’s result [Roll that the number of Whittaker models is encoded in the character of the

148


representation near the origin. Details of this proof are given in [F4;IV] in odd residual characteristic for the basechange lifting from U(3, E / F ) to GL(3,E). It implies that in a tempered packet of representations of U(3, E / F ) there is precisely one generic representation. We carried out this proof in the case of the symmetric square lifting from SL(2) to PGL(3) ([F3]) but not yet for our lifting from PGSp(2) to PGL(4).

7. Generic Representations of PGSp(2) We proceed to explain the result quoted at the end of the global proof of Proposition 5 above (after Lemma 6.7) and attributed to the theory of generic representations. We start with a result of [GRS] which asserts: the weak (in terms of almost all places) lifting establishes a bijection from the set of equivalence classes of (irreducible automorphic) cuspidal generic representations T H of the split group SO(2n 1,A), to the set of representations of PGL(2n, A) of the form 7r = I(7r1,. . . , rr),normalized induction from the standard parabolic subgroup of the type ( 2 n 1 , . . . , 2nr)r n = n1+. . . nr, where 7ri are cuspidal representations of GL(2ni, A) such that L( S, 7ri A2, s) has a pole at s = 1 and 7ri # 7rj for all i # j. The partial L-function is defined as a product outside a finite set S where all 7ri are unramified. Moreover, if T H is a cuspidal generic representation (in the space of cusp forms) of SO(2n 1,A) which weakly lifts to 7r as above, and 7rk is a cuspidal representation of SO(2n 1,A) which weakly lifts to 7r and is orthogonal to T H ,then 7rL is not generic (has zero Whittaker coefficients with respect to any nondegenerate character). Note that this result does not rule out the possibility that there exists a cuspidal representation 7rL of SO(2n 1,A) which is both orthogonal and equivalent to the generic cuspidal T H , and consequently is locally generic everywhere, but is not (globally) generic. Hence T H may occur in the discrete, in fact cuspidal, spectrum of SO(2n 1,A) with multiplicity ~ ( T H greater ) than one. Of course we are interested in the case n = 2, where

+

+

+

+

+

+

7.0 LEMMA.PGSp(2) 2 SO(5).

7. Generic Representations of P G S p ( 2 )

149

PROOF.This well-known isomorphism can be constructed as follows. Let U be the 5-dimensional space of 4 x 4 mastrices u such that tr(u) = 0 and t J t u J = u.Then PGSp(2) acts on U by conjugation: g : u H g u g - l , and the action preserves the nondegenerate form (u1,uz) H tr(u1uz) on U . The action embeds PGSp(2) as the connected component SO(5) of the 0 identity of the orthogonal group O(5) preserving this form. A related result is Theorem 8.1 of [KRS]. It asserts that if TO is a cuspidal representation of Sp(2, A) which is locally generic everywhere, and the partial L-function L ( S ,T O , id5, s) is nonzero at s = 1 then TO is (globally) generic. Here L is the degree 5 L-function associated with the 5-dimensional representation ids : SO(5, C)c-) GL(5, C)of the dual group SO(5,C) of Sp(2). When TO is generic this L-function is nonzero at s = 1 by Shahidi IShl], Theorem 5.1, since id5 can also be obtained by the adjoint action of the SO(5, C)-factor in the Levi subgroup GL(1, C)x SO(5, C)of SO(7,C) on the 5-dimensional Lie algebra of the unipotent radical. This is case (xx) of Langlands [L2]. Together, [KRS], Theorem 8.1, and [Shl], Theorem 5.1, although do not yet imply that a locally generic cuspidal representation of Sp(2, A) is generic, do assert that: Let TO) TI, be cuspidal representations of Sp(2,A). 7.1 PROPOSITION. Suppose that TI, i s generic, TO, i s generic f o r all v, and TO, N for almost all v. T h e n T O i s generic.

I wish to thank S. Rallis for pointing out to me [KRS], [GRS] and [Shl] in the context used above, and F. Shahidi for the reference to [L2], (xx). We need this result for PGSp(2,A): 7.2 COROLLARY. A n y generic cuspidal representation T occurs in the discrete spectrum of the group PGSp(2, A) = SO(5, A) with multiplicity one. In view of the results of [GRS] quoted above it suffices t o show that: 7 . 3 LEMMA.Let T H , nk be cuspidal representations of PGSp(2,A). Suppose that ~h is generic, T H , i s generic for all v, and T H , si rLv for almost all v. T h e n T H is generic.

To see this, let us explain the difference between the group PGSp(2, F ) (which is equal to GSp(2, F ) / Z ( F ) )and the group Sp(2, F ’ ) / { H } .

150

V. Lafiing from PGSp(2) to PGL(4)

Note that PGSp(2) = PSp(2) as algebraic groups (over an algebraic closure F of the base field F ) . We have the exact sequences

1 + G,

+ GSp(2)

I-+ {*I}

-+

-+

Sp(2)

PGSp(2)

-+

PSp(2)

-+

-+

1,

1,

since the center Z of GSp(2) is G, while that 2 s of Sp(2) is (*I}. Since H ( F,Gm) = (01 and H 1 ( F , Z / 2 ) = F X / F X 2 the , associate exact sequences of Galois cohomology give 1 + F X -+ GSp(2, F ) -+ PGSp(2, F ) + 1,

thus PGSp(2, F ) = GSp(2, F ) / F X ,and 1 -+ (61)-+ Sp(2, F ) -+ PSp(2, F ) + F X / F x 2 .

Hence Sp(2,F ) / { f I } = ker[PGSp(2,F ) + F x / F x 2 ](as PGSp(2, F ) = PSp(2, F ) ) . The kernel is induced from the map X : GSp(2) -+ G,, associating to g its factor of similitudes. Globally we have Sp(2,A)/Zs(A) = ker[GSp(2,A)/Z(A)

-+

AX/AX2],

where Zs(A)is the group of idiiles (2,) E Ax with z, E {*I} for all w. It will be simpler to work with the group Zp(2,A) = Z(A) Sp(2,A), with center Z(A), and Zp(2, F ) = Z ( F )Sp(2, F ) . Note that Zp(2,A)/Z(A) = Sp(2, A)/zs(A) and

An automorphic representation of Sp(2, A) with trivial central character is the same as an automorphic representation of Zp(2, A) with trivial central character. Let us also explain the passage from representations of GSp(2, F ) t o those on F X Sp(2, F ) .

7.4 LEMMA.Put H = GSp(2, F ) and S = Sp(2, F ) Z ( F ) . (i) Let 7r be an irreducible admissible representation of H . Then the restriction Res: 7r of 7r to S is the direct sum of finitely many irreducible representations.

7. Generic Representations of P G S p ( 2 ) (ii) Let

151

be a n irreducible admissible representation of S . T h e n there is a n irreducible admissible representation T of H whose restriction to S contains

T’

7r’.

PROOF.The map G S p ( 2 , F ) -+ F X associating to h its factor X(h) of similitudes defines the isomorphism H / S Z F X / F X 2= ( Z / 2 ) T ,r finite ( r = 2 if F has odd residual characteristic). By induction, it suffices t o show (i), (ii) with H , S replaced by HI, S’ with S c S’ c H’ c H , H‘IS‘ = 212. (i): Let ( T , V) be an admissible irreducible representation of HI. Then Re& T is admissible. If it is irreducible, (i) follows for 7r. If not, V contains a nontrivial subspace W invariant and irreducible under S’. For h E H’ - S’ we have V = W 7r(h)W. Since W n 7r(h)W is invariant under H’, it is zero, and so V = W @ r(h)W where W and 7r(h)W are irreducible S’-modules. (i) follows. (ii): Given an irreducible admissible representation ( T ” , W ) of S’, put T I = Ind$’(TS’). For h E H‘ - S’, if s H ~ ” ( h - l s h )(s E S’) is not equivalent to 7rs‘ then 7r1 is irreducible and Resg’(7rI) contains nS‘.0 t h erwise there exists an intertwining operator A : ( T ~ W ’ , ) + (7rs‘, W ) with 7rS’(h-’sh) = A-17rs’(s)A (s E S’) and A2 = 7rS’(h2)(by Schur’s lemma). We can then extend rrs’ to a representation T on the space W of 7r’’ by ~ ( h=) A . We have ( T , W )~f TI by w H fw(g) = 7r(g)w (g E H ’ ) , and T I pv 7r @ T W , where w is the nontrivial character of H‘IS‘ = 212.

+

REMARK. The restriction of a generic admissible irreducible T of H t o T’ with multiplicity > 1. Indeed, 7r is generic if 7r Indg7,b for some generic character 7,b of the unipotent radical N = N ( F ) of H . Note that N c S . Since H = Udiag(l,XI)S, X E F X / F x 2 ,and diag(1,Al) normalizes N , each 7r’ c Resg 7r is a constituent of Indz for some generic character of N . Now 7r1 = Indg(.rrs) c I n d g Ind; 7,bx = Ind; 7,bx. The uniqueness of the embedding (“Whittaker model”) of T in Ind; implies the uniqueness of the embedding of 7r in T I , hence of T’ in 7 r , since by Frobenius reciprocity: Homs(nS, Resg T ) = HomH(7rI,7r), and the complete reducibility (i) above, 7rs is contained in 7r with the same multiplicity that T is contained in T I . 0

S contains no irreducible representation ~f

+

PROOFO F LEMMA7.3. Let us then take a cuspidal representation

7r

=


152

of PGSp(2,A) which is locally generic. Thus for each ‘u there is a nondegenerate character $, of the unipotent radical N, of the Bore1 subgroup of H , = PGSp(2, F,) (and of S, = Zp(2, F,)/F,X) such that T, L-) Ind;; ($,). Applying the exact functor Resf; of restriction from H , to S, we see that Res;; 7rv -+ @-, Ind$’($J), where $2 are the translates of $, under H,/S, 21 F,X/F,X2. Thus each irreducible constituent T,” of Rest; 7rv is generic. Since 7r is a submodule of the space Li(PGSp(2, F)\PGSp(2,A)), the of restriction map 4 H $I(Zp(2, A)/Z(A)) defines a subspace @7rv

Choose an irreducible (under the right action of Zp(2,A)) subspace 7 r s of TO”. Then 7rs = @7rf is a cuspidal representation of Zp(2,A) whose components are all generic. The same construction, applied to the cuspidal locally equivalent to 7rs at almost generic d,gives a cuspidal generic ds, all places. By Proposition 7.1 (namely the results of [KRS] and [Shl] for Zp(2,A) = Z(A)Sp(2, A)), r s is generic. This means that for some nondegenerate character $ of N ( F ) \ N ( A ) , we have 7rs But

C

-+

Ind~(,~A)’A $.x

Indzp(2,A)/AX PGSp(24) (.rrs), and induction is transitive: Indg Ind:

and exact, hence

7r -+

Indz$’(2’A)$. In other words,

7r

= Indz,

is generic.

0

Once we complete our global results on the lifting X from the group PGSp(2, A) to the group PGL(4, A) in section 10, we deduce from [GRS] that each local tempered packet contains precisely one generic member, and each packet which lifts to a cuspidal representation of PGL(4,A), or to an induced I(nlI7r2) where 7r1, 7r2 are cuspidal on PGL(2,A), contains precisely one representation which is everywhere locally generic. The latter is generic if it lifts to I ( T ~ , T ~ ) .

8 . Local Lifting from PGSp(2) One more case remains to be dealt with. 8.1 PROPOSITION. Let 7rv, be a 8-invariant irreducible square integrable representation of G,, over a local field F,,, which is not a XI-lift. Then

8. Local Lifting from PGSp(2)

153

there exists a square integrable irreducible representation XH,,, of H,, which X-lifts to T,,, thus tr 7rv0 (f,, x 0) = t r TH,,, (fH,vo) for all matching fvo and f H , w o . I n particular the 0-character of 7rwo is 0-stable, and the character of TH,,, is a stable function o n H,,. PROOF.Since 7rwo is 0-invariant, square-integrable and not a XI-lift, its character is 0-stable by Proposition IV.4.4. We choose a totally imaginary global field F and a function f = 18 f, whose components f, at 4 places vi ( i = 0 , 1 , 2 , 3 ) with Fvt= F,, are pseudo matrix coefficients of 7rv, where at w = wo this 7 r , is the T,, of the proposition, at W I it is 7rwl = IG(7r1wl,7r~ where xiwlare distinct cuspidal PGL(2, F,,)-modules, while 7rwz and 7rw3 are the Steinberg PGL(4, F,)-modules. The other components f, at finite w are taken to be spherical and 3 0. Since the &orbital integrals of f,, (in fact also f,,, i = 1 , 2 , 3 ) are 0-stable functions (supported on the 0-elliptic set), the geometric part of the &trace formula is 0-stable: it is the sum of We choose fm = @,f,, w archimedean, 0-stable orbital integrals, to vanish on the non 0-regular set. Since G(F) is discrete in G(A) and f = @f, is compactly supported, @;t( f) # 0 for only finitely many 0-stable conjugacy classes (&elliptic and regular) y in G ( F ) . Restricting the support of fa we can arrange that # 0 for a single &stable class y. Hence the geometric side of the 0-trace formula is nonzero. Consequently the spectral side is nonzero. The choice of f,, (i = 0 , 1 , 2 , 3 ) as a pseudo-coefficient can be used now to show the existence of a 0-invariant 7r whose component at w1 is 7rvl = IG(~~~,~,TZ,~), and consequently 7rVz and 7rU3 are Steinberg (note that tr 7rwz(f,, x 0) # 0 does not assure us that 7rVz is Steinberg, but given that 7ru1 is I~G(7r~,~,7r2,,), the global 7r must be generic, hence 7rwz is the square integrable generic constituent in the fully induced

@T(f).

@v(f)

It follows that 7r is generic and cuspidal (it contributes to the sum I in the spectral side of the 8-trace formula, not to 1 ( 2 , 2 ) , etc.). Since the components nut ( i = 1 , 2 , 3 ) and by the same argument also T,,, are our 0-stable ones, 7r is not a XI-lift, nor it is a X-lift of the form I(7r1,7r2).Thus when we write the trace formula identity fixing all finite components to be

V. Laftang from PGSp(2) to PGL(4)

154

those of 7r at all u # vi ( i = 0,1,2,3),the only contribution other than would be from H , namely

V

RH

7r

V

Thus T H are discrete spectrum representations of H(A) whose components at each finite u # ui ( i = 0,1,2,3) are unramified and X-lift to 7rv. The products range over ui ( i = 0,1,2,3) and the archimedean places. Next we apply the generalized linear independence argument at the archimedean places. Consequently we can and do omit the archimedean u from the product, and restrict the sum to T H with X ( 7 r ~ = ~ )7rm. Evaluating at v1, u2, u3 with the pseudo coefficient f v i , which is 0elliptic, we can delete these u from the product, but now the sum ranges over the 7 r which ~ in addition have the Steinberg component a t u2 and v g , and 7rLV1or 7rkV1at u1. Omitting the index VO, we finally get for our 7r = 7rvo the equality

for all matching f and f H . Since 7r is square integrable and the m ( 7 r H ) are nonnegative, the theorem of [C] on modules of coinvariants implies that all T H on the right are cuspidal, except for one square integrable noncuspidal T H if 7r is the square integrable constituent of I ~ ( v ~ / V~ - 7~ /r ' ~ T ~, )for , a cuspidal 7rz = nz(p), where p is a character of E XI F X ,E f F being a local quadratic extension. Evaluating at fH = f& = f ( T H ~ )where , we list the T H and f ( n ~ i ) denotes a pseudo coefficient of T H ~ we , conclude from the orthonormality relations for twisted characters that the sum over T H is finite. The resulting character relation

c:=l

and the orthonormality relations for &characters of square integrable representations, i.e.: (x!, x!) = 1, imply that

is 1, thus Cm(7r~)' is 1. Hence there is only one term on the right with coefficient m = 1. 0

8. Local Lifting f r o m PGSp(2)

155

8.2 COROLLARY. Let 7r2 be a cuspidal (irreducible) representation of GL(2, F), F local, with ("2 = 7r2 and central character E # 1 = t2. The square integrable subrepresentation S(uE, ~ - ' / ~ 7 r 2 ) of the H-module v t x u-1/27r2, A-lifts to the square integrable submodule S ( L J ~ ~/ - ~' / I~ 7~r ~ 2 of ), the G-module I ~ ( v ~ / ~, V7 -r' 2/ ~ X Z ) .

PROOF.This follows from the proof of the proposition. Note that the only noncuspidal non Steinberg selfcontragredient square integrable representation of G ( F )is of the form S(v1/'7r2, v P ' / ' 7 r 2 ) , where 7r2 is a cuspidal representation of GL(2, F ) with central character El E2 = 1, and E7r2 = 7r2. The square integrable S ( U ' / ~ T ~ , where 7r2 is a cuspidal representation of PGL(2, F ) , is the XI-lift of sp2 X T Z . If the central character of 7r2 is E # 1 = E 2 , it is associated with a quadratic extension E of F , and since E7r2 = 7r2 there is a character p of E X ,trivial on F X ,such that 7r2 = 7r2(p). The only square integrable representations of PGSp(2, F ) not accounted for so far are S(v6,U - ' / ~ T ~ ) ,w,, = E # 1 = E 2 , 57r2 = 7 r 2 . Since uE x u-1/27r2 A-lifts to I ~ ( ~ ~ / ~ i r 2 , ~ - ~and / ~ 7772 r 2=) ,1x2, the decaying central exponents in these fully induced representations correspond, hence S(ua J sp,. If 7r1 = E sp2,J2 = 1, and 7r2 is cuspidal, then 7r; is the square integrable constituent 6 ( 7 r 2 [ ~ ' / ~J, v - l l 2 ) of the induced . r r ~ J u >a~ [/ ~v - ' / ~ while , 7r; is cuspidal, which we denote by 6- ( 5 ~ ~ / ~ 7Jrv2-, ' / ~ ) . If 7r1 = a s p 2 and 7 4 = [asp2, I(#1 = E 2 ) and a are characters of F X, then 7r; is the square integrable constituent 6([v1j2 spZl~ - l / ~ ofa ) the induced sp, [ Y ~ x/ (~T U - ' / ~ , while 7r; is cuspidal, which we denote by 6- ( & A 2 sp2,v-%). We made this explicit list in order to describe the character relations where in the last three paragraphs sp, is replaced by the nontempered trivial representation 1 2 of PGL(2, F ) .

8.4 PROPOSITION. For any cuspidal representation and character [ of F X with 5' = 1, we have

7r2

of PGL(2, F )

tr(Jl2 x 7r2)(fco> = t r L ( T ~ [ V ~~ v/- ~l / ,, ) ( f H )

+ tr 6 - ( 7 r 2 ~ v l /~v-'/~)(fH), ~,

tr I G ( E l 2 , n2; f x 6') = tr ~ ( 7 r , ~ va a , ~ , ~ / ~ We ) . also let 7 r i W be the cuspidal ~ - ( t w ~ , 1 ~ 2 s P 2 w , ~ w t w ” if ~ 1 ~E w2 # ) 1

and

cw

~ ( v ~ ~ ~ s p ~if ~ , = a 1. ~ v ~ ~ / ~ ) We conclude that for each Y the component of multiplicity is again determined by the formula 1 m(XH) = -(I 2

ITH is 7rffw

or

7rGw.

The

+ & ( a E 1 2 x a12)(-1)n(”H)),

where n ( r ~ is) the number of components in fact 1 since

7rZw of

TH. The sign E here is

7rff = @w7rffw= @wL(twvw,tw avv,1/2),

) , n ( 7 r ~= ) 0, is discrete which we denote also by L ( [ v , t x a ~ - l / ~ thus spectrum, in fact a residual representation, by [Kim], 7.3(2). The representations whose components are almost all 7rffw and which have a cuspidal component 7rGw are cuspidal (if they are automorphic). They make counterexamples to the generalized Ramanujan conjecture, as almost all of their components are the nontempered x i w . With the complete local results on the liftings A0 and A, as well as the full description of the global lifting A0 from Co(A) to H(A) and the global lifting X from the image of XO to the self-contragredient G(A)-modules of type 1 ( 2 , 2 ) (induced from the maximal parabolic of type (2,2)), we can complete the description of the lifting A.

10. Global Packets

169

10.7 DEFINITION. The stable discrete spectrum of L2(H(F)\H(A)) consists of all discrete spectrum representations T H of H(A) which are not in the image of the Xo-lifting (thus there is no Co(A)-module T I x 7r2 such that X H , is equivalent to Xo(7r1, x 7r2,) for almost all u). We proceed to describe the stable spectrum of H(A).

10.8 DEFINITION. Let F be a global field. To define a (quasi-) packet { T } of automorphic representations of H(A) we fix a (quasi-) packet {T,} of local representations for every place v of F, such that {xu} contains an unramified representation 7r,” for almost all v. The global (quasi-) packet { T } which is determined by the local { T ~ consists } by definition of all products @nu with rV in {nu} for all v and T, = T,” for almost all v. Put in other words, the (quasi-) packet of an irreducible representation T = of H(A) consists of all products @7rh where T ; is in the (quasi-) packet of T , and 7rh = T , for almost all v. If a (quasi-) packet contains an automorphic member, its other members are not necessarily automorphic, as we saw in the case of X O ( T ~x ~ 2 ) Thus . the multiplicity m(7r)of T in the discrete spectrum of H(A) may fail to be constant over a (quasi-) packet.

10.9 THEOREM.Every member of a (quasi-) packet of a stable discrete spectrum representation of H(A) is discrete spectrum (automorphic) representation, which occurs with multiplicity one in the discrete spectrum. Thus packets and quasi-packets partition the stable spectrum, and multiplicity one theorem holds for the discrete spectrum of H(A) ( a t least f o r those representations with at least three elliptic components). Every stable packet which does not consist of a one dimensional representation A-lifts t o a (unique) cuspidal self-contragredient representation of G(A). The quasi-packets in the stable spectrum of H(A) are all of the f o r m { L ( u < ~, - ‘ / ~ 7 r 2 ) } ,~2 cuspidal with central character 6 # 1 = E2. E v e y packet or quasi-packet in the discrete spectrum of H ( A ) with a local component which is one-dimensional or of the f o r m L(uvE,, v V ” ~ T ~ , ) , 7rlV cuspidal with central character Ev # 1 = E;, is globally so, and thus lies in the stable spectrum. In view of our global results we can write the remains of the trace


170

formula identity as the equality of the sums

The sum on the left, I / , ranges over all self-contragredient discrete spectrum representations of G(A) which are not XI-lifts from C(A). The sum on the right ranges over all discrete spectrum representations T H of H(A) which are not in packets or quasi-packets Ao-lifted from Co(A). Our test functions f = @fv and f H = @ f H v have matching orbital integrals and at least at three places their components are elliptic (the orbital integrals vanish outside the elliptic set). We first deal with the following residual case.

10.10 PROPOSITION. For every cuspidal representation 7r2 of GL(2, A) with central character [ # 1 = E2 (hence In2 = 7rz) there exists a quasipacket { L ( v [ ,U - ~ / ~ T Z ) }of representations of H(A) which A-lifts to the residual (discrete spectrum but not cuspidal) self-contragredient representation J(v1/27rz,u-1/27rz)

= @.vJ(u,l/27r~v, v,1/27rzv)

of G ( N . Each irreducible in such a quasi-packet occurs in the discrete spectrum of H(A) with multiplicity one, and precisely one irreducible is residual, namely ~ v ~ ( v vv-1/27rzv). v ~ v ,

PROOF.If

Cv # 1 and 7rzV is cuspidal, J

( U ~ / ~~ t T’ ” ~7 r~2 ~, )is the A-lift

of

L(vvc v ,v, If

1/27rzv

).

Ev # 1 and 7rzV is not cuspidal, 7rzv has the form I(pV,p&), p;

7raV = I(pv,pv[v),pi = 1, [,” quotient of the induced

If

namely

I ~ ( p ~ 1pu,Ev12). 2, This

= 1. = 1, then J ( u ~ ~ 2 7 r ~ V r v ~ 1is~ the 27r~v)

is the A-lift of the packet consisting of

10. Global Packets

171

If Ev = 1 then 7rzv is induced with central character Ev = 1, thus 7r2, = I(pv,p;'). We may assume that p: # 1 as the case of pi = 1 is dealt since 7r2, is with in the previous paragraph, and that 1 > 1pvl-2 > ~vv~-', a component of a cuspidal 7r2. Then J ( Y , ~ / ~ T~ i~" ,~,7 r 2 , ) is the quotient I ~ ( p , 1 2 , p ; ' 1 2 ) of I ~ ( ~ , 1 / ~ 7 r 2 ~ , v , ~ / It ~ 7isr 2 the ~ ) A-lift . of pG2 >a p,12, which is irreducible since p i z # 1,,':v v:2 (Proposition V.2.l(b)). This p i 2 x pv12 is the quotient of

= v,

x p i 2 x pvuv-'/2 = u, x Z p I 2 ( p v ,p,').

So we write L(v,Ev,v~1/27r2v) for p i 2 x pv12 (when Ev = 1 and 7r2, = l ( p v ,p ; ' ) ) ; it A-lifts to ~ ( u , 1 / ~ 7 r 2vi1/27r2v). ,, In summary, the quasi-packet of L(vvcv,~ i " ~ 7 r 2 which ~ ) A-lifts to

being a component of 7r2 as in the proposition, consists of one irreducible, unless 7rv = I(pv,p u t v ) , p: = E: = 1, when it consists of L, and X, (this includes all IJ where Ev # 1 and 7r2, is not cuspidal). Now to prove the proposition we apply the trace identity where the only entry on the side of G, having fixed almost all components, is the residual representation J ( u ' / ~ Tv~-,' / ~ T ~Generalized ). linear independence of 2) characters on H(F,) establishes the claim. Note that L ( ~ ( , v - l / ~ 7 r = @,L, is residual ([Kim], Theorem 7.2), but any other irreducible in the packet is cuspidal. 0

7rav

PROOF OF THEOREM. Since: 1. The one dimensional representations of H(A) A-lift to the one dimensional representations of G(A); and 2. The discrete spectrum quasi-packet { L ( v J ~, - ~ / ~ 7 r 2of) H(A) } A-lifts to the residual representation J ( ~ l / ~ 7 rv P21,j 2 7 r 2 ) (for every cuspidal representation 7r2 of GL(2, A) with central character 5 # 1 = E2, and E7r2 = 7 r 2 ) ; and 3. The only other noncuspidal discrete spectrum self contragredient representations of G(A) are the residual J(u1/27r2,v-1/27r2)where 7r2 is a cuspidal representation of PGL(2, A), in which case this J is the XI-lift of

172


1 2 x 7r2 from C(A); we may assume that I’ ranges only over cuspidal self contragredient representations of PGL(4, A). We pass to the form of the identity where almost all components are fixed. If there is a global discrete spectrum T H with the prescribed local ) nonnegative, components then the sum I;I is nonzero, since the m ( 7 r ~are by generalized linear independence of characters. Hence I’ # 0 and it consists of a single cuspidal 7r by rigidity theorem for cuspidal representations of G(A). Each component 7rw of the self contragredient generic T is a A-lift of a packet {TH,,} of representations of H(F,) (by our local results), hence our identity reads

Here V is a finite set, its complement consists of finite places where unramified 7rkwand 7r,” with A ( 7 r i w ) = 7r,” are fixed, and the sum ranges over the T H whose component at v $! V is T H ~ . Generalized linear independence of characters then implies that the right side of our identity has the same form as the left, hence the multiplicity m ( 7 r ~is) 1 and the X H which occur are precisely the members of the packet I3 1 & , { 7 r ~ ~ } , where {7rk,} = 7rkwfor all v outside V . Note that since we work with test functions which have at least three elliptic components, the only T H and 7r which we see in our identity have three such components. The unconditional statement would follow once the unconditional identity of the trace formulae is established. As explained in 1G of the Introduction, “three” elliptic components can be reduced to “two”, and even to “one real place”, with available technology. 10.11 PROPOSITION. (1) E v e y unstable packet Xo(7rl x7r2) of the group PGSp(2, A) , where 7r1 , 7r2 are cuspidal representations of PGL(2, A), contains precisely one generic representation. It is the only representation in the packet which is generic at all places. E v e y packet contains at most one generic representation. (2) I n a tempered packet {K&,T,} of PGSp(2,F), F local, 7r& i s generic and xITH is not. (3) I n a stable packet of PGSp(2,A) which lifts to a cuspidal representation

10. Global Packets

173

of PGL(4, A) there is precisely one representation which is generic at each place.

PROOF,(1) If 7rk and 7rL are generic, cuspidal, and lift to the same generic induced representation 1(7r1,7r2) of PGL(4, A), namely they are in the same packet, then they are equivalent by [GRS]. The second claim follows from this and Lemma 7.3. The third claim follows from the rigidity theorem for generic representations of GSp(2), see [So], Theorem 1.5. (2) Let F be a global field such that at an odd number of places, say w1,.. . ,us,its completion is our local field. Construct cuspidal representations 7r1, 7r2 of PGL(2,A) such that the set of places w where both 7rlW and n-zw are square integrable is precisely v1,. . . ,w g , and such that X0(7rlw, x 7 r z W , ) is our local packet, now denoted {7r$v,7rkw}, w = q. In Xo(7rl x 7 r 2 ) there is a unique cuspidal generic representation 7rg, by [GRS]. By our multiplicity formula the cuspidal members of Xo(7rl x 7 r 2 ) are those which have an even number of components 7rkv. Hence 7r; has a component T & ~ so , nt), must be generic. If both { ~ & ~ , 7 r i were ~ } generic, Lemma 7.3 would imply that the packet of the cuspidal generic 7rg contains more than one generic cuspidal representation (in fact, 2’ of them), contradicting [GRS]. (3) Every irreducible in such a packet is in the discrete spectrum. The packet is the product of local packets. When the local packet consists of a single representation, it is generic. If the local packet has the form {7r$w,~;w}, then 7rLv is generic but 7rGW is not. Hence the packet has 0 precisely one irreducible which is everywhere locally generic.

REMARK. Is the representation T H constructed in ( 3 ) above generic? By [GRS],it is, provided L ( S ,n-, R 2 ,s) has a pole at s = 1, where X ( 7 r ~ )= n-. We do not know to rule out at present the possibility that there is a packet { T H } containing no generic member and /\-lifting to a cuspidal 7 r , necessarily with L ( S ,7 r , A2, s) finite at s = 1. Note that the six-dimensional representation A2 of the dual group Sp(2, C)of PGSp(2) is the direct sum of the irreducible five-dimensional representation ids : Sp(2,C ) + SO(5, C ) (cf. Lemma 7 . 0 ) and the trivial representation (see [FH], Section 16.2, p. 245, for a formulation in terms of the Lie algebra of Sp(2,C)). Hence L( S, n-H , A2, s) = L( S, n,A2, s) has a pole at s = 1 provided L( S, 7 r ~ids, , s) is not zero at s = 1. This is guaranteed by [Shl], Theorem 5.1 (as noted after Lemma 7 . 0 ) when T H is generic. Thus the locally generic 7 - r ~of ( 3 ) is

174


generic iff L ( S ,7rl A’, s ) has a pole at s = 1, iff L ( S ,7 at s = 1. An alternative approach is to consider

ids, s) is not zero

r ~ ,

L ( S ,7r @ 7rl s ) = L(S,7 r , A2, s ) L ( S ,T , Sym2,s), which has a simple pole at s = 1since 7r N ir. If L ( S ,7r,A2, s) does not have a pole at s = 1, L ( S ,7 r , Sym2,s) has. One expects an analogue of [GRS] to show that 7r is then a XI-lift from SO(4,A). We shall then conclude that 7r is not a A-lift from PGSp(2, A).

11. Representations of PGSp(2,a) The parametrization of the irreducible representations of the real symplectic group PGSp(2, R) is analogous to the padic case, but there are some differences. We review the listing next, starting with the case of GL(2, IR). In particular we determine the cohomological representations, those which have Lie algebra (g,K)-cohomology,with view for further applications.

lla. Representations of SL(2, R) Packets of representations of a real group G are parametrized by maps of the Weil group WR to the L-group LG.Recall that

is 1 + Wc -+WR 4 Gal(C/R)

4

1

an extension of Gal(@/R) by We = C X . It can also be viewed as the normalizer C x U C x j of C X in W X , where W = R(1, i, j , Ic) is the Hamilton quaternions. The norm on W defines a norm on WR by restriction ([D3], [Tt]). The discrete series (packets of) representations of G are parametrized by the homomorphisms 4 : Wp. --+ G x WRwhose projection to WRis the identity and to the connected component G is bounded, and such that C$Z(G)/Z(G)is finite. Here C, is the centralizer Z & ( $ ( W , ) ) in G of the image of 4.

11b. Cohomological Representations

175

When G = GL(2,R) we have G = GL(2,R), and these maps are ( k 2 l), defined by

Since u2 = -1

H

(

(-l)k

)x

02, L

(jhk

must be (-l)k.Then det & ( a ) =

( - l ) k + ' ,and so k must be an odd integer (= 1 , 3 , 5 , .. . ) to get a discrete

series (packet of) representation of PGL(2,R). In fact 7r1 is the lowest discrete series representation, and (jho parametrizes the so called limit of discrete series representations; it is tempered. x CT define discrete series representations Even k 2 2 and CT c-)

( y i)

of GL(2, R) with the quadratic nontrivial central character sgn. Packets for GL(2,R) and PGL(2, R) consist of a single discrete series irreducible representation " k . Note that 7rk @ s g n N 7 r k . Here sgn : GL(2,R) + {*I}, sgn(g) = 1 if detg > 0, = -1 if detg < 0. The 7rk ( k > 0) have the same central and infinitesimal character as the kth dimensional nonunitarizable representation Symk-'

@'

z

I detgl-(k-1)/2 Sym"'

@'

into SL(k, C)* = {g E GL(2,C); det g E {fl}}.We have det Sym"'(g)

= det gk ( k - 1 ) / 2 I

and the normalizing factor is I det Sym"'

Then Symk-'

(; :)

In fact both 7rk and Symi-'C2 are constituents of the normalizedly induced representation I ( v k / 2sgnk-1v-k/2) , whose infinitesimal character is -$), where a basis for the lattice of characters of the diagonal torus in SL(2) is taken to be (1,-1).

(k,

l l b . Cohomological Representations An irreducible admissible representation T of H(A) which has nonzero Lie algebra cohomology H i j ( g , K ; 7r 8 V ) for some coefficients (finite dimensional representation) V is called here cohomological. Discrete series

176


representations are cohomological. The non discrete series representations which are cohomological are listed in [VZ]. They are nontempered. We proceed to list them here in our case of PGSp(2, R). We are interested in the (g, K)-cohomology H'j(sp(2, R),U(4); 7r I8 V), so we need to compute Hi3(sp(2,R),SU(4);7r@ V) and observe that U(4)/SU(4) acts trivially on the nonzero Hij, which are C. If Hij(7r 8 V) # 0 then ([BW]) the infinitesimal character ([Kn]) of 7r is equal to the sum of the highest weight ([FH]) of the self contragredient (in our case) V, and half the sum of the positive roots, 6. With the usual basis ( l , O ) , ( 0 , l ) on X * ( T ; ) , the positive roots are (1,-l), (0,2), (1,l), (2,O). Then 6 = Ca,OLY is ( 2 , l ) . Here T; denotes the diagonal subgroup {diag(z, y, l/y, 1/x)} of the algebraic group Sp(2). Its lattice X * ( T ; ) of rational characters consists of

4

( a ,b) : diag(x, y, l/y,

1/x) H xayb

( a ,b E

Z).

The irreducible finite dimensional representations V a , b of Sp(2) are parametrized by the highest weight ( a , b ) with a > b 0 ([FH]).The central character of Va,b is H j(sp(2,R),SU(4); 7r 8 & , b ) for some a 2 b 2 0 (the same results hold with Va,b replaced by V&), we first list the discrete series representations. Packets of discrete series representations of the group H = PGSp(2, R) are parametrized by maps $ of Ww to L H = H x W w which are admissible (pr204 = id) and whose projection to I? is bounded and C @ Z ( k ) / Z ( H ) is finite. Here C, is Zfi($(Ww)).They are parametrized $ = (bkl,kz by a pair ( I c l , Ic2) of integers with odd Icl > Ic2 > 0. The homomorphism

k2

0

x u

> 0)

(evenkl>k2>0),

factorizes via (LCo -+) L H = Sp(2,C) x WR precisely when ki are odd. When the ki are even it factorizes via LC = SO(4,C) x W,. When the ki are odd it parametrizes a packet r:::,,} of discrete series repis generic and rho'is holomorphic resentations of PGSp(2, R). Here rWh and antiholomorphic. Their restrictions to H o are reducible, consisting of r$!' and 7rHW.h o Int(L), and rk: o Int(L), L = diag(l,l, -1, -l), and rWh 8 sgn = 7rwh, rho' 8 sgn = rho'. To compute the infinitesimal character of r;l,kz we note that

{rzi2,

rki

r k

c I(u'/', sgnk"u-k'2)

(e.g. by [JL], Lemma 15.7 and Theorem 15.11) on GL(2,R). Via L C 4 ~ L H induced I ( u k 1 l 2u, P k 1 I 2x) I ( U ' ~ /u~- ,k z / 2 )(in our case the ki are odd) lifts to the induced IH(uk1/2,yk2/2) = J k l + k 2 ) / 2

x u(kl-k2)/2 >a

u-k2/2,

whose constituents (e.g. r ; 1 , k*2= , Wh, hol) have infinitesimal character

Here as k2 2 1 and k l > k2 and kl - k2 is even. For these a kl = a + b + 3 , k z = a - b + l , wehave

2 b 2 0, thus

Hij(sp(2, R), SU(4); 7rk"iiZ 8 K,b) = CC

if ( i , A = (2, I), (17%

Hij(sp(2,R),SU(4);r~:tk28 V & ) = CC

if

( z , j )= (3,0), (0,3).

Here kl > k2 > 0 and kl, k2 are odd. In particular, the discrete series representations of PGSp(2, R) are endoscopic.

V. Lzfting from PGSp(2) to PGL(4)

178

l l c . Nontempered Representations

Quasi-packets including nontempered representations are parametrized by homomorphisms $ : W, x SL(2,R) -+ ’ H and 411 : WR-+ ‘ H (see [A2]) defined by The norm 11.11 : WR t R x is defined by llzll = ZZ and I(a(1= 1. Then &,(a) = $(a,I ) and &,(z) = $(z,diag(r,r-’)) if z = rei8, T > 0. For example, $ : W n t ~ s L ( 2 , C-)+ SL(2,@), $lW,:

z d + + t ( - l ) j , qlSL(2,C) = i d ,

gives

(2;

411(c) = 5(-1)12 x hbk) = T o l ) x 2 , parametrizing the one dimensional representation k2 = a - b 1 > 0 are odd. Applications of the classification above in the theory of Shimura varieties and their cohomology with arbitrary coefficients are discussed in [F7]. 7rTkzI 7r,"pfk,.

+ +

+

VI. FUNDAMENTAL LEMMA The following is a computation of the orbital integrals for GL(2), SL(2), and our GSp(2), for the characteristic function 1~ of K in G, leading to a proof of the fundamental lemma for (PGSp(2), PGL(2)xPGL(2)), due to J.G.M. Mars (letter to me, 1997).

1. Case of SL(2) 1. Let E I F be a (separable) quadratic extension of nonarchimedean local fields. Denote by 0~ and 0 their rings of integers. Let x = T F be a generator of the maxima1 ideal in 0. Then e f = 2 where e is the degree of ramification of E over F . Let V = El considered as a 2-dimensional vector space over F . Multiplication in E gives an embedding E c EndR(V) and E X c GL(V). The ring of integers 0~ is a lattice (free 0-module of maximal rank, namely which spans V over F ) in V and K = Stab(0E) is a maximal compact subgroup of GL( V ) . Let A be a lattice in V. Then R = R ( A ) = {x E ElsA c A} is an order. The orders in E are R(m)= 0 x m O ~m, 2 0 of F . This is well-known and easy to check. The quotient R ( m ) / R ( m +1) is a 1-dimensional vector space over 0 / n . If R ( A ) = R ( m ) ,then A = zR(m) for some z E E X . Choose a basis 1, w of E such that 0~ = 0 Ow. Define d, E It follows GL(V) by d,(l) = 1, dm(w)= xmw. Then R(m) = d,OE. immediately that GL(V) = U E X d m K ,or, in coordinates with respect

+

+

m>O

to 1, w: GL(2, F ) = U T m20

with T =

{(

(i &) GL(2, 0 ) ,

a , b E F , not both = 0 , where w 2 = a+pw,a ,

p E 0. 182

1. Case of SL(2)

183

2. Put G = GL(V), K = Stab(0E). Choose the Haar measure dg on G such that dg = 1, and d t on EXsuch that d t = 1. Choose y E E X,

s

S

K

O E

y @ F X .Let 1~ be the characteristic function of K in G. Then

Now E X \ G / K is the set of EX-orbitson the set of all lattices in E . Representatives are the lattices R(m), m 0. So our sum is

>

rn2O,yER(m)

vOl(c3;) = vol(R(m)X)

c rn?O,yER(m)

(og:R(m)X). X

Note that (0; : R ( V L ) = ~ )1 if m = 0, = qrn+l-f& q-1 if m > 0, P u t M = max{m/y E R(m)X}. Then the integral equals

s

+

= 0). If y = a bw E O g , then M = o ~ ( b )the , (If y @ O g , then order-valuation at b. 3. Let G = SL(V), K = Stab(OE)nG,El = E XnG. Choose the Haar measure dg on G such that S d g = 1, and d t on El such that d t = 1.

s

Let y E E l , y

# 51. Then

K

E'

is the number of lattices in the G-orbit of OE fixed by y. Let A be a lattice in E . If R ( A ) = O E ,then A E G.OE @ A = O E . And T O E = 0~ if y fixes A. If R ( A ) = R(m) with m > 0, then A = zR(m) E G . O E @ N E / F ( z ) n mE O x @ f D E ( Z ) = -m and y A = A w y E R(m)X. Suppose e = 1. Then m must be even and rn

A =w-TuR(m),

u E OgmodR(m)X.

If y E R ( m ) X ,this gives (0; : R ( m ) X )= qm-'(q

+ 1 ) lattices,

VI. Fundamental Lemma

184

Suppose e = 2. Then A = r,"uR(m), u E 0;mod R(m)'. If y E R(m)' this gives ( U g : R(m)') = qm lattices. Put N = max{mly E R(m)', m = O(f)}. Then the integral equals

N'f1-l

For K = Stab(R(1))n G one find 9 q - l with N' defined as N , but with m f l(f). 4. Notations as in 3. Choose r = N E / F ( T Eif) e = 2. The description of the lattices in G . OE above gives the following decomposition for SL(2, F ) . Choose a set A , of representatives for NE/Fog/NE/FR(m)' and for each E E A , choose b, such that NE/p(b,) = E . For m = 0 we may take A0 = {l},bl = 1.

SL(2,F)=

u

u E ' b ; l r i m ( A : ) ( l0 7 rO m ) K

m/O € € A ,

if e = 2.

REMARK.If e = 1, m > 0, then

(two elements, when 121 = 1). If 121 = 1 and e = 2, then

for all m.

2. Case of GSp(2) 2a. Preliminaries 1. Let V be a symplectic vector space defined over a field F of characteristic # 2. We have G = Sp(V) C GL(V) c End(V) = A . Let y be a regular

2a. Preliminaries

185

semisimple element of G ( F ) , T the centralizer of y in G, C the conjugacy class of y in G. If L denotes the centralizer of y in A, we have L ( F ) = Li, a direct product of separable field extensions of F . The space V ( F ) is isomorphic to L ( F ) as L(F)-module and V ( F ) = $ V , ( F ) , where V , ( F )is a 1-dimensional vector space over Li. We denote the symplectic form on V by (2,y) and define the involution L of A ( F ) by

n

From 'y = y-' it follows that L stabilizes L ( F ) . The restriction LT of L to L ( F ) is a F-automorphism of L ( F ) of order 2. It may interchange two components Li and Lj ( a # j) and it can leave a component Li fixed. If T is F-anisotropic we have a ( L i ) = Li for all 2. Note that T ( F ) is the set of u E L ( F ) Xsuch that U L T ( U )= 1. If a ( L i ) = Li, then V , l & for all j # i.

{G(F)-orbits in C ( F ) }

H

G(F)\{h E A(F)'lhyh-'

E C}/L(F)'

{u E L ( F ) X l L T ( U ) = u}/{ua(u)luE L ( q X } 2. If we take G = GSp(V) instead of Sp(V), we have 'yy E F X and T ( F ) is the set of u E L ( F ) Xsuch that U L T ( U )E F X .Now

{G(F)-orbits in C ( F ) }

-

G(F)\{h E A(F)'lhyh-l

E C}/L(F)'

{u E L ( F ) X l a ( u= ) u)/FX{uLT(u)Iu E L(F)X} In this case consider T such that T / Z is F-anisotropic (2= center of G and of A X ) .


186

Notation:

L' = {u E L(F)la(u)= u}, N u = u a ( u ) if u E L ( F ) X .

3. Assume F is a nonarchimedean local field. If A is a lattice in V ( F ) , the dual lattice is

A* = {x E V(F)I(x,y) E 0 for all y

E

A}

= Homo(A,O).

Properties:

(.A)* = L ~ - l A *

(uE GL(V (F ))),

in particular (cA)* = c-lA* if c E FXand (gX)* = gA* if g E Sp(V(F)). Further, A** = A. The lattices which are equal (resp. proportional by a factor in F X )to their dual form one orbit of Sp(V(F)) (resp. GSp(V(F))). We want to compute the following numbers. Orbital integral for Sp(V(F)): Card{AIA* = A , y A = A}. Stable orbital integral for Sp(V(F)): Card{AlA* = vA, yA = A}.

c

V E L t XI N L ( F ) / L / L ( F ) x

Orbital integral for GSp(V(F)): Card{AlA*

N

A , y A = A } / F X=

Card{AlA* = crA,yA = A}. a E F X/ F x 2 0 X

Stable orbital integral for GSp(V(F)):

c

c

Card{AlA* = avA, y h = A}

v E L ' ~ I FN ~ L(F)X aEFXIFX20X

So the stable orbital integrals for Sp(V(F)) and GSp(V(F)) differ by a factor, which is a power of 2, when y E Sp(V(F)). 4. Let L / F be a quadratic extension of nonarchimedean local fields. ( n 2 0 ) . We can find w E L The orders of L are 0 ~ ( n=) OF + n g 0 ~

+

such that 0 ~ ( n=)OF O F T ~ Zfor U all n 2 0. Any lattice in L is of the form z O L ( n ) , z E L x , n 2 0. Let a symplectic form on the F-vector space L be given.

2a. Preliminaries

187

If (1,w) E 0:, the lattice dual to zOL(n) is T’n,”o~(n)

(0,” : C3L(n)X)= 1 ( n = o),

qn-l(q

+ 1)

( n > o), if L / F is unramified,

= qn

Here q

=

(n2

o>,

if L / F is ramified.

number of elements of the residual field of F .

5 . Let V = V1 @

be a direct sum of two vector spaces over a nonarchimedean local field F . Let A be a lattice in V. Put Mt = A n V, and N, = pri(A). Then Mi and Ni are lattices in V,, and Mi c Ni. The set V2

{(vi

+ Mi,vz + M2)lvi + vz E A}

is the graph of an isomorphism between N1/M1 and N2/M2. The lattices in V correspond bijectively to the data: M I c N1, lattices in V1; M2 c N2, lattices in V2; NI/MI 4 N2/M2, an isomorphism of (finite) 0-modules. Assume a symplectic form is given on V and V = Vl@V2is an orthogonal direct sum. If the lattice A corresponds to the data

then the data of the dual lattice A* are:

One may identify Mr/N,* with Homo(Ni/Mi, F I 0 ) using (v,v’), v E Ni , v’ E M:. Then cp* is defined using this identification. 6. In the notation of section 1 assume that L ( F ) is a field. For brevity write L for this field. Let L’ be the field of fixed points of 0,so [L:L’]= 2. We identify V ( F )with L ( F ) = L and have then ( x , y ) = trL/F (aa(x)y), with some a E L x such that a ( a ) = -a. Put (x,y)’ = trL/l/(aa(z)y). This is a symplectic form on L over L’. We have (x,y) = trL,/F((x, y)’) and ( z z , ~=) (x,zy)if z E L’. Assume now that F is local, nonarchimedean. If M is an QLI-lattice in L, then M*= {x E LI(x,y) E 0 for all y E M }


188

is also an 0L)-lattice. The dual

-

M = {x E LI(x,y)’ E

c3L1

for all y E M }

E

of M as an 0Lt-lattice is related to M * by the formula = VLl/FM*, where 2)Ll/F is the different of L‘IF. We have = det(g)-lgM if g E GLtt(L), in particular u M = a ( u ) - l E if u E L X . In the remainder of this section we assume dim V = 4. The nonidentical automorphism of L‘IF is denoted by T or by z H 7.

-

5

-

Let A be an 0-lattice in L. Put M = O p A , N = 0 ~ i A * . Then M * = {x E L I ~ L , c z A*} is the largest OLI-lattice contained in A* and N = DL~/F.largest OLj-lattice contained in A. We have M 3 A 3 N . If ai E 0 L l , xi E A, then

Indeed, u E N @ (u,y)’ E 0 L I for all y E A*. If C a i x i E N , then Cai(xiy)’ E 0 L l for all y E A*. Since

it follows that C ~ ( a i ) ( xy)’ i,E 0 ~ ’ . So we can define a homomorphism

whenever ai E 0 L 1 , xi E A. The homomorphism cp is 0Lr-semilinear and cp2 = id. The set A / N is the set of fixed points of cp. Indeed, if C.(ui)xi C a i x i E N , then

-

hence ( C a i x i , y ) E 0 for all y E A*, i.e. C a i x i E A. Conversely, let M 3 N be two OLi-lattices in L and cp : M / N + M / N an 0 LI-semilinear homomorphism.

2b. L(F) is a Product

189

Necessary conditions for ( M ,N, cp) to correspond to a lattice A are:

cp = id

on D,:lFN/N.

These conditions are also sufficient when L'IF is unramified (in which case the only condition is cp2 = id) and when L'IF is tamely ramified. If A exists, it is unique, since A/N is the set of fixed points of cp. The lattice can be identified with Homo,, ( M ,OL,) using (m,h)' (this gives M

-

6= M , m

H

-m). If M 3 N , then

--

Homo=,( M I N ,L'IOL,) = N / M If cp : M / N + M/N is OLI-semilinear, then f H I-fcp-is a semilinear endomorphism (p of Homo,, (MIN, L'IOp), which on N / M is given by

(cp(m),fi)' = ~ ( ( r n , $ ( E ) ) ' modOLt. ) If A

H

( M ,N, cp) then A*

--

H

( N ,M , -@).

7. In the following computations F is a nonarchimedean local field. Notations are as in section 1, dimV = 4. We have either L ( F ) is a field or L ( F ) is the product of two quadratic fields. 2b. L ( F ) is a Product 1. Assume L(F) = L1 x L2, [Li:F] = 2. Then V(F) = V1 @ VZ, V , a 1-dimensional vector space over Li, VlIV2. We identify V , with Li. Then

We compute the number of lattices A in V ( F ) which satisfy A* = uA and y A = A, for a given regular element y of T ( F ) and a set of representatives u Of FX/NLIIFLr X FX/NL,,FL,X. By section 2a.5 the lattice A is given by lattices Mi c Ni in Li (i = 1,2) and an isomorphism cp : N l / M l + Nz/M2. Let y = (t1,tZ) and v = (ul,24).


190

The condition A* = vA means that N, = V,-~M: (z = 1 , 2 ) and v~pul-l = -(p*)-'. Then rA = A is equivalent to taMa= Ma (i = 1 , 2 ) and t2(ptc1= p. Put M, = ~ , O L(m,) , with z, E L,X,m, 2 0. Choose w, E L, such that OL, = O + O w , . On each V, = L, there is only one symplectic form, up to a factor from F X ,and in order to compute our four numbers we may assume that ( 1 , w l ) = (1,wa) = 1. Then M: = Z ; l ~ m t O ~(m,) , and

Moreover, vFIM:/M1 and uT1M;/M2 have to be isomorphic. This means ? r ~be , ) independent of i. So put that ~ ( v i N ~ , / ~ ( z i )must

Then

N / M z = vLIM:/Ma21 OL,(mz)/~mC3Lt(rna). With respect to the bases 1,?rmaw,of O ~ , ( r n , )the , isomorphism p is given by a matrix p E GL(2, O / r m O ) satisfying (from v2pv;1 = - ( p * ) - l )

The conditions with respect to y are: ti E OLi(mi), The number to compute is the sum over m 2 0 of

c

c

t2cpt;l

=~

p .

Card{p E GL(2, O/?rmO)Io}

where 0 stands for t2p = p t l , det(p) = ZL and

Here Card is 1 when m = 0. 2. We now assume that 121 = 1 in F . Then wi can be so chosen that w: = ai E 0. Put ti = ai bi?rmiwi with ai,bi E 0.

+

2b. L ( F ) is a Product

Let cp =

(zi :)

191

E GL(2, C3/nmO). The matrix corresponding to

ti is

We have a: - aibpnarni= 1 (i = 1,2). Assume m > 0. The conditions on cp are:

where u is an element of O x . This system cannot be solvable unless a1 implies tr(t2) = tr(t1). Assume this. Then (1)

b2Xl E b 1 X q

(2)

albl?r2m1x1

(3) biz:! E (

u

a2b2R2m2x4

Z

~

~

(4)

b 2 ~ 2E ~

l

b

(5)

21x4 - ~ 2 x 3

u

~

= a2(7rrn),since t 2 = c p t l c p - I

T l

~

~

~

~

~

~

X ~ modnm 1

~

3

This system is unsolvable when a ( b 2 ) > ~ ( b l and ) m > ~ ( b l )as, (1) and (3) would imply that x4 f x2 O(a);and also when a(b1) > n(b2) and m > b ( b 2 ) , as (1) and (4) would imply that x1 = 2 2 z O(n).It remains to consider: m 5 a ( b i ) (i = 1 , 2 ) or m > b ( b 1 ) = a ( b 2 ) . Suppose m I v(b1) and m I v(b2). Then 21x4 -x2x3 = u(nrn)has q3m-2(q2 - 1) solutions. Suppose m > o ( b 1 ) = a ( b 2 ) . Put k = a ( b 1 ) = o(b2). Put ci = aibgrami (i = 1,2). Then (1)-(4) imply that

=

for all i, so we must necessarily have b l c l _= b2cZ mod nrn+k. This is equivalent to U: E a: mod?rrn+kand implies that either ~ ( c i 2) m for i = 1 , 2


192

or a(c1) = a(c2) < m. From (3) and (4) it follows that 2 2 = O(.rr), unless ~ ( c I= ) b(c2) = Ic, i.e. b(ai) = 0, mi = 0 (i = 1,2). Assume we are not in this case. Then a(ci) > k for i = 1,2. Also 2 2 O(?r), hence 2 1 E O x and (5) gives 2 4 = 2 y 1 2 2 2 3 z,lumod?r"(*). (3) and (4)give 1 22 = b, ~ 2 x 3 mod?rmPk(**). (2) is a consequence of (1). After substitution of (*) and (**) the congruence (1) reads

--

+

221 = -

~ T ' C ~+ X ;blb,lurnodnm-k.

Here b;'c2 = O(.lr). We find 2qm+2ksolutions when blb,'u is a square in F , and otherwise no solution. Now suppose that a(ai) = 0, mi = 0 (i = 1 , 2 ) . Then L1 = L2 = the unramified quadratic extension of F . Take a1 = a2 = a . We have b: == bg mod?rmfk. Now (1) and (2) are equivalent, (3) and (4) are equivalent. From (1), (3) , (5) one deduces that 2: - ax: = bl b;'u mod.rrm-k. This 1) solutions modulo .rrmPk. For each solution congruence has qm-"'(q 2 1 E O x or 2 3 E O x . If z1 E O x we have

+

22

c;t'b2by1z3

( x ~ - ~ ) ,24

--

= ~ r ~ ( 2 2 +2 3U ) (rm).

1 If z3 E O x we have 2 4 bzb,lzl (nmPk), 2 2 = 2; (21x4 - u)(+). So there are qm+2k-1 ( q 1) solutions for the system in this case.

+

3. Recall that O L = ~ O+Ow,, wp = a, and 121 = 1. Let t, = a,+ bzwzr a: - azb: = 1 (z = 1,2). As t = ( t 1 , t ~is) supposed to be regular, we have bl # 0, b2 # 0, and in case L1 = L2, a1 # a2. Let us be given: v1,v2 E FX ( v , m o d N ~ , / ~ L ; ) ; m 2 0; rnl,m2 2 0 such that t, E OL,(m,),i.e. m, 5 a(b,); z1 E L r , Z ~ L,EX ( ~ , m o d O ~ ~ ( m w ,i )t h~ f), a ~ ~ ( z , ) = m - m , - ~ ( v , ) . ) P 1 .U E Put U = - V ~ V ~ 1 ? r m 2 - m 1 N ~ 2 / ~ ( Z 2 ) N ~ I / ~ ( Z 1Then By section 2 we have that

Card{cp E GL(2,0/?rmO)I is given by

t2cp

= cptl,

ox.

det(cp) = u}

2b. L ( F ) is a Product

193

m = 0; if m > 0, m m, 5 a(b,)(i = 1 , 2 ) , a1 = a z ( n ) ; if a(b,) = m , + k , 0 5 k < m, and ezther a(a,) 2 m , k 2 m ( z = 1 , 2 ) , a1 z a2(n), and a, where we put a for -u2u1-'b2 by N ( Z Z ) N( z i ) - l E F 2 , or a1 = 122, ml = m2, k < a(&,) f 2 m , k < m, a, and a1 = a2(nm+'); qm+2k- 1 (q 1) if a1 = a2 E O x ) ml = m2 = 0, 0 5 o(61) = ~ ( 6 2 ) = k < m, and al = a 2 ( n m f k ) . It is zero in all other cases. We are computing

if

1 q3m-2(q2 - 1 ) 2p+2k

+

+

+

+

+

c

cc mlo

V 1 ~ 2

c

Card{cp).

O<m,5u(bz) Z:EO:~/O~,(~,)X m,=m+u(v,) mod f z

We put z, = ~a ' Ln, ~ - ~ ' - ~ (The " ' )condition '~'.

N L, F (4) N LI F~ E - v ~ T- O,( V z ) ~1-1

a

becomes

(4

)-l

rl O ( ~ I ban2 ) -O(bz)b-llrU(h) 1 (lr4)k+mOx2) 1

where NL,,F(?TL,) = rp. Notice that Card{cp} is independent of z i , zh) except for the cases where the condition a plays a role. In those cases one has m, > 0 when L , / F is unramified, so that O L , ( ~ ,c) O ~ i $ . This is used in the following. Our sum is the sum of the following sums.

11)

C

ele2

q3"-'(q2

-

1)AB

rn>O,mi>O m+mila(bi)

if

a1 E az(n); here

if a1

E a z ( n ) ; here

A

= (02, : C ? ~ , ( r n l ) ~ ) ,

A = (O;,

:

B = (02, : O L 2 ( m 2 ) x ) ;

O ~ , ( a ( b l-) k)'),

( O i 2: O ~ ~ ( b ( b-2 k) ) ' ) .


194

a2 and b ( h ) = b ( b 2 ) , put A = (OEl : o L , ( b ( b l ) - I C ) ~ ) ~ :

If

01 =

If

a1 = a2

E O x and b(b1) = b ( b 2 ) :

c

V)

qm+2~(bi )-1

(Q

+ 1).

o ( b l ) < m l o ( a l- a z ) - a ( b i )

Put Mi = ~ ( b i )M, = max(M1, M 2 ) , N = min(M1, M2). The sub-sums are:

if

a1

--

a2(7r).

' qM+N-l

(Q

2qM+Y4

III) 2qM+yq

+ 1)2 ( 4N - l ) ( q N + l

+ +

l ) ( q- 1 ) - 2 if e l = e2 = 1;

-

-

1)(qN+1 - l ) ( q - 1 ) - 2

if el = 1,e2 = 2,M1 5 M ~ ; l ) ( q N f l - 1 ) 2 ( q - 1)-2

if e l = l , e 2 = 2,M1 > M2 ; 4 q ~ + ~ + 1 ( 9~ + 12

) (9-

if el = e2 = 2.

2c. L ( F ) is a Quartic Extension

if L1 = Lz is unramified and a1 E az(7r). The formulas IV) and V) hold even when b(b1) have then b ( a l - u2) = 2N b(cr1).

+

# ~ ( b z ) ,because

195

we


1. Assume L ( F ) = L is a field. We identify V ( F ) with L(F). A quadratic subfield L' of L is given and T ( F ) = {t E L x INL/L,(t) = 1). We compute the number of lattices A in L which satisfy A* = v A and tA = A, for a given regular element of T(F) and a set of representatives v of L " / N L / L , L ~ . That t is regular means that F ( t ) = L. We use the L'-bilinear alternating form (z, y)' = trL/L/ (aa(x)y) introduced in section 6 of Case of SL(2) (we shall choose a later). Assume that L'IF is unramified or tamely ramified. A lattice A is given by OL,-lattices M 3 N and a map cp : M / N 4M / N satisfying certain-conditions (see section 6). Moreover, A* = U A is equivalent to M = vP1N and -(p = vcpu-'. The identity tA = A is equivalent to t M = M, Put N

= uOL(n),u E

tN =N,

tcpt-' = cp.

L x , n 2 0. Assuming that min b ~ / ( ( z , y ) ' = ) 0 X,YEOL

(cf. section 4 of Case of SL(2); we come back to this later) we have

M = v-'Z = v-'~(~)-~TLPOL(~).

196


Put

m = DL/(V)

+ ~ L / L / D L ( +U )n 2 0.

Then

M/N

2~

OL(n)/rEOL(n)

and we consider 'p as a semilinear endomorphism of this OLr-module. We choose rp = T when L ' / F is unramified, x i , E F when L ' / F is tamely ramified. In any case 'p must satisfy 'p2 = id. When L'/F is tamely ramified ( D L , / F= ~ L / O L /there ) , are more conditions, namely: 1) N c DL 'IF M, i.e. m 2 1; 2) 'p = id mod r p ; 3) p = id on rI;I",-1C3L(n)/nzOL(n). When 2 ) holds, condition 3) means that m is odd. The condition -@ = v'pv-' translates to:

*

-c('p(s), y)'

= (2,' p ( ~ ) ) ' m o d r Y ? ~ O ~ t

for all s,y E OL(n),

where c = VNL/L,(U)/T NL/L'(U).

Write V N L / L ' ( U ) = c 1 rm-2 p

,

c1 E O,x/.

Then c = cl/i?l when L'IF is unramified or n is odd and c = -cl/Zl, when L ' / F is ramified and n is even. Now * is:

(+ when L'IF is ramified and n even, - otherwise). Choose W L E L such that OL = OL, O L ~ W LThen . O L ( n )= OLt + O L ~ & W LIn. (2,~)' = trl/L/(aa(z)y) the element a is such that .(a) = --a. We may take a = al(wL - O ( W L ) ) - ' with any a1 E L t X . Note that ( l , n & w ~ ) '= alr&. So, when we take a unit for a l , we have (1,W L ) ' E O:,, which was used above. A possible choice is: a1 = 1 if L ' / F is ramified, a1 E 0:' such that El = -a1 if L'IF is unramified. Then (l,'lrR,,wL)'/(l,r&wL)' is just the sign in **. With respect to the basis (1, r z I w ~ } the map cl'p is given by a matrix 2 in GL(2,OL//rFOL/)

+


197

satisfying

I

tZJ =JZ,

J =

( i) , :l

zz= c l q l t Z = 21, m is odd and Z

E

c1 m o d n p

if L'IF

is (tamely) ramified.

It is perhaps better to say that the map is given by 27:if Z = and z = (zt), then

(:: f:)

2. We now assume that 121 = 1 in F . Then we can take W L such

that w i

E

O L ~Suppose . t E OL(n). Put t = tl

+ t 2 r E , w ~with , ti,t2 E

OL,. The matrix corresponding to multiplication by t is X

(::):

with

E OLI. We have tf - At; = 1. Let

=n s w ;

The conditions on Z are (all = modr;):

and if L ' / F is ramified: z1 that z2

G

c1 modrLt.

= .z3 = O ( T L ' ) ,

It follows from (l),in this case,

z1

=fc1(np).

When (1)holds, (2) is equivalent to

Necessary for solvability of the system is that tl = Tl(n;). We treat the cases L'IF ramified, resp. unramified, separately.


198

Suppose m > ap(t2). Put t2 = s i , b l , bl E O:,, 0 5 lc congruences (2) are blz2 = (-l)kXblzg,

< m. The

Introduce the new variable z; = blzl. Then

zi

= (-l)'f',

Xbff',

= (-1)'X 6:zi modsE-kOLr.

And from (1):

and z{ = blcl modnLtOLf. From the last congruence and z{ even. Then

= (-l)'Z',,

with X I , yi E 0 and (5) blbly2 = Xbfy3 m o d n ~ , - k - l O L ~ ,

(6) xf - nm-'y:

we see that lc must be

Xbfxl =

x 6:xl

-+I

-

bl&sy2y3 E bl&clcl m o d s 2 O , x1

modn?,-'OL,, -

b i c i + b ?LmodnO.

m+l

Here x1 has to be taken m o d ? r z , y1 m o d s $ , y2 and y3 m o d n q . The congruences (5) are equivalent to

(Xb: -x61)x1

G

Omodny,2,-kOLt,

(Xb: - x 6 8 ) y 3

=Omod?r~~k-l

m-k-1

2blbly2 = y3 trLj,F(Xb:) m o d n z O . 2

We must necessarily have Xbf = 5; & modny,-kOLl, since x1 E O x by (6). Then there are q?+' solutions (blblclZl is always a square in F , because L'IF is ramified).

REMARK. The condition Xb: t 2 --2 = t,

5

x 61modnyTkOL/,is equivalent to

rnodn:/+'OL/

in both cases (L'IF ramified or not).


3. Recall that

As t is regular, we have t z # 0 and L/

E L”

tl

# T I . Let us be given:

(vmodNL,LtLX),

m 2 0, n 2 0 such that t E O L ( ~ i.e. ) , n I~ ~ / ( t 2 ) , u E L X ( u m o d O L ( n ) Xsuch ) that ~ L / L / D L ( = u )m - n - D L ( ( Y ) .

By section 2 the number of corresponding cp is: If L’IF is unramified:

If L’/F is ramified and m odd

It is 0 in all other cases. First, we consider the case where L’IF is unramified. We have OLI = O

+ Ow’,

wr2 = a! E ox,

7rL’ = x.

Suppose m I b ~ j ( t 2 ) . Only the congruences (1) are left. We have

with

XI,

y1, y 2 , y3 E U(modnm). Further

199


200

+

There are q3"(l q-2) solutions. . Suppose m > a ~ l ( t 2 )Put t2 = a k b l , The congruences (2) become

Introduce the new variable z{

= blzl.

bl E O;,,

0 5 Ic < m.

Then

Moreover we have, from (1):

,

+nrn-'ylw', Y Z , y3 E 0 and z' = -

XI

z2

= Y~w',

z3

= y3w'modnm0Lt

with xl,yl,

- -2

blbly2 = Xb:y3, Xb:xl s X b,xl modn"-'"OLt, xf - a7r2m-2' y1 - ablbly2y3 = blblclE1 modn"0. (4) The elements X I , y2, y3 are to be taken modulo nm and y1 modulo n k . The congruences (3) are equivalent to

(3)

- -2

(Xbf - X bl)zl = 0,

2blbly2

(Xb; - X Z;)y3

= 0,

= y3 trLt/F(Xbf)rnodn"-'Op.

We must necessarily have Xb? = X 5; rnodnm-kOLI, since X I and y3 cannot be both = O(n)because of (4 . It follows from Xb? = b, rnodIr"-'C?L/ that Xbf is congruent to an element of 0 , which must be in ~ 0for,otherwise X would be a square in L'. So X E 7 r 0 ~ and t y2 E TO. Hence, for (4) to be solvable, blblclE1 must be a square in F . The number of solutions is

x

2qm+2k

if

and 0 otherwise.

a

---' blmodnm-kOLt

Xb2 = X

and

blblclE1 E F x 2 ,


201

Next, consider the case where L'/F is ramified. , , = 7r, a uniformizing element of F . Now We have OLI = 0 O ~ L 7r& m is odd and n z O L , = O n Y + &+aL,.

+

Suppose m 5 a ~ l ( t 2 ) Then .

with

XI,

yi E 0. Further

Here x1 is to be taken modulo 7rw and the yi modulo n w . There are 4 solutions. We compute

cc u

m>O

c

c

O 0, SO O L ( ~ ) 'c O i 2 .


202

Our sum is the sum of the following sums. If L'IF is unramified:

eamified

+

LEMMA.a) A L 2B O~l(6). b) A = 2B bLl(b) except for the cases where L / F is the noncyclic Galois extension.

+

PROOF.a) follows from t: -7; = 6?;(t;?g2 - 6-'8). Note that tl +?l E 02, if 2B ap(6) > 0. b) If A > 2B + bL/(b), then f b-l?modnL,. One checks case-bycase that this is impossible when L J F is not the composite of the three quadratic extensions of F . 0

+


203

The sums (I)-(V) are:

qq2B+2

-

(L/L’ ramified);

q2 - 1

1.1)

q2B-1

(q2 f

42-1

q2

+1

q2B+1(q2

42 1 q3B - 1 2q42-1 43-1

q-1 B

q-i

111)

+

8

---

~

-

-}

(L/L‘ ramified);

q3-1

+ l ) ( q B - 1)(qB+1

-

( 4 - 1)2

- 1)(&B (Q -

2q2B+l(qB+1

Here A = 2B

(L/L’ unramified);

- 1)

1)

(L/L’ unramified); (L/L‘ ramified);

+ 1 if L / F is cyclic.

(L/L’ ramified).

I. PRELIMINARIES 1. Introduction Eichler expressed the Hasse-Weil Zeta function of a modular curve as a product of L-functions of modular forms in 1954, and, a few years later, Shimura introduced the theory of canonical models and used it to similarly compute the Zeta functions of the quaternionic Shimura curves. Both authors based their work on congruence relations. Ihara introduced (1967) a new technique, based on comparison of the number of points on the Shimura variety over various finite fields with the Selberg trace formula. He used this to study forms of higher weight. Deligne [Dl] explained Shimura’s theory of canonical models in group theoretical terms, and obtained Ramanujan’s conjecture for some cusp forms on G L ( ~ , A Q namely ), that their Hecke eigenvalues are algebraic and all of their conjugates have absolute value 1 in C x , for almost all components. Langlands [L3-51 developed Ihara’s approach to predict the contribution of the tempered automorphic representations to the Zeta function of arbitrary Shimura varieties, introducing in the process the theory of endoscopic groups. He carried out the computations in [L5] for subgroups of the multiplicative groups of nonsplit quaternionic algebras. Using Arthur’s conjectural description [A2-41 of the automorphic nontempered representations] Kottwitz [K3] developed Langlands’ conjectural description of the Zeta function to include nontempered representations. In [K4] he associated Galois representations to automorphic representations which occur in the cohomology of unitary groups associated to division algebras. In this anisotropic case the trace formula simplifies. To deal with isotropic cases, where the Shimura variety is not proper and one has continuous spectrum on the automorphic side, Deligne conjectured that the Lefschetz fixed point formula for a correspondence on a variety over a finite field remains valid if the correspondence is twisted by a sufficiently high power of the Frobenius. 207

208

I. Preliminaries

Deligne’s conjecture was used with Kazhdan in [FK3] to decompose the cohomology with compact supports of the Drinfeld moduli scheme of elliptic modules, and relate Galois representations and automorphic representations of GL(n) over function fields of curves over finite fields. Deligne’s conjecture was proven in some cases by Zink [Zi], Pink [PI, Shpiz [Sh], and in general by Fujiwara [Fu]. See Varshavsky [Val for a recent simple proof. We use it here to express the Zeta function of the Shimura varieties of the projective symplectic group of similitudes H = PGSp(2) of rank 2 over any totally real field F and with any coefficients, in terms of automorphic representations of this group and of its unique proper elliptic endoscopic group, CO= PGL(2) x PGL(2). Moreover we decompose the cohomology (&ale, with compact supports) of the Shimura variety (with coefficients in a finite dimensional representation of H ) , thus associating a Galois representation to any “cohomological” automorphic representation of H(A). Here A = Ap denotes the ring of adgles of F , and AQ of Q.Our results are consistent with the conjectures of Langlands and Kottwitz [ K o ~ ] We . make extensive use of the results of [ K o ~ ]expressing , the Zeta function in terms of stable trace formulae of PGSp(2) and its endoscopic group 170,also for twisted coefficients. We use the fundamental lemma proven in this case in [F5] and assumed in [Ko4] in general. Using congruence relations Taylor [Ty] associated Galois representations to automorphic representations of GSp(2, AQ) which occur in the cohomology of the Shimura three-fold, in the case of F = Q. Laumon [Ln] used the Arthur-Selberg trace formula and Deligne’s conjecture to get more precise results on such representations again for the case F = Q where the Shimura variety is a %fold, and with trivial coefficients. Similar results were obtained by Weissauer [W] (unpublished) using the topological trace formula of Harder and Goresky-MacPherson. However, a description of the automorphic representations of the group PGSp(2, AF)has recently become available [F6]. We use this, together with the fundamental lemma [F5] and Deligne’s conjecture [Fu], [Val, to decompose the ag-adic cohomology with compact supports and describe all of its constituents. This permits us to compute the Zeta function, in addition to describing the Galois representation associated to each automorphic representation occurring in the cohomology. To use [F6] when

2. Statement of Results

209

F = Q we work only with automorphic representations which have an elliptic component at a finite place. There is no restriction when F # Q. We work with any coefficients, and with any totally real base field F . In the case F # Q the Galois representations which occur are related to the interesting “twisted tensor” representation of the dual group. Using Deligne’s “mixed purity” theorem [D6]we conclude that for all good primes p the Hecke eigenvalues of any automorphic representation 7r = @7rp occurring in the cohomology are algebraic and all of their conjugates lie on the unit circle for 7r which lift ([F6])to representations on PGL(4) induced from cuspidal ones, or are related by lifting - in a way which we make explicit - to automorphic representations of GL(2) with such a property. This is known as the “generalized” Ramanujan conjecture (for PGSp(2)).

2. Statement of Results To describe our results we briefly introduce the subjects of study; more detailed account is given in the body of the work. Let F be a totally real number field, H = GSp(2) the group of symplectic similitudes (whose Bore1 subgroup is the group of upper triangular matrices), H’ = Rp/QGSp(2) the 0-group obtained by restriction of scalars, AQ and AQf the rings of adkles and finite adhles of Q, K f an open compact subgroup of H ’ ( A Q ~ ) of the form K p , K p open compact in H ’ ( Z p ) for all p with equality for almost all primes p , h : R@/RG, -+ Hk an R-homomorphism satisfying the axioms of [D5] and S K the ~ associated Shimura variety, defined over its reflex field E,which is Q. The finite dimensional irreducible algebraic representations of H are parametrized by their highest weights ( a ,b; c) : diag(x, y, z/y, z/x) H xaybzc, where a , b, c E Z and a 2 b 2 0. Those with trivial central character have a b = -2c even, and we denote them by (Pa$, Va,b). For each rational prime e, the representation

np k2, > 0 odd integers for all w E S. This packet A-lifts to the (normalizedly) induced representation I(7r1, 7r2) of PGL(4, Ap). Here (av,b,)= (+(kl, k 2 v ) - 2, ;(h, - k 2 v ) - 1). (5) X H is in a quasi-packet { L ( [ V ~ /[~v -Xl l~2 ,) }which is the Xo-lift of [ x 7r2, where [ is a character A:/FXAg2 + {fl} and 7r2 is a cuspidal 7r," = ~ 2 k , + 3 , k , 2 0, w E S . Here representation of P G L ( ~ , A F with ) (a,, bv) = ( k v , kv). A global (quasi-)packet is the restricted product of local (quasi-)packets, which are sets of one or two irreducibles, pointed by the property of being unramified (the local (quasi-) packets contains a single unramified representation at almost all places). The packets (1) and ( 3 ) and the quasipacket (2) are stable: each member is automorphic and occurs in the discrete spectrum with multiplicity one. The packets (4) and quasi-packets

-

---f

,A.{

+

I. Preliminaries

212

( 5 ) are not stable, their members occur in the discrete spectrum with multiplicity one or zero, according to a formula of [F6] recalled below. We now describe the semisimplification H , * ( X H f ) " of the representation H,*(xH,) of Gal(o/Q) associated to each of these X H f . From now The ~ . Chebotarev's density theoon, we write H,'(rHf) for H , ' ( r ~ f ) ~ rem asserts that the Frobenius elements Fr, for almost all p make a dense subgroup of Gal(o/Q). Hence it suffices to specify the conjugacy class of H , * ( r ~ f ) ( F r , for ) almost all p . This makes sense since H , * ( x H ~is) unramified at almost all p , trivial on the inertia subgroup I, of the decomposition group D, = Gal@,/Q,) of Gal@/Q), and D p / I , is (topologically) generated by Fr,. The conjugacy class H,*(nHf)(Fr,) is determined by its trace, and since H,*(7rHf)(Frp)is semisimple it is determined by H , ' ( r ~ f ) ( F r i ) for all sufficiently large j . We consider only p which are unramified in F, thus the residual cardinality q, of F, at any place u of F over p is p n u , nu = [F, : Q,]. Further we use only p with K f = K,KP, where K, = H'(Z,) is the standard maximal compact, thus S K has ~ good reduction at p . Note that dimSKf = 3[F : Q]. Part of the data defining the Shimura variety is the R-homomorphism h : R@/wG, + H' = RF/QH. Over C the one-parameter subgroup p : C x 4 H ' ( C ) , p ( z ) = h ( z , l ) factorizes through any maximal C-torus TL(C)c H ' ( C ) . The H'(C)-conjugacy class of p defines then a Weyl group Wc-orbit p = p r in X,(T&)= X*(?&). The dual torus = TH in g' = g, 0 E Emb(F,R), can be taken to be the diagonal subgroup, and X*(?H) = 2'. See section 10 for more detail. We choose p7. to be the character ( 1 , O ) : diag(a, b, b - l , a - ' ) H a of ?H. Thus the If(@)-orbit of the coweight p r determines a Wc-orbit of a character - again denoted by p, - of ?H, which is the highest weight of the standard representation r: = st of Sp(2,C). Put r: = Brr:. It is a representation of H'.

n,

n,

A

n,

A

h

The Galois group Gal(Q/Q) acts on Emb(F,Q). The stabilizer of p , Gal(Q/IE), defines the reflex field IE. In our case IE = Q. An irreducible admissible representation X H , of H(F@Q,) = H'(Q,) = H(F,) has the form @ u ~ ~ Suppose u . it is unramified. Then T H , has the form ~ ~ ( p lpz,), , , a subquotient of the normalizedly induced representation I(pl,, pzu) of H(F,) = PGSp(2, F,), where pi, are unramified characters of F,". Write pmu for the value p m U ( r u ) at any uniformizing parameter A, of F,". Put t , = t ( r H , ) = d i a g ( p ~ , , p z , , p ~ ~ , pNote ~~).

nu,,


+

+

213

+

that tr[ti] = pi, p$,, pi: p;:. The representation 7rp is parametrized by the conjugacy class of t(7rp) = t, x Fr, in the unramified dual group L H I = fi[F:Q] P

x (Fr,). A

Here t, is the [ F :Q]-tuple (t,; ulp) of diagonal matrices in H = Sp(2,C), where each t, = ( t , ~ ,... ,tun,) is any n, = [F,: Q,]-tuple with t,i = t,. The Frobenius Fr, acts on each t, by permutation to the left: Frp(tu)= ( t U 2 , . . . , tUnu, t,1). Each 7r, is parametrized by the conjugacy class of t(7rHu) = t, x Fr, in the unramified dual group LHh = Hnu x (Fr,), or alternatively by the conjugacy class oft, x Fr, in L H , = H x (Fr,), where Fr, = Frp"". ~ in Our determination of the Galois representation attached to T H is terms of the traces of the representation r: of the dual group LHb =

ni

-

h

A

H' x WEat the positive powers of the n,th powers of the classes t(7rHp) = ( t ( 7 r U ) ; ulp) parametrizing the unramified components TH, = @J,~,TH,. The representation H , * ( 7 r ~ fis) determined by tr[Fri IH,*(nHf)] for every integer j 2 0, prime p unramified in E , and lE-prime p dividing p . As lE = Q here, p = p and n p = 1. The following very detailed statement describes the Galois representaattached to the cohomological T H . tion H:(TH~)(TH) THEOREM 2. (1) Fix 7 r of~ type (1) which occurs in the cohomology with coeficients in Va,b, a, 2 b, 2 0 , even a, - b,. Thus T H has archimedean components 7ril,,k2v * = Wh or hol, k l , = a , b, + 3 > ka, = a, - b, 1 > 1 are odd. It contributes to the cohomology only in dimension 3[F : Denote by T H , = 7 r ~ ( p 1 , ,p2,) the component of the representation T H of H ( A F ) at a place u of F above p . It is parametrized by the conjugacy class t ( r H , ) = diag(tl,,tz,,t,-,',t;;') in fi = Sp(2,C), where t,, = p,,(x,), m = 1 , 2 . Then H , * ( T H ~is) 4[F:Ql-dimensionaZ7and with j , = ( j ,nu),

+

+

a].

tr[Frj, IH,*(THf)] = p f dim SKf trrE[(t(7rp)x ~ r * ) j = ] n(tr[ti/j~])ju. "IP

Namely H,*(7rHf)(Frp) is @J,~pv~1'2r,(Fr,), twisted tensor representation (r,, (C4))"u) as t(rH~)

t(THu)

where r,(Fr,) acts on the

= ( t l , .. . , t n u ) ,

I. Preliminaries

214

tm diagonal with ITllmln,

t , = t ( 7 r H u ) . Here v, is the unramified character of Gal(o',/Q,) with v,(F'r,) = qC1. The eigenvalues of H,*(7rHf)(Frp) are ~ $ I F : Q I u , p ,,t,),L(,( " ) m(u) E {l,2}, ~ ( uE) {fl}. I n our stable case (l),the representation H , * ( T H ~depends ) only on the packet of I T H , and not o n T H itselj. The Hecke eigenvalues tl,, t 2 , are algebraic and each of their conjugates has complex absolute value one. (2) Representations T H in a quasi-packet { L ( E Vv-1/27r2)} , of type (2) occur in the cohomology with coeficients in v2k,,O, k , 2 0, thus T H has At a place archimedean components T H , = L ( v sgn, v-1/27r2k,+2)1 v E u of F overp the component 7 r t of 7r2 is unramified of the form 7 ~ ' ( z 12~2 , ) . The Hecke eigenvalues zl,, 22, satisfy zlUz2, = [,(nu) E {fl}. The component T H , has parameter

s.

t,

= diag(q;/'zl,,

-1/2

Q ~ / ~ ~ . 4z ,u ,

-1 22,

-1/2

4,

z -1i ~

in H = Sp(2, C ) . The associated representation H,*( 7 r ~ f has ) dimension 4 ~ and~ H,*(7rHf)(Frp) 1 is the same as in case (1) but with tl, = q;/'z1,, t z u = q ~ / 2 z 2 u . The zl,, z2, are algebraic, all their conjugates lie o n the unit circle in C. (3) The case of type (3) of the one dimensional representation T H = = 1, occurs in the cohomology with coeficients in VO,Oonly. The o A, parameter t ( 7 r H U ) is

) again 4[F:Q1-dimensional and The associated representation H z ( 7 r ~ f is H,*(7rHf)(Frp)is the same as in case (1) but with tl, = [uqi'2, t 2 , = tu&/2,ttl E ( 5 1 ) . (4) The ITH of the unstable tempered case (4) occur in the cohomology with coeficients in v a , b , a, 2 b, 2 0, even a, - b, . Thus the archimedean k,,= } , a, b, 3 > kzv = components T H , are in { 7 r ~ ~ , k z v , 7 r ~ ~ ~ ,k1, a, - b, 1 > 0 are odd. The component T H , of T H at a place u of F over p is unramified of the form T H ~= ~ ~ ( p l , , p parametrized ~~), t-l t-1 by t , = diag(tiu, t z U , 2, , 1, ), tmu = p m U ( n u ) ,m = 1,2, in H . The packet { T H } of ITH is the XO-lift of 7r1 x 7r2, where 7r1, 7r2 are cuspidal representations of PGL(2,Ap). It is defined b y means of local packets

+

+ +


215

are singletons unless 7rL and IT; are discrete series, in which case { . i r ~ ~ = }{ 7 r ~ w , 7 r ~ w } with , indicating generic and - nongeneric. I f { T H ~ }consists of a single term, it is 7rLw)and we put 7rhw = 0 . W e say that 7rHf lies in { T H ~ } ’ if it has an even number of components 7rGW (w < w), and in (7rHf)- otherwise. Write n(7r’ x 7 r 2 ) for the number of archimedean places v E with (7rA,7r,”) = (7rkzv,7rk1,) (recalk { T H , } = &(7rkl, X Irkzv), k l , > k2, > 0). Then the dimension of H,*(“Hf) is 1 2 . 4[F:Q1and the trace of H,*(7rHf)(Fri) is f p i dim s K f times { T H ~ } which )

+

v

iff H f { Hf} . The t1u, t2u are algebraic and their conjugates lie on the unit circle. Thus H*8c( Hf)(Frjp is

where r$(Fru) acts o n the twisted tensor representation (rur(C4)[Fu:Qu1) as s*t(7rH,)xFru where s+ = I and s- = ( s , I , .. . , I ) , s = diag(1, -1, - l , l ) . (5) The T H of the unstable nontempered case (5) occur in the cohomology

with coeficients in V k , k , k = ( k , ) , k , L 0. Its archimedean components Wh 7rhol X H , are ~ 2 k , + 3 , i , 2k,+3,1 or the nontempered L([V1/’7r2kU+3,[ U - l / 2 ) , [ = 1 or sgn. It lies in a quasi-packet { L ( [ V ~ / ~ ~ ~ ~ , [ 7r2 U -cuspi~/~ dal representation of PGL(2, A p ) , whose real components are 7r2kV+3, and E is a character A g / F x A g 2 { i l } . The unramified components T H are ~ rGU= L ( [ u d / 2 7 r : , [ u ~ i 1 / 7r:2 ) ,= 7 r : ( z 1 ~z,z z L ) , = 1, parametrized t-1 t-1 by t , = diag(tl,,tz,, z U , z U ) , ti, = tUqA/2ziUJ t 2 = ~ E4h/2zlu . The quasi-packet {TH} is the Xo-lifl of 7 r 2 x [ l z , defined using the local quasi~2 [wuw -‘I2) , 7 r-H w is 0 unless 7 r i is packets {xiw,7 r H w ) , T i w = ~ ( [ , u i /xu,, square integrable in which case 7rkWis square integrable (in the real case --$

-’

-

.

7rH~

hol

7r2k,+3,1).

I. Preliminaries

216

We write 7rHf E (7rHf)' if the number of components 7 r i w ( w < cm) of is even, and THf E {THf}- if this number is odd. Then the dimension of H,"(THf) is ; . ~ ~ F : Q and I the trace of H,*(THf)(Frjp) is ; p d d i r r r S f ~times

7rHf

f

n(,,,

+t

- p

-

(G,, + t z u ) - m l ) A I

4 P

with + if 7rHf E (THf}' and - if T H f E {7rHf}-. Here zlu is algebraic, its conjugates are all on the complex unit circle. Thus H,*(nHf)(Fr,) has the same description as i n case (4), except f o r the values of tl, and t z U . Note that the Hodge types of 7 r for~ each ~ ZI E S are (1,2), (2,1), (0,3), (3,O) in types (1) and (4); ( W ) , (0,2), (1,3), (3,1) in type (2); (U)and (2,2) in type (5); and (O,O), (1,1),(2,2), (3,3) in type (3), specifying in which H:ij ( S K @Q ~ Q, va,b)each 7rH f may occur. In particular H,"j is 0 if i # j and i j < 2 [ F : Q]or i j > 4 [ F : Q]; H: has contributions only from one dimensional representations TH (of type (3)) if i < [ F : Q]or i > 2 [ F : Q]; H:[F'Q1 (and H:[F'Q1)has contributions only from representations of type (2), (3), (5). For example, H:[F:Q1>O (and H:'2[F:Q1)has contributions only from representations of type (2). The representations of types (2) and (5) are parametrized only by certain representations of GL(2) (and quadratic characters); these have smaller parametrizing set than the representations of type (4)(two copies of PGL(2)) or of type (1) (representations of PGL(4)). In stating Theorem 2 we implicitly made a choice of a square root of p . For unitary groups defined using division algebras endoscopy does not show and Kottwitz [K4] used the trace formula in this anisotropic case to associate Galois representations H," (7rH f ) to some automorphic T H and obtain some of their properties. However, in this case the classification of automorphic representations and their packets is not yet known. For GSp(2), in the case of F = Q and trivial coefficients a , = b, = 0 , in particular trivial central character (PGSp(2)), Laumon [Ln], Thm 7.5, gave a list of possibilities for the trace of H,*(THf) at Fr, for TH in the stable spectrum, removing Eisensteinian contributions, see [Ln], (6.1). His

+

+


217

Thm 7.5 (l),(2) says T H might be Eisensteinian (our cases (2), (3), (5)) or endoscopic (our cases (4), (5)), his (3) corresponds to our case (l),but our cases (2), (5) are included again as a possibility in his Thm 7.5 (4). That is, by [F6] the T H in his Thm 7.5 (4) are already included in his (1) and (2). Using the results of [F6], namely classification of and multiplicity one for the automorphic representations of the symplectic group, as well as the fundamental lemma of [F5] and Deligne’s conjecture of [Fu], [Val, makes it possible for us to obtain more precise results, namely specify the H , * ( T H ~ such that T H f @3 H , * ( T H f ) occurs in H,*, for all T H , and list the T H which occur. Also, knowing the structure of packets and quasi-packets from [F6] lets us state and deal with the general case of F # Q. Laumon [Ln] works with F = Q and uses very extensively Arthur’s deep analysis of the distributions occurring in the trace formula, together with the ideas of the simple trace formula of [FK3],[FK2] (the test function has an elliptic ([Ln], p. 301: “very cuspidal”) real component and a regular component). This lets him put no restriction on the test function, but leads to very involved usage of the spectral side of the Arthur trace formula. A simple trace formula (for a test measure with no cuspidal components) is available for comparisons in cases of F-rank 1 (see [F2;I], [F3;VI], [F4;III])but not yet in F-rank 2. Hence in [F6] we use instead the trace formula with 3 discrete components (in fact 2 suffice, as explained in [F6], 1G). Using the results of [F6] leads us to the restriction (elliptic component at a finite place) we made here when F = Q. This can be removed, to get unconditional result also for F = Q, on using Arthur’s deep analysis of the distributions occurring in the trace formula, as [Ln] explains. Results similar to [Ln] have been obtained by Weissauer [W] (unpublished), who used the topological trace formula of Harder and GoreskyMacPherson. This trace formula applies to “geometric” representations only, namely those with elliptic (in fact cohomological) components at the real places. Previously some results (for a dense set of places) were derived by Taylor [Ty] from the congruence relations.

218

I . Preliminaries

3. The Z e t a Function The Zeta function 2 of the Shimura variety is a product over the rational primes p of local factors 2, each of which is a product over the primes p of the reflex field IE which divide p of local factors 2,. In our case E = Q and g~ = p but we keep using the symbol p to suggest the general form. Write q = q, for the cardinality of the residue field F = R p / p R , ( R , denotes the ring of integers of IE,; q is p in our case). We work only with “good” p , thus K f = KpKfp,K p = H ’ ( Z p ) ,S K is ~ defined over R, and has good reduction mod p. A general form of the Zeta function is for a correspondence, namely for a Kf-biinvariant Q-valued function :f on H(APf),(A is AF and we fix a field isomorphism 2 C ) , and with coefficients in the smooth ae-sheaf Va,b;e constructed from an absolutely irreducible algebraic finite dimensional representation Va,b = @,E~Vav,b,of H’ over F , each Va,,bv with highest weight (a,, b,), a , 2 b, 2 0, even a, - b,. The standard form of the Zeta function is stated for = lH(Al;), and for the trivial coefficient system ((a,, b,) = (0,O) for all w). In this case the coefficients of the Zeta function store the number of points of the Shimura variety over finite residue fields. Thus the Zeta function, or rather its natural logarithm, is defined by

fg

The subscript c on the left emphasizes that we work with Hc rather than H or I H ; we drop it from now on. One can add a superscript i on the left to isolate the contribution from H:. Our results decompose the alternating sum of the traces on the cohomology for a correspondence f;projecting on the subspace parametrized by those representations T H of H ( A F )with at least 2 discrete series components. We make this assumption from now on. The coefficient of l / j q F is then equal to the sum of 5 types of terms. The first 3, stable, terms, are

3. The Zeta Function

219

of the form

The 4th, unstable tempered term, is the sum of

AND

4P

The 5th, unstable nontempered term, is the sum of

and

The representation (r:, (C4)IFu:Qp]) is the twisted tensor representation of L ( R F , / Q p H )= k [ F = : Q p l x1 Gal(Fu/Qp). Here C4 is the standard representation of H c GL(4,C) and the generator Fr, of Gal(F,/Qp) acts by permutation Fr,(zl @ z2 @ . . . @ z n u ) = z2 @ . . . @ znu @ 5 1 , n, = [F, : The class t(7rH,) is ( t l , .. . ,t,,), t , is diagonal in H with t , = t(nHu) being the Satake parameter of the unramified ~ . s = (s, I , . . . , I ) , s = diag( 1,-1, - l , l ) . component 7 r ~ Further, The three stable contributions to the first sum are parametrized by: (1) Stable packets ( 7 r ~ ) . These A-lift to cuspidal &invariant representations 7r of G(AF), G = PGL(4). The infinitesimal character of each

a,].

nl<m 0 odd, specify the packet { 7 r H } = X 0 ( d x 7 r 2 ) . This occurs when a, = ( k l , k2,) - 2, b, = ( k l , - k2,) - 1 for all u E S. In = this case the t(7rHu) are as in (1). The number of 2, E S with (n:,~,”) ( 7 r k 2 v 1 7 r k l v ) l h, > h,, is denoted by n ( ~ x 1 7r2). (5) Pairs 7r2 x € 1 2 , where 7r2 is a cuspidal representation of P G L ( ~ , A F ) with discrete series archimedean components 7r: = x 2 k v + 3 , k, 2 0, all u E S , unramified components 7 r 2 , ulp, with Satake parameters z:‘, and character : A g / F x A g 2+ {fl}. Such pair specifies the quasi-packet of 7rG = [v-~/’) = Ao(7r2 x 5 1 ~whose ) ~ archimedean components have infinitesimal characters ( 2 , l ) plus (a,,b,) = (k,, k , ) for all w E S. Thus this case occurs only for (a,b) with a, = b, for all u E S . The diagonal entries of ~ ( T H , ) , ulp, are (a 7r1 from the Heisenberg parabolic subgroup, 7rL is the generic constituent. If 7r1 = 7r2 = a s p 2 where a is a character of F X with o2 = 1, then 7r; and 7rk are the two tempered inequivalent constituents T ( u ' / spa, ~ a~-l/~) and ~ ( u ' / ~ 1 2 , a u - of ~ /1~x) asp2. If 7r1 = osp2, o2 = 1, and 7r2 is cuspidal, then 7rA is the square integrable constituent of the induced x uu-'/' from the Siege1 maximal parabolic subgroup of H ( F ) (with abelian unipotent radical). The 7 r i is cuspidal, denote by S - ( ~ U ~ / a~ ~T -~ l, / ~ ) . If 7r1 = osp, and 7r2 = Josp2, J (# 1 = J2) and D (a2 = 1) are characters of F X, then 7r$ is the square integrable constituent S(JV1/2

sp2,a v - q

of the induced [ u ' / ~sp2 x o ~ - ' / ~The . 7 r i is cuspidal, denoted by

( 2 ) For every character o of F X / F X and 2 square integrable exists a nontempered representation 7rff of H ( F ) such that

7r2

there

tr(n 2 x a12)(fco)= trxG(fH) + t r ~ g ( f ~ ) tr1G(7r~,al2;f x 0) = tr7rG(fH) for every triple (f,f -

7rH

-

tr7rg(fH)

~fc,) , of matching functions. Here

= 7rh(asp2 x7r2)

and

7rG

=L(a~~/~7r',au-~/~).

3. Local Packets

233

(3) fOR ANY CHARACTEERS , OF fX/fX2 AND MATCHING F, Fh, Fc0 WE HAVE

Here X = 6- if # 1 and X = L if J = 1. (4) Any &invariant irreducible square integrable representation 7r of G which is not a XI-lift is a A-lift of an irreducible square integrable representation X H of HI thus tr7r(f x 0) = tr7rH(fH) for all matching f , f H . In particular, the square integrable (resp. nontempered) constituent s ( < ~u,- 1 / 2 7 r 2 )(resp. L([u,~ - ' / ~ 7 r ' ) ) of the induced representation Eu x u-1/27r2 of HI where 7r2 is a cuspidal (irreducible) representation of GL(2, F ) with central character # 1 = t2and En2 = 7r2, A-lifts to the square integrable (resp. nontempered) constituent

a c r ~ - ' / ~ )to , consist of { 7 r G 1 7 r ; } , and of { L ,X}, x = X( k2. When k l = k2, IT; is the generic and 7rg is the nongeneric constituents of the induced 1 XI 7r2k1+1. There is no special or Steinberg representation of GL(2,R); the analogue is the lowest discrete series 7r1. It is self invariant under twist with sgn. In (2) with 7r2 = ~ 2 k + 3 ( k 2 0), 7rg is ~ ( a v 1 / 2 ~ 2 k + 3 , ~ ~ - 1 / 27rG ) , is 7r;i:3,1. In (3), if = sgn then X is 7rG c 1XI 7 r ' , if = 1 then X is L ( ~ ' / ~ 7 ~ r ' v, - ' / but both of these X Ias well as L(ua WE is defined on the [ F : Q]-fold tensor product @sC4,as follows. For h = (h,;v E S ) E I?' we have r,(h) = @uEsrl(hu). The Weil group WEacts by permuting the factors, thus by left multiplication on S . Then dim r , = 4[F:a1 and r p is the twisted tensor representation.

2. Local Terms at p Let p be a rational prime which is unramified in F . The Q-group H' = R F / Q H is 0,-isomorphic to HL, where HL = R F , / ~ , H and u ranges

nu,,

over the primes of F over Q. The set S of embeddings L of F into

0(or R,C

2. Local Terms at p

24 1

0,)

or is parametrized by the homogeneous space Gal(a/Q)/ G a l ( a / F ) , once we fix such an embedding. The Galois group Gal(o,/Q,) acts on the left. If p is unramified in F this action factorizes via its quotient Gal(QEr/Q,) by the inertia subgroup. The orbits of the Frobenius generator Fr, are the places u of F over Q.The group of Q,-points of H’ is

nUlp

nu/,

= H m J >= H(Fu). An irreducible admissible representation TH, of H’(Q,) has the form @,TH,. If T H is ~ unramified then each T H , has the form ~ ~ ( p l , , p ~ where p l U 7pzU are unramified characters of F,” . We write pmu ( m = 1,2) also for its value p , , ( ~ , ) at any uniformizing parameter xu of F,”. Put tu = t ( T H U ) = diag(p.lu,p2u7p i t 7p;;t). The representation TH, is parametrized in the unramified dual group L H i = fiIF:Ql XI (Fr,) by the conjugacy cla.ss of t , x Fr,. Here t , is the [ F : Q]-tuple (t,;ulp) of diagonal matrices in H = Sp(2,@), each t, = ( t u 1 7 . . . tun,) is any n, = [F, : Q,]-tuple with t,i = t,. The Frobenius Frp acts on each t, by permutation to the left: fip(tu)= (t,Z,. . . , t,,, t,1). Each TH, is parametrized by the conjugacy class of t, x Fr, in the unramified dual group LHL = I?IFU:Q~IXI (Fr,), or alternatively by the conjugacy class of t, x Fr, in L H u = H x (nu), where Fr, = Fr[Fu:%I. f w P )

ni

P

The representation r = @ S r L of LHh can be written as the product @,lPru7where r , = BLE,rL.A basis for r is given by @seL, where eL lies in the standard basis {el, e2, e3, eq} of C4. A basis for r, is given by BLEUeL. The representation r, is called the twisted tensor representation. The vectors fixed by Fr, are those which are homogeneous on each orbit of S, in the sense that eL = ei for a fixed i = i ( u )for all L E u.In particular

More generally let us compute the trace tr r p [ ( t ,x Frp)j]=

-

n

tr r u [ ( t Ux

Frp)j]

UIP

We proceed to describe the action of Fr, on Emb(F,R). Fixing a 00 : F f l R (c R), we identify Gal(a/Q)/ Gal@/F) with Emb(F7 n R) = (01,. . . ,a,}. The decomposition group of Q at

III. Local Terms

242

p , Gal(~,/Q,), acts by left multiplication. Suppose p is unramified in F .

Then F’r, acts, and the Fr,-orbits in Emb(F,R) are in bijection with the places u1, . . . ,u, of F over p . The F’robenius Fr, acts transitively on its orbit u = Emb(F,,Q,). The smallest positive power of Fr, which fixes each rs E u is n,. The action of Fr, on G: = Gnu is by F’rp(tu) = ( t u 2 , . . . ,tunU, t , ~ ) ,where t, = ( t , l , . . . , tun,). Then F’I-,”~(~,) is t,, h

h

and

A basis for the 4”~-dimensionalrepresentation r, =

&To,

u E

u,is

given by @ o E u e ~ ( owhere ), e q a ) lies in the standard basis {e1,ez,e3,e*} on these vectors it is of C4 for each u . To compute the action of convenient t o enumerate the o so that the vectors become

and Fr, acts by sending this vector to

Then F’rp”” fixes each vector, and a vector is fixed by Fri iff it is fixed by I+$, 0 5 j o < nu, j = jo(modn,). A vector is fixed by iff

F’ri

it is equal t o BieiTz3) = @,ei-j0 e ( i ) , thus l ( i )depends only on i m o d j (and imodn,), namely only on imodj,, where j , = (j,n,). Then (tUx

Fr,p

=

(. . . ,

n Oa 7r:) = I(T:,~F:) so that t , = diag(z,,~;~,z,,z;~). In summary, as noted in the last section, the factor at p of each of the summands in S T F H ( ~ Hhas ) the form (where j , = (n,,j)) P i dim sK ~

f

trrp[(t(THp) x

H(t.77.

p $ dim S K ~

~ r ~ > = jp]i d i m S ~r -f( t r l t u

+ t;;/ju

+

g u ,;;/j. +

x

fiPl9

UlP j,

) .

UlP

REMARK. As p splits in F into a product of primes u with F,/Qp un: Q,p ] , and the dimension of the symmetric ramified with [F : Q]= ~ , I p [ F space H(R)/KH(w)is 3, we note that dimSKf = 3[F : Q]= 3 1 [ F , : Q p ] UlP

4.

Terms at p for the Endoscopic Group

247

4. Terms at p for the Endoscopic Group The other trace formula which contributes is that of Co(A) = PGL(2, A) x PGL(2, A). The factors at p of the various summands have the form

"IP

where s = diag(1, -1, -1,1) is the element in I ? = Sp(2, C ) whose centralizer is C o = SL(2,C) x SL(2,C). We need to specify the 4-tuples t , again, according to the three parts of STFco(fo). For the first part, where the summands are indexed by pairs d x 7r2 of cuspidal representations of PGL(2,A), the t, is the same as in the second part of STFH(fH) if 7r1 # 7r2, and as in the fifth part if 7r1 = 7r2. For the second part of STFca(fo), the t, for the term indexed by (a17r2)is the same as for the third part of S T F H ( ~ H )For . the third part of STFc,(fO), the t, for the term indexed by (a,to)is the same as for the fourth part of S T F H ( ~ Hif) # 1 or the fifth part when 5 = 1.

IV. REAL REPRESENTATIONS 1. Representations of SL(2,a) Packets of representations of a real group G are parametrized by maps of the Weil group WR to the L-group LG. Recall that WR = ( z ,o;z E C X , o2 E R x - N @ / n C Xo,z = to) is

an extension of Gal(@/R) by Wc = C x . It can also be viewed as the normalizer C x U C x j of C x in Wx,where W = R(1, i, j , Ic) is the Hamilton quaternions. The norm on W defines a norm on WR by restriction ([D3], [Tt]). The discrete series (packets of) representations of G are parametrized by the homomorphisms 4 : WR4 G x Ww whose projection to WR is the identity and to the connected component G is bounded, and such that CdZ( G) /Z(G)is finite. Here Cd is the centralizer Z ~ ( + ( W R in G of the image of 4. When G = GL(2,R) we have G = GL(2,R), and these maps are c$k (k 2 l),defined by

Since 0 2

=

-1

c-f

(

( ~ 1 ) 0~ 0

(-1)k

)x

0 2 ,L

must be (-l)k. Then det &(o) =

( - l ) k + l ,and so k must be an odd integer (= 1 , 3 , 5 , .. . ) to get a discrete series (packet of) representation of PGL(2,R). In fact 7r1 is the lowest discrete series representation, and $0 parametrizes the so called limit of discrete series representations; it is tempered. Even k _> 2 and CT H x o define discrete series representations of GL(2, R) with

( y i)

the quadratic nontrivial central character sgn. Packets for GL(2,R) and PGL(2, R) consist of a single discrete series irreducible representation nk. Note that "k €9 sgn N 7 r k . Here sgn : GL(2,R) + {fl},sgn(g) = 1 if detg > 0, = -1 if detg < 0. 248

2. Cohomological Representations

249

The 7rk ( k > 0) have the same central and infinitesimal character as the kth dimensional nonunitarizable representation

into

SL(k, C)* = {g E GL(2, C); det g E {kl}}. Note that det Sym"'(g)

= det gk(("-l)i2.The normalizing factor is

In fact both " k and Symt-' C2 are constituents of the normalizedly induced representation I(U"~, sgnk-' whose infinitesimal character is ( -$), where a basis for the lattice of characters of the diagonal torus in SL(2) is taken to be (1,-1).

i,

2. Cohomological Representations An irreducible admissible representation 7r of H ( A ) which has nonzero Lie algebra cohomology H'J (g, K ;7r @ V) for some coefficients (finite dimensional representation) V is called here cohomological. Discrete series representations are cohomological. The non discrete series representations which are cohomological are listed in [VZ]. They are nontempered. We proceed to list them here in our case of PGSp(2,R). We are interested in the (g, K)-cohomology H Z J(sp(2,R),U(4); 7r @ V), so we need to compute HzJ(sp(2,R),SU(4); 7r @ V) and observe that U(4)/SU(4) acts trivially on the nonzero H ' J , which are C. If H'J (7r 8 V ) # 0 then ([BW])the infinitesimal character ([Kn])of 7r is equal to the sum of the highest weight ([FH]) of the self contragredient (in our case) V, and half the sum of the positive roots, b. With the usual basis (1,0), ( 0 , l ) on X*(Ti), the positive roots are (1,-l), (0,2), (1,l ) ,(2,O). Then 6 = COl,,, (Y is ( 2 , l ) . Here T: denotes the diagonal subgroup {diag(z, y, l / y , l/z)} of the algebraic group Sp(2). Its lattice X * ( T i )of rational characters consists of

( a ,b ) : diag(z, y, l / y , l / z ) H zayb

( a ,b E Z).

IV. Real Representations

250

The irreducible finite dimensional representations of Sp(2) are V&, parametrized by the highest weight ( a ,b) with a 2 b 2 0 ([FH]).The central character of V a , b is C H E {fl}. It is trivial iff a + b is even. Since GSp(2) = Sp(2) x {diag(l, 1,z , z ) } , such V a , b extends to a representation of PGSp(2) by (1,1,z , z ) H z-(a+b)/2.This gives a representation of H ( R ) = PGSp(2, R), extending its restriction to the index 2 connected subgroup H o = Psp(2, R). Another - nonalgebraic - extension is V& = V a , b 63 sgn, where s g n ( l , l ,z , z ) = sgn(z), z E R x . Va,bis self dual. To list the irreducible admissible representations 7r of PGSp(2, R) with nonzero Lie algebra cohomology Hif(sp(2,R), su(4);7r I8 V a , b ) for some a 2 b 2 0 (the same results hold with V a , b replaced by v&), we first list the discrete series representations. Packets of discrete series representations of H(R) = PGSp(2,R) are parametrized by maps 4 of WR to L H = H x Ww which are admissible (pr2 04 = id) and whose projection to H is bounded and C$Z(fi)/Z(fi is finite. Here C+ is Zfi(~$(Ww)). They are parametrized 4 = & , , k z by a pair ( k l , k2) of integers with k l > k2 > 0 and odd 51, k2. The homomorphism $kl,kz : WR --+ LG = G x WR, G = sL(4,C), given

k2 > 0)

(T

or x

0

(even I C ~> 1c2 > o),

factorizes via (LCo +) L H = Sp(2,C) x WR precisely when ki are odd. When the ki are even it factorizes via LC = SO(4,C) x WR. When the ki are odd it parametrizes a packet {7rz:k2, of discrete series representations of H(R). Here 7rwh is generic and 7rho1 is holomorphic and antiholomorphic. Their restrictions to H o are reducible, consisting of 7rpt and n$' o Int(L), and o Int(L), L = diag( 1,1,-1, -l), and 7rWh 8 sgn = n W h , rho' 18sgn = 7rho1. To compute the infinitesimal character of r l l , k z we note that 7rk C 1 ( v k / 2 , sgnk-1 v-k/2 ) (e.g. by [JL], 15.7 and 15.11) on GL(2,R). Via (in our case the LC~ + L H induced I ( u k 1 f 2 , u - k 1 f 2 )x

TLP,'~,}

7r2:

3. Nontempered Representations

251

ki are odd) lifts to the induced

whose constituents (e.g. n-;l,kz, * = Wh, hol) have infinitesimal character ( k l + k z , kl-kz ) = ( 2 , l ) - 2 2 b = - 1 20 ( a , b ) . Here a = 2 as k2 2 1 and k l > k2 and kl - k2 is even. For these a 2 b 2 0, thus kl = a b 3, k2 = a - b 1, we have

9

+

+ +

+

Hi’(sp(2,R),SU(4);.rr,hp!kz @va,b) = @

if

( i , j )= (3,0),(0,3).

Here kl > k2 > 0 and k l , k2 are odd. In particular, the discrete series representations of PGSp(2, R) are endoscopic.

3. Nontempered Representations Quasi-packets including nontempered representations are parametrized by homomorphisms 1c, : WR x SL(2,R) + L H and & : W, 4 L H defined

([All by

4 d W ) = N W ? ( llWIl1’* 0

l,wll-l,z)). 0

R x is defined by llzll = zZ+ and ~~o~~ = 1. Then The norm 11.11 : W, $+(a)= G(o,1) and $+(z) = $(z,diag(r,r-l)) if z = reie, T > 0. For example, T,!I : WR x SL(2,C) --+ SL(2,C),

gives

~ / ~ , of parametrizing the one dimensional representation i3 = J ( ~ v av-l/’7rZk induced from the Heisenberg parabolic subgroup of H . It A-lifts to J(V’/27r2k, V-1/27r2k), the Langlands quotient of the induced representation I ( V ’ / ~ Tv-’/’7r2k) ~~, of PGL(4,R). Note that the discrete series 7r2k 2 s g n m2 k N %2k has central character sgn(# 1). Now

WITH

4. and

($1

SL(2,C))

The Cohomological L(u sgn, V-1/27T2k)

253

(E 1) = (E; ii), defines

which factorizes via

fi = Sp(2, C)

L-$

SL(4, C)and parametrizes

Note that when 2k is replaced by 2k diag(1, -l), then

$&)=$(o,I)=(

bJ(z)=

( 'El

+ 1, 4 2 k + l ( c 7 )

=

EW

x

g, E

=

E0 w ) = I @ E w E c ,

EW

@ 42k+l(z) E

1~r-1)

c,

thus $+ defines a representation of C(R) (which X1-lifts to the representation J(v1'27T2k+1, v-1/27T2k+1)

of PGL(4,W)), but not a representation of H ( R ) . As in [Ty] write T:k,o for L(SgnV,v-1/27T2k+2). We have that 7&,0 N and 7rik,olHo consists of two irreducibles. In the Grothendieck sgn group the induced decomposes as

To compute the infinitesimal character of u sgn >aY-1/27r2k, note that it is a constituent of the induced u sgn >av-1/21(uk,sgnv-k) N sgnu2k x sgn v x z.-~-'/' sgn (using the Weyl group element (12)(34)), whose infinitesimal character is (2k,l) = ( 2 , l ) (a,O), with a = 2k - 2 > 0 as IC >_ 1. For k 2 1 we have Hz~(sp(2,R),SU(4);7~~~,, V2k,0) = C if ( i , j ) = (2,0), (0,2), (3,1), ( L 3 ) .

+


254

5. The Cohomological L

( ~ u ~ /, a < Y - ’ / ~ induced from the Siege1 parabolic subgroup of H(R). It is the Aolift of T 2 k + l x J 1 2 and A-lifts to the induced 1 ( 7 r ~ k + 1 , ( 1 2 ) of PGL(4,R). The central character of 7r2k+1 is trivial, but that of 7r2k is sgn. Hence 1 ( n 2 k 1 t2)defines a representation of GL(4, IR) but not of PGL(4, EX). The endoscopic map 7f!i

:

w,x SL(2,C) --+

LCO = SL(2,C) x

SL(2,C)

3 I?,

defines

I?

+

c SL(4,C ) since 2k 1 is odd. which lies in As in [Ty] we write 7 r i f l , k - l for L ( E U 1 / 2 7 r 2 k + l , E v - 1 / 2 ) , k 1 0 . Now ’ = 7rZiE and 7r2,EIHois irreducible. In the Grothendieck group the induced decomposes as

Here

7rz:l,l

a < Y - ~ /is~ a

which is equivalent to J v k + l x ( v k >a ( v - ~ - ’ / ’ (using the Weyl group element (23)). Its infinitesimal character is ( k f l , k) = ( 2 , 1 ) + ( k - l ) k-1). We have

6. Finite Dimensional Representations

255

In summary, H i j ( , 8 Va,b) is 0 except in the following four cases, where it is @. (1) One dimensional case: ( a ,b ) = (0,O) and 7r is xrf, -irk$, JH, 7ro,o 1

= L ( U sgn, u - ’ / ~ T ~ ) , xi;$= L ( c u ’ / ~~ ~ u -~l l,‘ ) .

(2) Unstable nontempered case: ( a ,b) = ( I c , Ic) ( k 2 1) and 7rhol

7r::3,1,

2k+3,1>

7r

is

2,E - L u 1 / 2 “ k , k - (< 7r2k+3> b 2 1, a b even, and x is 7rky,’k2. Here kl = a b t 3 > 1c2 = a - b + 1 > 0 are odd.

+

,z:k2,

6. Finite Dimensional Representations The Q-rational representation ( p , V) of H’ = RF/QH has the form (h,) H @p,(h,), where H‘ = H,, H , = H , over and p, is a representation (irreducible and finite dimensional) of H,. Here L ranges over S = G a l ( a / Q ) / G a l ( a / F ) , = Hom(F,R) and so H‘ = { ( h , ) ; h, E H } . The Galois group Gal(a/Q) acts by ~ ( ( h ~ = )( )( ~ h ‘ ) ~=, )( ( ~ h ~ - t , The fixed points are the the (h,) with h, = ~ h lwhere , hl ranges over H ( F ) (the “1”indicates the fixed embedding F R). Thus H’(Q) = H ( F ) and H’(R) = H(R) with I S1 = [ F : Q]since F is totally real; S is the set of embeddings F c_t R. Now the representation p is defined over Q,namely fixed under the action of Gal(a/Q). Thus B L p L ( h L= ) @ ~ ~ p ~ ~ The (~h, element h = ( h l ,1,.. . ,1) (thus h, = 1 for all L # 1) is mapped by T to (1,.. . ,1,~ h l1,. , . . , 1) (the entry ~ h isl at the place parametrized by T ) . Hence p l ( h 1 ) equals pT(7h1) (both are equal to p(h)(= p ( 7 h ) ) ) . Hence PT = % I ( : hl &) ‘ h l ) ) ,and the components pT of p are all translates of the same representation p1. For (h,) = ( ~ h lin) H’(Q) = H ( F ) , p ( ( h L ) )= @ A ( L ~ I ) = @ L P I ( ~ I )= m ( h 1 )@ ... 8 p ~ l ( h l () [ F: Q]times). However, over F we have H‘ N H , with H , = H . An irreducible representation ( p , V ) of H’ over F has the form (pa,b = @,~spa,,b,, Va,b = @,E~Va,,b,),where a, 2 b, 2 0, even a, - b, for all w E S.

a,

n,

n,

-

nuES


256

7. Local Terms at 00 Next we wish to compute the factors at m of each of the terms in the trace formulae STFH(f H ) and TFc, (fc,).The functions f H , c o (= h, of [ K o ~ ] ) and fc,,m are products @f H v and €9fc,, over u in S. We fixed a finite dimensional representation

even a, -b, for all u E S , over F of the group H’ over Q. Denote by {,,TH,} the packet of discrete series representations of H(R) with infinitesimal character (a,, b,) (2,l). For any (p,, Va,,b,), the packet { p ~ ~ vconsists } of two representations, 7r+ = Wh ~ k l , , k z V and p r i , = TiTjchpAI,..,, where k l , = a, b, f 3 > k2, = p H, a, - b, 1 > 0 are odd. It is the &,-lift of the representations T k I v x 7rkzv and T k z v x T k l , of Co(R)= PGL(2,iR) x PGL(2,R). Denote by h(riTjchp;, k z v ) a pseudo coefficient of the indicated representation. Then

+

+

+

~(TZ:,~

Put

Note that if 7rkl = T k z , then X o ( T k l X T k z v ) = 1 % 7Tk1 I which is not discrete series but properly induced. In particular, the fifth term I ( H , 5) of sTFH(fH), and the corresponding terms of I ( C 0 , 2 ) in T F c o ( f c a ) those which are parametrized by r 2 x 012 where r2 is cuspidal whose components at 03 are 7r1, vanish for our functions f H , fc,. Moreover, as explained at the end of section 6, I ( H , 4 ) and I ( C 0 , 3 ) are 0 for our f H ,

fc,.

7. Local Terms at

257

DC)

Note that q(H’) = [ F : Q ] q ( H )is half the real dimension of the symmetric space attached to H’(R), and q ( H ) is that of H ( R ) ,thus q ( H ) = 3 in our case. ( h2~v, , = ) tr rr,hy: , k z v ( h ~ , , = ) f (- l ) q ( H ) = When Then tr 7-rz:,k (a, b) = ( 2 k , 0 ) , k L 0, we have in addition tr L ( v sgn, ~ - ~ / ‘ n 2 k + 2 ) ( h ~ , , = ) 1. When (a, b) = ( k , k ) , k 2 0, we have tr L(Iv1/2.2k+3,E2/-1/2)(h~,,) = f. When ( a , b ) = ( O , O ) , we have in addition t r [ ~ ( h H , = ~ )1, 1’ = 1. Note that if T H contributes to I ( H , 1)’ then its archimedean components 7 - r have ~ ~ infinitesimal characters of the form ( 2 k , , O ) , k , 2 0, for all u E s. If T H contributes to I ( H , 3) then there is a contribution to I ( C o , 2 ) , and the archimedean components T H , have infinitesimal characters of the form (k,, kv),k , L 0, for all u E S. If 7 - r ~contributes to I ( H , 1)s the infinitesimal characters of its archimedean components are (0,O).

4.

V. GALOIS REPRESENTATIONS 1. Tempered Case We apply the Lefschetz formula in Deligne's conjecture form to the &ale cohomology

H,*( S K f @E

IF,va,b;X)

with compact supports and coefficients in the representation (Pa$, Va,b), even a, - b,, for all v E S . Suppose 7 r occurs ~ in the stable spectrum, namely in I ( H , 1)l. The choice of the function f m H guarantees that the components T H , lie in the packet 7 r ~ ~ ~ , k z , k} lr , = a, b, 3 , ka, = a, - b, 1, at each archimedean place v E S. We start by fixing a cuspidal representation T H with TH, in the set {7rz:,kz, } for all w in S and with 7KrHi# 0. In particular the component a t p of such T H is unramified, of the form @ u l p ~ ~ ( p pl ,~, , ) . We use a correspondence which is a KfP-biinvariant function on H(APf).Since there are only finitely many discrete series representations of H ( A ) with a given infinitesimal character (determined by p ) and a nonzero K Kf-fixed vector, we can choose to be a projection onto {T,;}. Writing

{~z;,~~~,

7

+ +

+

4p:>,,,

fg,

fg

K

t,,

for p m U ( r U )m , = 1, 2, the trace of the action of Fri on the { r H f f isotypic component of H , ' ( S K ~@IFF, V p )(which vanishes outside the middle dimension 3[F : is multiplication by (we put j, = (j,n u ) )

a]),

Note that H,$F:Q1occurs in the alternating sum H,' with coefficient (-l)3[":41.This sign is canceled by the sign ( - l ) q ( H ' ) of the definition of the functions f H f , m = B u E S h H w . Thus the {.rrz}-isotypic part of H$F:Q1(namely the 7rZ-isotypic part for each member of the packet) is of the form { r z }@ p ( { r ~ } ) .Here 258

1. Tempered Case

259

p( ( 7 r ~ ) ) is a 4[F:Ql-dimensionalrepresentation of Gal(a/Q). The 4#{”IP) nonzero eigenvalues t of the action of Fr, are p$[F’Ql t$:),”, m(u) E (1,a}, L ( U ) E {fl}. This we see first for sufficiently large j by Deligne’s conjecture, but then for all j 2 0, by multiplicativity. Deligne’s “Weil conjecture” purity theorem asserts that the Frobenius eigenvalues are algebraic numbers and all their conjugates have equal comi/2 plex absolute values of the form q , (0 5 i 5 2 d i m s ) . This is also referred to as “mixed purity”. The eigenvalues of Fr, on I H i have complex absoi/2 lute values equal q, , by a variant of the purity theorem due to Gabber. We shall use this to show that the absolute values in our case are all equal

nUlp

In our case E = Q,the ideal p is ( p ) , the residual cardinality to q$ q , is p , and n, = [IE, : Q,] is 1. Note that the cuspidal 7r define part not only of the cohomology

H$%,

@E

a,V)

but also part of the intersection cohomology IH’(SL, 8~0 , V ) . By the Zucker isomorphism it defines a contribution to the L2-cohomology,which is of the form 7ryf 8 Hi(g,K,;7r, @ V,(C)). We shall compute this (g,K,)-cohomology space to know for which i there is nonzero contribution corresponding to our 7 r j . We shall then be able to evaluate the absolute values of the conjugates of the Frobenius eigenvalues using Deligne’s “Weil conjecture” theorem. The space HaJ (8, K ; 7r 8 V a , b ) is 0 for 7r = 7r;t,kz, * = Wh or hol, kl > k2 > 0 are odd (indexed by a 2 b 2 0) except when (i,j) = (2, l), (1,2), (3,0), (0,3) (respectively), when this space is C From the “MatsushimaMurakami” decomposition of section 2, first for the L2-cohomology H ( z ) but then by Zucker’s conjecture also for IH*, and using the Kiinneth formula, we conclude that I H i ( 7 r f ) is zero unless i is equal to dimSK, = 3[F : Q], and there dimIH2[F:Q](7rj)is 4[F:Q1 (as there are [F : Q]real places of F ) . Since 7rj is the finite component of cuspidal representations only, ~f contributes also to the cohomology H:(SK, V a , b ; ~only ) in dimension i = 3 [ F :Q ] , and dimH2[F:Q1(7rf)= 4[F:Ql. This space depends only on the packet of 7rj and not on 7 r j itself. Deligne’s theorem [D6] (in fact its IH-version due to Gabber) asserts that the eigenvalues t of the Frobenius Fr, acting on the t-adic intersection cohomology I H i of a variety over a finite field of qp elements are algebraic

a,

V. Galois Representations

260

and “pure” , namely all conjugates have the same complex absolute value, of hence the eigenvalues the form q$2. In our case i = dimSKf = 3[F : Q], 3 [F:Q]

in of the F’robenius are algebraic and each of their conjugates is q; absolute value. Consequently the eigenvalues pl,, p2, are algebraic, and all of their conjugates have complex absolute value 1. Note that we could not use only “mixed-purity” (that the eigenvalues are powers of qk’2 in absolute value) and the unitarity estimates lpm,l*l < q:I2 on the Hecke eigenvalues, since the estimate (less than (6) away from (&JdimS) does not define the absolute value uniquely. This estimate does suffice to show unitarity when d i m s = 1. In summary, the representation p = p(7rHf) of Gal(o/Q) attached to THf depends only on the packet of T H f , its dimension is dFzal.Its restriction to Gal(a,/Q,) is unramified, and the trace of p(Fr,) on the is equal to the trace of @vc1/2r,(t(7rHu) x {7rz}-isotypic part of H2[F:Q1 Fr,). Here (r,, (C4)[Fu:Q~]) denotes the twisted tensor representation of LRFu/QpH = f i [ F u : Q p l x Gal(F,/Q,), F’r, is Frp:Qplland v, is the character of LRFu/Qp H which is trivial on the connected component of the identity and whose value at Fr, is qL1, where qu. = pIFu:Q~l. The eigenvalues of t(7rHu) and all of their conjugates, lie on the complex unit circle. We continue by fixing a cuspidal representation T H with T H in ~ the set 7rLpt,k,, } for all v in S and with 7rHKf # 0. But now we assume it occurs in the unstable spectrum, namely in I ( H ,2). We fix a correspondence f&which projects to the packet {7rLf}. Since the function f& is chosen to be matching f&, by [F6] the contributions to I(C0,l) are precisely those parametrized by 7r1 x 7r2 and 7r2 x 7r1 , where 7rm are cuspidal representations of PGL(2,A) whose real components are {7ri17r,”} = {7rklv,7rk2,}, a set of cardinality two. Write {THf}’ for the set of ?rHf = @ w 7 r ~ w rw < m, which are the finite part of an irreducible T H in our packet { T H } , such that T H is ~ 7rGw for an even number of places w < a.Similarly define { T H ~ } - by replacing “even” with “odd”. The contribution of { T H } to I ( H ,2) is {7r?2,k2,

1

1. tEMPERED cASE

261

Here and below f& indicates - as suitable - its product with the unit element of the H'(Zp)-Hecke algebra of H ' ( Q p ) . The corresponding contribution to I(C0,l) is twice (from 7r1 x 7rz and 7r2

x

7r1)

.P5 d i m SK

f

n(g7 + t;-./j" - glu

-

t;-p)j,

UlP

By choice of jgo we have that tr(7r; x 7r;)(f,?,) = tr{7rHf}+(fG) tr{nHf}-(f&). The choice of h H v is such that tr{7rrgv}(hHv) = (-1) q ( W , and tr(7rh x 7r,2)(hCov)is ( - ~ ) q ( ~if7rA ) x 7r,2 is 7rkl, x 7 r k Z v , and - ( - l ) q ( H ) if it is ?rkzv x rk,,. K we conclude that for each irreducible 7rHf E {THf}', the 7rH;-isotypic part of H: is zero unless i = 3[F : Q] (middle dimension), in which case it is 7 r g P({7rHf}+), ~ and Fri acts on P({xHf}+) with trace

+(-1)WXT?

.,7.t(JJ

+t

; - p - (gtu + t ; p " ) ) 3 " ] .

"IP

We write 1 (Tv,

n(7r'

= ("kz,

x Y

7r2)

for the number of archimedean places u of F with

rkl,).

Similarly, for each irreducible THf E {7rHf}-, the 7r:;-isotypic part of H: is zero unless i = 3[F : Q] (middle dimension), in which case it is 7rHf Kf 8 p ( { n H f } - ) , and Fri acts on p ( { 7 r H f } - ) with trace


262

As usual, we conclude from Deligne’s mixed purity [D6] that the Hecke eigenvalues t,, are algebraic and their conjugates all lie in the unit circle in C.

2. Nontempered Case Next we deal with the case of T H which occurs in I ( H , 1 ) 2 , namely in a quasi packet { L ( [ v ,v-1/27r2)} which A-lifts to the residual representation J(v1l27r2,v-1/27r2) of G(A), G = PGL(4). Here 7r2 is a cuspidal representation of GL(2,A) with quadratic central character [ # 1 and En2 = 7 r 2 , and 7r: = 7r2kV+2 at each v E The infinitesimal character of T H is ~ (2k,, 0) (2, l ) , k , 2 0, for all ‘u E S . Choosing :f to project on 7rZ for such T H , we note that there are no contributions from the endoscopic group CO,thus I(C0, i ) are zero. Namely the contributions to I ( H , 1 ) 2 are stable. The result for Shimura varieties associated with GL(2) assures us that the Hecke eigenvalues, or Satake parameters, of each component 7rE of 7r2 a t ulp are algebraic and their conjugates have complex absolute value one. Alternatively we can conclude that the components above p are all - unramified - of the form L([,v,, where 7r: = 7r(plu,t U / p 1 , ) , [:= 1, p1, unramified and equals 1 or -1 if [, = -1. As noted in section 12,

+

s.

with zl, = PI,, z2, = [,/p1,, where we write p1, and [, also for their values at x u . Using the estimate qZL1I2 < Iz,,l < q:I2 we conclude from Deligne’s theorem [D6] that plU. is algebraic and the complex absolute value of each of its conjugates is equal to one. On the 7r$-isotypic part of the cohomology, Fr, acts with the 4#{uIP) eigenvalues p i where

dim sK

nu,, a,,

Note that T H , = L(sgnv,, ~ ~ ~ / ~ 7 r ~ has k , +H 2i 3) ( ~ ~ , € 3 V a , , b ,# ) 0 only when a, = 2k,, b, = 0, and ( i , j ) = (2,0), (0,2), (3, l ) ,(1,3).

2. Nontempered Case

263

Next we deal with the case of T H which occurs in I ( H , 3 ) , namely in a quasi-packet Xo(7r2 x 5 1 ~ where ) ~ 7r2 is a cuspidal representation of PGL(2,A) whose real components are r 2 k v + 3 , and E is a character of A x / F X A x 2 . There are corresponding contributions in I(C0,2) from 7r2 x 5 1 2 and from E l 2 x 7r2. The infinitesimal character of T H , is ( k , , k , ) , k , 2 0, for all 21 E 5'. We fix a correspondence f$ which projects to the packet {7rgf}.Since the function f :, is chosen to be matching by [F6j the contributions to I(C0,l) are precisely those parametrized by 7r' x E l 2 and E l 2 x 7r2. Write { T H ~ } ' for the set of THf = @ w ~ ~ w w ,< 00, which are the finite part of an irreducible T H in our quasi-packet {TH}, such that n~~ is T;, for an even number of places w < 00. Similarly define { n ~ f }by replacing "even" with "odd". The contribution of { T H } to I ( H ,3) is

fg,

H Here

and

and we abbreviate

The corresponding contribution to I(C0,2) is twice (from 7r2 x 6 1 2 and from ( 1 2 x r 2 )

By choice of

we have that The choice of is such that and if it iss and

is

is


264

We conclude that Fri acts on the r;;-isotypic pic past of H,*, for each irreducible 7rH f E

(7rH f

} ’, with trace $ p $ dim s K f times

K Similarly, I3-iacts on the 7rH;-isotypic past of H,*, for each irreducible

T H E ~ ( T H ~ } - , with

trace i p $ dim S Kf times

Note that I T H , = L ( ~ , u ~ / 2 7 r ~ k + 3 , ~ , uhas ~ 1 fH2 i) j ( 7 r ~ C,3 Va,,b,) # 0 only when a, = k,, b, = k,, and ( i , j ) = (1,l)or (2,2). As usual, we conclude from Deligne’s mixed purity [D6] that the pmu are algebraic and their conjugates all lie in the unit circle in @. Finally we deal with the case of a one dimensional representation T H =

&, which occurs in I ( H , l ) 3 . We can choose f; to factorize through a projection onto this one dimensional representation T H = 5 such that # 0. The infinitesimal character of T H , is (0,O) for all w E S. In particular the component at p of such T H is unramified, and the trace of the action of Fr: on the 7rHf-isotypic component of H, * ( S K~ 8~ V,) is

7 r z

a,

2. Nontempered Case

265

Note that H ~ ( s p ( 2 , R ) , S U ( 4 ) ; Cis) C for ( i , j ) = (O,O), (111),( 2 , 2 ) , ( 3 , 3 ) and (0) otherwise. Thus T H = J H contributes only to the (even) part CB ( S K f (8%IF,1). O<m 0, and -1 H jW,", where W i is the unit circle C1 = { z / F ; z E C"}= ker N c p . The norm map N : Wx --$ R:o induces a norm z1 j z 2 F+ zlTi1 z2Z2 on W R . When F is local nonarchimedean, for each finite extension E of F in F let k. = R E / ( r E )be the residual field of E and q E its cardinality. P u t k = U E k E and k = k F . Then

+

+

1

--f

IF

--$

Gal(F/F)

-+

Gal(z/k)

-+0,

I . Class Field Theory where I F is the inertia subgroup, consisting of the Ic. The Galois group Gal(x/k) is the profinite group

-

273

E Gal(F/F) fixing

2 = limZ/nZ, c

topo-

logically generated by z t-) xqF. Any element of Gal(F/F) which maps to this generator is called Frobenius and denoted by FrF. Then WF is the dense subgroup of Gal(F/F) generated by the Frobenii. Thus the sequence 1 4 I F 4 WF -+ Z4 1 is exact. The subgroup I F is a profinite subgroup of Gal(F/F), and open in WF, making WF a topological group. Then p : WF 4 Gal(F/F) is the inclusion and T E : E X 4 WEb are the reciprocity law homomorphisms, r E ( a ) acts as z H zIalEon the valuation being normalized by I T E I E =qil. When F is a global function field the situation is similar to the previous case, with “residual field” replaced by “constant field”, “inertia group IF” by “geometric Galois group Gal(F/Fx)”, and the absolute value l a l ~of the idkle class a = (a,)E CE is lawlv. When F is a number field Weil gave an abstract, cohomological construction of WF, and asked for a natural construction. He showed that p : WF 4 Gal(F/F) is onto. Its kernel is the connected component of the identity in WF. The isomorphism r F : CF WFb and the absolute value CF -+ z = (z,) t-) 1 x 1 ~= Izvlw,define the norm WF 4 I R:,, w H 1wJ. Since WEb C W$b corresponds via r E and T F to N E I F : CE 4CF and INEIFUIF= l a l ~the , restriction of WF 4 R:, to WE coincides with the and we write simply (w( instead of ( w ( F .The kernel Wk norm WE 4 of w H IzuJ is compact. The image of w H JwI is & and WF is Wk >a Z in the nonarchimedean and function field cases, while in the archimedean : R and WF is Wk x R:,,. and number field cases the image is , Finally there are commutative squares of local-to-global maps, for each v, WF~ Gal(F,/F,)

z,

n,

n,

-

+

1 WF

1 Gal(F/F).

Class Field Theory, which asserts that WFb N C F ,can then be phrased as an isomorphism between the set of continuous, complex valued characters of WF, and the set of continuous, complex valued characters of CF (= h X / F Xglobally, F X locally). One is interested in all finite dimensional

274

I. On Automorphic Forms

(continuous, over C)representations of the Weil group W F ,as by the Tannakian formalism these determine WF itself as the "motivic Galois group'' of their category. The hypothetical reciprocity law would associate to an irreducible n-dimensional representation X : WF -+ GL(n, C)a cuspidal representation 7r of GL(n, A) if F is global and of GL(n, F) if F is local, and to $~='=,X, the representation Ip(7r1,. . . , 7 r T ) normalizedly induced from the . .@T, . of the parabolic P (trivial on its unipocuspidal representation ~ 1 8 tent radical) of type (dim XI,. . . , dim A,.), where A, H T,. We postpone the explanation of the new terms, but note that this new correspondence is defined similarly to the case of CFT, which is that of n = 1. The local analogue has recently been proven (by Harris-Taylor [HT], Henniart [He]) in the nonarchimedean case, and by Lafforgue [Lf] in the function field case. Once the connection between n-dimensional representations of W F and admissible (locally) or automorphic (globally) representations is accepted, one would like to include all admissible and automorphic representations. For that the group WF has to be replaced by a bigger group, which is the Weil-Deligne group WF x SU(2,R), an extension of W F by a compact group (see [D2], [Tt], Kazhdan-Lusztig [KL], and Kottwitz [Ko2], 312) when F is nonarchimedean (when F is archimedean the group remains W F ) . This is necessary for inclusion of the square integrable but noncuspidal representations of GL(n, F) in the reciprocity law. The representations X of Wp x SU(2,R) of interest are analytic in the second variable, thus extend to SL(2,C). We embed Wp in WF x SL(2,C) by w H w x diag()w)1/2, )wJ-~/~ where ), 1.1 : W p 4 F X + Cx is the composition of the usual absolute value with W$b pv F X . For example, the Steinberg representation of GL(n, F ) is parametrized by the homomorphism X which is trivial on WF while its restriction to SL(2, C)is the irreducible n-dimensional representation Sym"-l; it maps diag(a, u-')

to diag(a("-l)l2,

1 " .

,

and

(i :> to the

regular unipotent matrix exp( ( ~ 5 ( ~ , ~ + ~ ) ) The ). nontempered trivial representation of GL(n,F) is the quotient of the normalizedly induced repn+1 resentation I(p1, . . . , p n ) of G L ( n , F ) , with pt = v z - ' , v(x) = 1x1, while the square integrable Steinberg is a subrepresentation. The quotient is parametrized by X trivial on the second factor, SU(2, R), and with X(W) = diag(pl(w), . . . p n ( W ) ) , w E W F . 7

2. Reductive Groups

2 75

If the reciprocity law holds, the category of the representations 7r should be Tannakian, with addition 7r1 EE . . . EE rr being normalized induction Ip(7r1,. . . , 7 r T ) , and multiplication 7r1 I x I ~ ~ + I x 1 7 rand r , fiber functor. At least when considering only those representations formed by twisting tempered representations, and assuming the Ramanujan conjecture ( “cuspidal representations of GL(n,A) are tempered, i.e. all their components are tempered”), if the category is Tannakian, its motivic Galois group is expected to be the correct substitute for W F ,for which the reciprocity law holds. This hypothetical group is denoted L F , named the “Langlandsgroup”. We often write WF below for what would one day be L F . See Arthur [A51 for a proposed construction.

2. Reductive Groups Since progress on the global reciprocity law for GL(n) is not expected soon, one looks for a generalization to the context of any reductive connected F-group G. This is not a generalization for its own sake, as it leads to two practical developments. The first is the theory of liftings of representations of one group to another. Reflecting simple relations of representations of Galois or Weil groups, one is led to deep relations of automorphic and admissible representations on different groups. The second is the use of Shimura varieties (see [D5]) to actually prove parts of the global reciprocity law for groups which define Shimura varieties (symplectic, orthogonal and unitary groups, but not G L ( n ) and its inner forms if n > 2 ) , and for LLcohomological’l representations, whose components at the archimedean places are discrete series or nontempered representations with cohomology # 0. The reciprocity law for G is stated in terms of the Langlands dual group LG = 3 x W F ,where 3 is the connected component of the identity of LG, a complex group, and W Facts via its image in Gal(F/F). The law relates homomorphisms X : WF + LG whose composition with the projection to WF is the identity, with admissible and automorphic representations of G ( F ) or G(A), in fact with packets of such representations. It was proven by Langlands [L7] for archimedean local fields, as part of his classification of admissible representations of real reductive connected groups, and for

276


tori in [L8]. Globally it is compatible with the theory of Eisenstein series [L3], [MW2]. For unramified representations of a padic group it coincides with the theory of the Satake transform, and for representations with a nonzero Iwahori fixed vector it was proven by Kazhdan and Lusztig [KL]. These results, and those on liftings and cohomology of Shimura varieties, in addition to the local and function field results for GL(n), give some hope that the reciprocity law is indeed valid. Of course, a final form of this law will be stated with LF replacing the Weil group W F ,once LF is defined. We proceed to review the definition of the connected dual group and the L-group LG, following Langlands [Ll], Borel [Boll, Kottwitz [KO217 51. Books on linear algebraic groups include Borel [Bo2], Humphreys [Hu], Springer [Sp]. Associated with a torus T defined over F is the characters lattice X * ( T ) = Hom(T, G,) and the lattice X , ( T ) = Hom(G,, T ) of 1-parameter subgroups, or cocharacters. These are free abelian groups, dual in the pairing

(., .) : X , ( T ) x X * ( T ) -+ Z = Hom(G,,G,)

T^

The connected dual group of T is the complex torus = Hom(X,(T), C " ) . Then X * ( @ = X , ( T ) , and by duality X,(?) = X * ( T ) . Thus T H ? interchanges X , and X * . As T is defined over F, Gal(F/F) acts on X , ( T ) , hence on ?. An action of Gal(F/F) on ?, or LT = ? x W F ,determines T as an F-torus (up to isomorphism), since the F-isomorphism class of T is determined by the Gal(7S;lF)-module X , ( T ) ( = X*(!?)). The Gal(F/F)action is trivial iff T is an F-split torus. Let X , X v be free Z-modules of finite rank, dual in a Z-valued pairing < .,. >. Suppose V c X , Vv c X v are finite subsets and a H a", V 4 V v , is a bijection with < a , av > = 2. The 4-tuple ( X ,V, X " , Vv) is a root datum if the reflection sol(.) = z - (z,a")a (z E X ) maps V to itself, and sav(y) = y - (a,y)a" ( y E X v ) maps Vv to itself. Then V is the set of roots and V" the set of coroots. The root datum is called reduced if a and na in V ( n E Z) implies that n = fl. The set V defines a root system in a subspace of the the vector space X @ R. Thus one has the notions of positive roots and simple roots. If A = { a } is a set of simple roots, put Av = {a"}. The 4-tuple Q = ( X , A , X V , A V ) is called a bused root datum (it determines the root datum). The dual

2. Reductive Groups

277

based root datum is S" = ( X V , A V , X , A ) and , the dual root datum is ( X V ,V", X I V). A Borel pair ( B , T )of a reductive connected F-group G is a maximal torus T of G and a Borel subgroup B of G containing T , both defined over F . If G has a Borel pair defined over F , it is called quasisplit. It is split if there is such a pair with T split over F . Any pair ( B ,T ) defines a reduced root datum Q ( G , B , T )= ( X * ( T ) , A , X , ( T ) , A " ) .Here A = A ( B , T ) c X * ( T )is the set of simple roots of T in B , and Av = Av(B,T ) c X , ( T ) is the set of coroots dual to A. Any two Borel pairs are conjugate under the adjoint group Gad = G/Z(G) of G (here Z(G) denotes the center of G). If Int(g) (x H gxg-l) maps ( B ,T ) to (B',TI), it defines an isomorphism S(G,B , T ) 2 @(GIB', TI), independent of g . Using this, we identify the based root data, to get S ( G ) . Then Aut(G) acts on S ( G ) , with Gad acting trivially. A connected dual group for G is a complex connected reductive group with an isomorphism S(z) S(G)". The map G H S ( G ) defines a bijection from the set of F-isomorphism classes of connected reductive groups G to the set of isomorphism classes Ga determines an of reduced based root data 9.An isomorphism G1 isomorphism Q(G1) 2 Q(Gz), which in turn determines G1 Gz up to an inner automorphism. This classification theorem implies that a connected dual group of G exists and is unique up to an inner automorphism. It depends only on the F-isomorphism class of G. If ( B I T )is a Borel pair for G and ( B , S ) is a Borel pair for G, there exists a unique isomorphism ? (defined from T ) 4 s^ inducing the chosen isomorphism S ( C )= ( X * ( s ^ ) , & X * ( 3 ) , P 1 )

z

r

z

A

h

s(G)" = (x,(T) = X * ( ? ) , A ~ , X * ( T= ) x*(?),A). If f : G -+ G' is a normal morphism (its image is a normal subgroup), and ( B ,T ) is a Borel pair in GI there exists a Borel pair ( B ' ,T ' ) in G' with f ( B ) c B', f ( T )c T'. Hence there is a map S(f): S ( G ) + S(G') and a dual map a"(f) : S(G')' 4 Q(G)",and so a map f : G' -+ Any other such map has the form Int(t) . f .Int(t') (t E T^,t' E ?'), mapping ?' to ?, E' to 6. h

h

e.


2 78

The simplest example is that of G = GL(n). Then X * ( T )= Z"has the standard basis {ei;1 5 i 5 n } , and X , ( T ) = Z"the dual basis {ey(= ei)}. Also A = {ei - ei+l; 1 5 i < n } and Av = {e," - e:++,; 1 5 i < n}. Then 9 ( G ) = Q(G)" and = GL(n,C). A more complicated example is G = PGSp(2) = {g E GL(4), t g J g =

AJ}/Gm, J =

yk

(_",

w

=

( y i) , the projective symplectic group of

similitudes of rank 2. ith the form J , a Bore1 subgroup B is the upper triangular matrices, and a maximal torus T is the diagonal subgroup. The simple roots in X * ( T ) = Z2 are a = e l - e2, ,8 = 2e2, and the other positive roots are a+P = e1+e2, 2a+P = 2el. Then Av = {av = el-ez, Pv = e2}. The isomorphism from the lattice

where

(2, y,

z , t ) takes diag(a,b, b-', a-') in

is given by (z, y, z , t ) H (z

5;to c ~ " - ~ b Y - ' ,

to the lattice

+ y, z + z , y + t , z + t ) ,with inverse

The isomorphism L* : X,(T^) 2 X * ( T ) dual to L : X , ( T ) 2 X*(T^) is defined by ( L ( u ) , v )= (u,~ ' ( w ) ) . Thus w = ( a ,b, -b, -a) E X , ( T ) maps to the character A

of T . The character 77 : diag(a,P,y,S) H p 1 ( a / y ) p 2 ( a / P ) of T corresponds to the homomorphism

By an isogeny we mean a surjective homomorphism f : G + G' of algebraic groups whose kernel is finite and central (in G). The finite kernel is always central if char F = 0, and if char F > 0 our f is usually named central isogeny.

2. Reductive Groups

279

A connected (linear algebraic) group is called reductive if its unipotent radical (the maximal connected unipotent normal subgroup) is trivial, and semisimple if its radical (replace "unipotent" by "solvable" in the definition of the unipotent radical) is trivial. A semisimple group G is called simply connected if every isogeny f : H -+ G, where H is connected reductive (as is G), is an isomorphism, and adjoint if every such f : G -+ H is an isomorphism. The adjoint group of a reductive G is Gad = G/Z(G),where Z(G) is the center of G. This Gad is adjoint. The derived group Gder of a reductive G (the closure of the subgroup generated by commutators [x,y] = zys-ly-') is semisimple, denoted also by G"". Let G be semisimple and Q(G) = ( X ,A, X " , A"), V the root system with basis A and Vv the root system with basis A". Then G is simply connected iff the lattice of weights P ( V )c X @ Q of V is X , and adjoint iff the group Q ( V ) generated by V in X is X . Since P ( V ) = {A E X @ Q ;(A,VV)c Z} and

P ( V v ) = {A

E

X" @ Q;(A, V)

E

Z},

e

G is simply connected iff is adjoint, G is adjoint iff 6 is simply connected. A simple group G (one which has no nontrivial connected normal subgroup) is characterized - up to isogeny by its type A,, . . . ,Gz. The map Q(G) -+ Q(G)" interchanges B, with C,, and fixes all other types. Thus the connected dual of a simple group is a simple group of the same type unless G is of type B, or C,, and duality changes simply connected to adjoint. The classical simply connected simple groups and their duals are in type A, : S L ( n ) , PGL(n,C); B, : Spin(2n+1), PGSp(2n,C); C, : Sp(2n), SO(2n 1,C); D, : Spin(2n), P0(2n, C). The dual group, or L-group, LG = x W F ,is the semidirect product of the connected dual group with the Weil group W F ,which acts on G^ via its image (by p) in Gal(F/F). To explain how Gal(F/F) acts, note that we have a split exact sequence ~

+

e

1 4 InnG

-+

e

AutG

-+

OutG

-+

1,

where Inn G = Int G P Gad is the subgroup of inner automorphisms of G, and the group Out G = Aut G/ Inn G of outer automorphisms is isomorphic to the group Aut Q(G)= Aut Q(6)of automorphisms of Q(G) (or Q(G^)).

I. O n Automorphic Forms

280

A splitting for 2 is a triple C = (5, T^,{Xa; Q E A}), where X , is an aroot vector in L i e 6 for each simple root Q of T^ in 6. The set of splittings is a principal homogeneous space for (the action of) Gad (by conjugation). A choice of a splitting C determines a splitting Aut !I!(@ + Aut G of our exact sequence: an element of Aut Q(G)maps to the unique automorphism of G fixing C. The action of Gal(F/F) on !I!(G)" = then lifts to an action on which fixes the fixed splitting C. The L-group LG = >a WF depends on the choice of C, but a different choice gives rise to an isomorphic L-group. If v is a place of the number field F there is a natural WF-conjugacy class of embeddings W F ~ W F ,hence such a class of embeddings LG/F,, L G I F which restrict to the identity 6 + 6. A

Q(e)

e

e

~f

3. Functoriality The purpose of the principle of functoriality is to parametrize the admissible representations of G ( F ) in the local case, and automorphic representations of G(A) in the global case, in terms of L-parameters. These are the (continuous) homomorphisms X : LF + LG = x W F ,where LF is W , if F is archimedean and W F x SU(2, R) if F is nonarchimedean, such that X followed by the projection to WF is the natural map LF + W F ,przoX is complex analytic if F is archimedean, and prE(X(w)) is semisimple for all w in LF. Two parameters A, A' are called equivalent if z. A' = Int(g)X for some g in and z : LF -+ Z ( 6 ) such that the class of the 1-cocycle z in H 1 ( L ~ , ~ ( 6 is) locally ) trivial. If Gal(F/F) acts trivially on the center Z ( 6 ) of then

e

e

e

H 1 ( L ~z(6)) , = Hom(LF, z(6)). In this case Chebotarev density theorem for L$b = WFb implies that any locally trivial element of H 1 ( L F , Z ( e ) )is trivial. Thus X(w)= $(w) x 6(w) where 6 denotes the projection L F + W F followed by cp : WF + Gal(F/F), and $ is a (continuous) 1-cocycle of LF in G. The cocycle X'(w) = #(w)x S(w) is equivalent to X iff $ and 4' are cohomologous. Hence the set of equivalence classes, denoted A(G/F), is the quotient of the A

3. Functoriality

281

group H 1 ( L F , of (continuous) cohomology classes by ker[H1(LF,z(G)) @,H1(LF,, Z(G))l.

G)

+

-

F’unctoriality for tori T over F concerns then (continuous) homomorphisms X : WF 4 ‘T = T M WF with prWFOX = idw,, thus X which factorize through the projection LF -+ W F . Langlands [L8] shows that when F is local, H1( W F?) , is canonically isomorphic to the group of E X ) where , E is a finite Galois characters of T ( F ) = HomGal(E/F)(X+(T), extension of F over which T splits. If F is global the group of characters of T ( A F ) / T ( F )is the quotient of H 1 ( W p ,T^) by the kernel of the localization maps , -+ eUH1(W F ?)I. ~, ker[H1(W F ?) Let 7r : G ( F ) 4 Aut V be a representation (which simply means a homomorphism) of the group G ( F ) of F-points of the connected reductive F-group GI on a complex vector space V . In other words, V is a G(F)-module. If F is a nonarchimedean local field, 7r is called algebraic (Bernstein-Zelevinsky [BZl])or smooth if for each vector w in V there is an open subgroup U of G ( F ) which fixes w (thus 7r(U)u = u).Such 7r is called admissible if moreover, for every open subgroup U of G ( F )the space V U of U-fixed vectors in V is finite dimensional. Admissible representations ( T I ,Vl) and ( ~ 2 V2) , are equivalent if there exists a vector space isomorphism A : Vl -+ V2 intertwining IT^ and 7 r 2 , thus A(nl(g)w)= 7r2(g)Aw. In the next few paragraphs we abbreviate G for G(F ) (same for a parabolic subgroup P , its unipotent radical N, its Levi factor M ) , where F is a local field. Put b p ( p ) = 1 det(Ad(p)I Lie N)I for p E P. A useful construction in module theory is that of induction. Let (7, W ) be an admissible M-module. Denote by 7r = I ( T ) = 1(7;G, P ) the space of all functions f : G + W with f(nmg) = bk’2(m)T(m)f(g)( m E MI n E N , g E G). It is viewed as a G-module by ( 7 r ( g ) f ) ( h )= f(hg). Another useful construction is that of the module ITN of N-coinvariants of an admissible G-module IT. Thus if V denotes the space of IT, put ‘VN for V/(7r(n)v- w;n E N,w E V ) . Since the Levi factor M = P / N of P normalizes N , ‘VN is an M-module, with action ‘ T N . Put I T N = b i 1 / 2 t j 7 N . The functor IT +-+ T N of N-coinvariants is exact and left-adjoint to the exact functor of induction. Indeed, this is the content of Frobenius reciprocity ([BZl], 3.13): HomM(7rN,7 ) = HomG(IT, Ind(r; G, P ) ) . Let N be the unipotent radical of the parabolic subgroup opposite to P

282

I. O n Automorphic Forms

(thus P n p = M ) . Unpublished lecture notes of J. Bernstein show that the functor 7r H nw is right adjoint to induction r H I ( r ;G, P ) , namely HomG(Ind(r;G, P ) ,n) = HomM(r, nv). An irreducible admissible representation 7r is called cuspidal if T N is zero for all proper F-parabolic subgroups P of G. Related notions are of square integrability and temperedness. Thus T is square integrable, or discrete ser i e s , if its central exponents decay. A n is tempered if its central exponents are bounded. A cuspidal n is square integrable. A square integrable n is tempered. Cuspidal 7r exist only for padic F . Langlands classification parametrizes all irreducible 7r as unique quotients of induced I ( w S ;G, P ) where r is tempered on M and us is a character in the “positive cone” (see [L7], [BW], [Si]). As for central exponents, they are the central characters of the irreducibles in the T N for proper P . Decay means that these exponents are strictly less than 1 on the positive cone (defined by the positive roots being positive on the center of M ) , and bounded means that these exponents are 5 1 there. All three definitions can be stated in terms of matrix coefficients of 7r. Harish-Chandra used the term “supercuspidal” for what is termed in [BZl] and above “cuspidal”. He used the term “cuspidal” for what is currently named “square integrable” or “discrete series”. If F is R or @, let K be a maximal compact subgroup of G ( F ) . By an “admissible representation of G(F)” we mean a (g, K)-module V , thus a complex vector space V on which both K, and the Lie algebra g of G ( F ) act. The action is denoted n. The action of t obtained from the differential of the action of K coincides with the restriction to t of the action of g, n(Ad(k)X) = n ( k ) ~ ( X ) ~ ( k -( ’k )E K , X E 8). As a K module, V decomposes as a direct sum of irreducible representations of K , each occurring with finite multiplicities. A (g, K)-module ( T I , V1) is equivaIent to (nz,Vz)if there is an isomorphism V1 + fi which intertwines the actions of both K and g. Denote by I I ( G ( F ) )the set of equivalence classes of irreducible admissible representations of G ( F ) ,namely (8, K)-modules when F is IR or C. The local Langlands conjecture, or the local Principle of Functoriality, predicts that there is a partition of the set I I ( G / F ) of equivalence classes of irreducible admissible representations of G ( F ) into finite sets, named (L-)packets, which are parametrized by the set A ( G / F ) of admis-

3. Functoriality

283

sible homomorphisms X of L F into LG, the “L-parameters”. When F is R or C the partition and parametrization were defined by Langlands [L7]. When F is padic, a packet for G = G L ( n , F ) consists of a single irreducible, and the parametrization II(GL(n)/F) = A(GL(n)/F) is defined by means of (identity of) L- and E- (or y-) factors. The parametrization for GL(n, F ) has recently been proven by Harris-Taylor [HT] and Henniart Pel ’ Packets for G = SL(n, F ) can be defined to be the set of irreducibles in the restriction to S L ( n , F ) of an irreducible of GL(n,F). This is done for G = SL(2, F ) in Labesse-Langlands [LL]. Alternatively, packets for G ( F ) = SL(n, F ) can be defined to be the Gad(F)-orbit7rg (where 7rg(h)= r(g-’hg)) of an irreducible 7r, as g ranges over Gad(F) = PGL(n, F ) . Other cases where packets were introduced are those of the unitary group U(3, E / F ) in 3-variables ([F4])and the projective symplectic group of similitudes of rank 2 ([F6]). Although the Gad(F)-orbitof an irreducible representation is contained in a packet, in both cases there are packets which consist of several orbits. In both cases the packets are defined by proving liftings to representations of GL(n, F) for a suitable n, by means of the trace formula and character relations. Such an intrinsic definition is given in [F3] for SL(2). There are several compatibility requirements on the packets II,I and their parameters A. Some are: (1) One element of II,I is square integrable modulo the center Z(G)(F) of G ( F ) iff all elements of II,I have this property, iff X(LF)is not contained in any proper Levi subgroup of LG. (2) One element of II, is (essentially) tempered iff all elements are, iff X ( L p ) is bounded (modulo the center Z ( e ) of resp.). A representation r is “essentially *” if its product with some character is *. (3) A packet should contain at most one unramified irreducible, and be parametrized in this case by an unramified parameter (which is trivial on the factor SU(2, R) and the inertia subgroup I F of W F ) see , below.

e,

The parametrization is to be compatible with central characters. We proceed to explain this (for details see [Ll]).


284

Given a parameter X : L F 4 LG, we define a character of the center Z ( G ) ( F )of G ( F ) as follows. Suppose Z is the maximal torus in Z ( G ) . The normal homomorphism Z -+ G defines a surjection LG -+ L Z , hence a map R ( G / F ) R ( Z / F ) . Duality for tori associates to X E A ( G / F ) a character W A of Z ( F ) . If Z ( G ) is a torus, this is the desired character. If not, choose a connected reductive F-group G1 generated by G and a central torus, whose center is a torus. The normal homomorphism G c-f GI defines a surjection A ( G l / F ) 4 A ( G / F ) . We get a character of the center of G l ( F ) ,and by restriction one of the center of G ( F ) ,independent of the choice of G I . Given a parameter : L F + Z ( e ) x W F ,equivalently E H1(L~, Z ( G ) ) ,where Z(e)is the center of we define a character of G ( F ) as follows. Let H be a z-extension of G (see [Kol]), namely an extension 1 -+ D 4 H 4 G 4 1 of G by a quasitrivial torus D (product of tori RE,FGm,obtained by restriction of scalars from G m ) ,H and D are defined over F and the derived group of H is simply connected, equal to G"". Then the commutative diagram -+

0 and a homomorphism J, : S , --+ {fl}such occurs in the discrete that the multiplicity rn(7r) with which 7r E spectrum of L 2 ( G ( F ) / G ( A )is)

n,

n,

n,

In particular, if 3, and each ZaU are abelian then the multiplicity of 7r is d , if (.,7r) = E,, and'O otherwise. If 3, consists of a single element then the multiplicity rn(7r) is constant on and we say that is stable. In case the quasipackets have nonzero intersection, the multiplicity r n ( 7 r ) will be the sum of the expressions displayed above over all a such that

n,,

7r

E

n,

n,.

7. Endoscopy An auxiliary notion is that of an endoscopic group H of G. It comes up on stabilizing the trace formula, which permits lifting representations from H to G. We recall its definition following Kottwitz [ K o ~ ] .


296

Let G be a connected reductive group over a local or global field F. An endoscopic datum for G is a pair ( s , p ) . The s is a semisimple element of G/Z(G). Put for the connected centralizer ZZ(S)’ of s in The p :

c.

I?

W F -+ Out(%) is a homomorphism (which factorizes via WF + Gal(F/F). We may work with Gal(F/F) instead of W F ) . For each w in WF the element p ( w ) is required to have the form n x w E G x w and it normalizes %. In particular p induces an action of W F on Z(%).Of course W F acts on G and on its subgroup Z ( 6 ) . The map Z(2) ~f Z(%)is a WF-map. The exact sequence 1 4 Z ( e ) + Z ( @ + Z ( % ) / Z ( @+ 1 gives a long exact sequence ( [ K o ~ ]Cor. , 2.3)

The element s E Z(%)/Z(@is required to be fixed by W F ,and its image in

7 r O ( [ Z ( ~ ) / Z (w”) ~)l is in the subgroup .E(s,p), consisting of the elements whose image in H1(F,Z ( @ ) is trivial if F is local, and locally trivial if F is global. An isomorphism of endoscopic data (s1,p1) and ( s 2 , p2) is g E with

e

((Intg)’ is the isomorphism Out(%l) 4 Out(fi2) induced by Intg; Int(g)sl and s 2 have the same image in ff(s,p). Write Aut(s,p) for the group of automorphisms of ( s , p ) . It is an algebraic subgroup of with identity component %. Put

R(s,p ) = Aut(s, p ) / % . An endoscopic datum ( s , p ) is elliptic if (Z(@wF)o c Z(@. Then the 3rd condition in the definition of an isomorphism can be replaced by Int(g)sl = s 2 . An endoscopic group H of G is in fact a triple (H,s,q), where H is a quasisplit connected reductive F-group, s E Z(@, and q : fi 46 is an embedding of complex groups. It is required that (1) q(%) is the connected centralizer Z$(q(s))Oof q ( s ) in GI and that

7. Endoscopy

297

(2) the e-conjugacy class of q is fixed by WF (that is, by c p ( W ~ C ) Gal ( F / F ) ) . We regard Z ( g ) as a subgroup of Z ( g ) . By (2), the WF-actions on Z ( @ and Z ( @ are compatible. Define a subgroup f i ( H / F ) of

analogously to fi(s,p ) above. It is further required that (3) the image of s in Z ( @ / Z ( @ is fixed by W F and its image in T O ( [ Z ( H ) / Z ( G ) ] W Flies ) in B ( H / F ) . An isomorphism of endoscopic groups ( H I, s1, q l ) and ( H z ,s2, q z ) is an F-isomorphism Q : H I + H z satisfying: (1) 171 0 6 and 7 2 are G-conjugate. (6 is defined up to HI-conjugacy; it induces a canonical isomorphism fi(H2 / F ) 2 fi( H I / F ) ) . (2) The elements of fi(Hi/F) defined by si correspond under h

The group Aut(H, s , q ) of automorphisms of ( H ,s, 7 ) contains

Had( F )(= (Int H )( F ) )as a normal subgroup. Put R ( H ,s, 7 ) = Aut(H, S,q)/Had(F). An endoscopic group ( H ,s, 17) determines an endoscopic datum ( q ( s ) ,p ) , where p is the composition

W F+ Aut(E) 2 Aut(Z,-(q(s))o) + OUt(Z,-(~(s))O). Every endoscopic datum arises from some endoscopic group. There is a canonical bijection from the set of isomorphisms from an endoscopic group ( H I ,s l 1 q 1 ) to another, ( H z ,s 2 , q z ) , taken modulo Int(H2), to the set of isomorphisms from the corresponding endoscopic datum ( q ( s 1 ), p 1 ) to ( ~ ( s z )p, z ) , taken modulo Z~(q(s2))'.Thus there is a bijection from the set of isomorphism classes of endoscopic groups to the set of isomorphism classes of endoscopic data. Moreover, there is a canonical isomorphism 31 17) 2 N r l ( s ) ,P ) . We say that ( H ,s , ~ )is elliptic if ( q ( s ) p, ) is elliptic.

NHl

298


Twisted endoscopic groups are defined, discussed and used to stabilize the twisted trace formula in [KS]. For further discussion of parameters and (quasi) packets see [A4]. Let f : G* --+ G be an F-isomorphism of F-groups. It defines a map f : Q(G*)+ Q ( G ) . It is called an inner twist if for every c in Gal(F/F) there is go in G ( F ) with f ( a ( g ) )= Int(g,)(a(f(g))). In this case G* is called an inner form of G. The L-group LG depends only on the class of inner forms of G. In each such class there exists a unique quasisplit form. The L-group determines the F-isomorphism class of the quasisplit form. The Galois action on is trivial iff G is an inner form of a split group. The L-parameters of G are only those which factorize through L P for an F-parabolic subgroup P of the quasisplit form G* of G , provided P is relevant, namely is an F-parabolic subgroup of G itself. The group G is defined over a field F, and the theory for G depends on the choice of F. What would happen if we replace the base field by a finite extension E of F ? For this it is convenient to recall the theory of induced groups. Let A’ be a subgroup of finite index in a group A. The example of interest to us will later be A = Gal(F/F) and A’ = Gal(F/E). Suppose A’ acts on a group G. The induced group I i , ( G ) = Ind;,(G) is defined to consist of all f : A + G with ~ ( u ’ u = ) u ’ f ( u ) ( u E A, a’ E A’). The group structure is ( f f ’ ) ( a ) = f ( a ) f ’ ( u ) . The group A acts by (r(a)f)(x)= f(m) (a,. E A ) . For a coset s in A’\A put

G, = {f E I i , ( G ) ;f(a)= 0 if a @ s}. It is a group and I i , ( G ) is n s E A , , A G S . The groups G, are permuted by A. The subgroup Ge: is stable under A’, and f H f ( e ) , G, -+ G , is an A’-module isomorphism. Shapiro’s lemma asserts H 1 ( A ,I i , ( G ) ) =

H 1(A’,G ). Let B be a group, p : B

---f A a homomorphism, put B’ = p-’(A’), and suppose p induces a bijection B‘\BGA‘\A. Then B’ acts on G via p : b’ . g = p(b’)g. The map f H p o f is a p-equivarient isomorphism p’ : I ~ , ( G ) ~ I ~ , W ( eGhave ) . r(CL(a))(l-l0f)(.) = P O f(.a). If E/F is a finite field extension, we have

WE\WF = Gal(F/E)\ Gal(F/F)

= HomF(E,F).

7. En do s copy

If G is an E-group, its restriction of scalars G‘

299

=

RE/FG is the F-group

~ Thus G’(F) ~ = I~( G ( F ) )= ~ n , G ( F~) , where~ a

I G where I = 1

ranges over H o ~ F ( E , F ) .Let oi E W F (1 5 i 5 [ E : F]) be a set of representatives for WE\WF. Define an action of T E W F on ( 2 ; 1 5 i 5 [E: F])by W ~ a i 7 - l= W ~ q ( i Put ) . ~i = g7(i)7oi1E W E .Then (yT)Z = o(?T)(Z)yTgL1= g ( T T ) ( Z ) ? g L ( i )

The group

W F

acts on G’(F) =

’

gT(i)ToF1

= yT(i)‘i

WE’

ni G ( F )by (gi)y = (ycl(gY(il)).Indeed,

( g i ) ( y r )= ( ( Y 7 ) i 1 ( g ( Y T ) ( i ) ) ) = ((yT(i)7Z)-1( g Y ( T ( i ) ) ) ) = (7F1k77(i)))T= ( ( d Y ) T .

nu

In particular G’(F) = G ( E )and if E/F is Galois then G’(E) = G(E). Suppose G is reductive connected. Then Q(G’) = ( X I ,V’, X’*, V’”) is related to q ( G ) = ( X , V , X * , V “ ) by X’ = I X , V = U,Va (a E G a l ( F / E ) \ G a l ( F / F ) ) . Similarly, bases A’ and A of V’ and V are related by A’ = U,Aa. In particular we have a natural isomorphism e ’ s I ( @ , thus 6’ 2 e [ E : F 1 . The map P H R E / F P induces a bijection from the set of E-parabolic subgroups of G to the set of F-parabolic subgroups of G’, P is a Bore1 subgroup of G iff R E / F P is one of G’. Hence G is quasisplit over E iff G’ is quasisplit over F . If (Y : LE x SL(2,C) + LG is an A-parameter for G , then the corresponding parameter (Y’ : L F x SL(2,C) + LG’ is defined by Q’(7X S)

=

( Q ( T 1 X S),

. . . , Q ( T ~ [ E : FX] S ) )

X 7.

e e‘

The diagonal embedding ~f induces S, + Sat, and by Shapiro’s lemma gives kerl(LE, + kerl(LF, Z(e’)), where ker’ denote the set of classes in H 1 which are locally trivial. We have a commutative square S, Saf

z(@) I

kerl(LE, Z ( e ) )

---f

1 kerl(LF, Z ( 2 ) ) .

Hence S,, = Z(G’) . Im(S,), and the diagonal map yields an isomorphism S , = 3a,.In other words, the representation theory of G ( E ) is the same as that of G’(F).

-

~


300

8. Basechange As an example, let us consider the case of basechange lifting. It concerns an F-group G , and “lifting” admissible representations of G ( F ) to such representations of G ( E )if F is local and E / F is a finite extension of fields, or automorphic representations of G(AF) to such representations of G(AE) if E/F is an extension of global fields. We need to view G ( E ) (or G ( A E ) ) as the group of points of an F-group in order to compare L-parameters of the F-group G with those of what should describe G ( E ) . Such a group is given by G’ = R E / F G ,which is an F-group with G’(F) = G ( E ) . As for L-parameters, we have that the composition of X : WF 4 LG with the diagonal embedding h

bCE/F : LG =

X

WF4 LG‘ = G’

A

X

W F = (G X . . . X

e)

X

WF

gives an L-parameter A’ = bCE/F(X) : WF + LG’. In particular, the group W F permutes the factors in The parameter A‘ can be viewed as the restriction XE : W E 4 LG = x W E of x from W F to W E . As a special case, suppose G is split, thus the group WF acts trivially on but it permutes the factors in 2‘. Suppose E/F is an unramified local fields extension. Then an unramified representation 7r of G ( F ) is This image determined by the image t(7r) = X(Fr) of the F’robenius in is determined up to conjugacy. The image of t(7r) in LG’ is the conjugacy class of t(7r’)= (t(7r)x . . . x t(7r))>a Fr = (t(7r)lE:F], 1 . .. ,1) x F’r, which is the conjugacy class of t(.rr)[E:F] in the L-group L ( G / E )of G over E . For example, the unramified irreducible constituent 7r in the normalizedly induced representation I ( p 1 , . . . , p n ) of GL(n, F), where pi : FX4 C X are unramified characters, lifts to the unramified irreducible constituent T E in the normalizedly induced representation I ( p l o N E I F , . . ON^/^), NE/F : E X --f F X being the norm. If w is a place of a global field F which splits in E , thus E, = E @ FF, = F, @ . . . @ F,,, then bcEIF(7r,) = 7rv x . . . x 7rv is a representation of G ( E v )= G(F,) x . . . x G(F,). The problem of basechange is to show, given an automorphic T of G(AF), the existence of an automorphic T E of G(AE) = G’(Ap) with t(7rE,v) = bcE/F(t(7rv))for almost all w. For G = GL(n), if X E exists it is unique by rigidity theorem for GL(n).

e

e,

2’. G

6

z.

8. Basechange

301

A related question is to define and prove the existence of the local lifting. In any case, basic properties of basechange are, suitably interpreted: 0 transitivity: if F c E c L then bcLIE(bcElF(7r)) = bcL/F(7r) for 7r on G ( F ) . 0 twists: bCE/F(T@X) = bCE/F(T)@XE,XE = XONE/F(if G = GL(n), x : A:/F" + C"). 0 parameters compatibility: bcE/F(7r(X)) = 7 r ( x ~ ) . For G = GL(n), cyclic basechange (thus when EIF is a cyclic, in particular Galois, extension of number fields) was proven by Arthur-Clozel [AC]. A simple proof, but only for 7r with a cuspidal component, is given in [F2;II],where the trace formula simplifies on using a regular-Iwahori component of the test function. The case of n = 2 had been done by Langlands [L6], using ideas of Saito and Shintani (twisted trace formula, character relations). A simple proof of basechange for GL(2), with no restrictions, is given in [F2;I],again using regular-Iwahori component to simplify the trace formula. Basechange for GL(n) asserts (see [AC]): Let EIF be a cyclic extension of prime degree C. 0 Given a cuspidal automorphic representation of GL(n, A,) there exists a unique automorphic representation T E = bcE/F(?'r) of GL(n, AE) which is the basechange lift of T . It is cuspidal unless C divides n and 7rw = 7r for some character w # 1 of A;/FX NEiFA;. 0 If 7r and 7r' are cuspidal then bcE,F(T) = bcElF(7r') iff 7r' = 7rw for some character w of A i / F XNE/pA; 0 A cuspidal representation XE of GL(n, A E )is the basechange bCEIF(7r) of a cuspidal T of GL(n,AF) iff u7rE = T E for all 0 E Gal(E/F). Here uTE(g) = TE(Cg)* 0 If n = em and 7r is a cuspidal representation of GL(n, AF) with 7rw = 7 r , w # 1 on A;/FXNE~FAg (thus w has order C = [E : F]),then there is a cuspidal representation T of G L ( ~ , A Ewith ) u~ # T for all 0 # 1 in ,e-1 Gal(E/F) such that bcEIF(7r) is the representation I ( T , ~ T , " ' T.,.. , T) T @ ..' 7 on the parabolic of type normalizedly induced from T (m,. . . , m). The last statement can also be stated as T H 7 r , as follows. Let EIF be a cyclic extension of prime degree e. Let T be a cuspidal representation of GL(m, AE) with u~ # T for all 0 # 1 in Gal(E/F). Conjecturally this T is parametrized by an L-parameter X E : W E + GL(m, @).

302


Consider X = IndE F X E . It is a representation of WF in GL(n, C ) , n = me. The group G L ( m , E ) , or R E / F G L ( ~ ) can , be viewed as an w-twisted endoscopic group of GL(n) over F, where w is a primitive character on AG/FXNEIFAg. At a place v of F which stays prime in E, an unramified representation I ( p % 1 ; 5 i 5 rn) of GL(m, E,) would correspond to I(C3p;le;O 5 j 5 ,!. 1 5 i 5 m ) on GL(n, Fu).Here C is a primitive !th root of 1. At a place v of F which splits in E, Xu = @ X,:l, and r, of GL(m, E u ) ,which is @,jvr, of GL(m, F,) corresponds to I(@,~vr,). The last result stated above, as part of basechange for GL(n), asserts that endoscopic lifting for GL(n) exists. Denote it be endE/F(T). Namely Let E/F be a cyclic extension of prime degree el and r and a cuspidal representation of G L ( ~ , A E ) .Then x = endE/F(T) exists as an automorphic representation of GL(n, AF),which is cuspidal when Or # T for all c # 1 in G a l ( E / F ) . Moreover xw = 7r for any character w of AG/FXNEIFAg. Any cuspidal x of GL(n,AF) with w x = x for such w # 1 is x = e n d ~ / ~ (for 7 ) a cuspidal T of G L ( ~ , A Ewith ) OT # T for all in G a l ( E / F ) . Further, endE/F(r’) = endEIF(r) iff 7’ = u r . This result was first proven for m = 1, thus != n, by Kazhdan [Kl] for 7r with a cuspidal component, and by [Fl;I] without such restriction, and by Waldspurger [W3] and [Fl;I]for all m. This technique, of endoscopic lifting (twisted by w, into GL(n, F)),has the advantage of giving (local) character relations which are useful in the study of the metaplectic correspondence ([FKl]). The theory of basechange gives other character relations, and lifts x with x w = x to I(r,“r,. . . ). The endoscopic case of n = 2 = !had been done by Labesse-Langlands [LL]. See also [F6]. The basechange and endoscopic liftings described above were proven using the trace formula, and they apply only t o cyclic (Galois) extensions EIF. By means of the converse theorem, Jacquet, Piatetski-Shapiro, Shalika ([JPS])showed Let E/F be a non-Galois extension of degree 3 of number fields. If 7r is a cuspidal representation of G L ( ~ , A F )then the basechange lift bcEIF(7r) exists and is a cuspidal representation of G L ( ~ , A E ) . Again, the lifting is defined by means of almost all components 7r,, and bcE/F(n) is unique - if it exists - by rigidity theorem for GL(2).

n,,,

11. ON ARTIN'S CONJECTURE Let F be a number field, F an algebraic closure, and X : Gal(F/F) -+ Aut V , dimc V < co,an irreducible representation. Define L ( s ,A) to be the product over all finite places u of F of the local factors L(s,X,) = d~t[l-q,".(X,IV'~)(E,)l-~, where V'v is the space of vectors in V fixed by the inertia group I , at u,and A, is the restriction of X t o the decomposition group D, at u. Artin's conjecture asserts that the L-function L ( s , X ) is entire unless X is trivial (= 1). Langlands proposed approach to it is to show that there exists a cuspidal representation .(A) of GL(dim V, AF) with L ( s ,A), = L ( s ,.(A),) for almost all u. In this case, the holomorphy follows from the fact that L ( s ,7 r ) = L ( s ,T,) is entire for a cuspidal 7r = @7r, # 1. Thus 7r = .(A) is related to X by the identity t(7r,) = X(Fr,) of semisimple conjugacy classes in GL(n, C)for almost all u. If this relation holds, X is uniquely determined by Chebotarev's density theorem, and 7r is uniquely determined by the rigidity theorem for cuspidal representations of GL(n, AF).The case of dimc V = 1 is that of Class Field Theory, which asserts that .(A) exists as a character of A g / F X . Suppose dimX (i.e., dimV) is two. Denote by Sym2 : GL(2,C) 4 GL(3, C)the irreducible 3 dimensional representation of GL(2, C)which maps g to Int(g) on LieSL(2). Its image is SO(3,C) and its kernel is the center of GL(2,C) (thus it gives

n,

PGL(2, C)2; SO(3, C) c SL(3,C)). The finite subgroups of SO(3,C)are cyclic, dihedral, the alternating groups A4 or As, or the symmetric group S, on four letters; see, e.g., Artin [A],Ch. 5, Theorem 9.1 (p. 184). If Im(Sym2OX) is cyclic then Im(X) is contained in a torus of GL(2, C) and X is reducible, the sum of two characters. This case reduces to the case of CFT. Let X : G 4 GL(2, C)be an irreducible two dimensional representation of a finite group. 303

304

II. O n Artin’s Conjecture

1. PROPOSITION. Im(Sym2 OX) is dihedral ifl X = Indg x is induced from a character x of an index two subgroup H of G , and g x # x f o r all gEG-H. PROOF. Assume X is faithful by replacing G with G/ ker A. Let T be the cyclic subgroup of Im(Sym20X) of index two. Since the kernel of Sym2 is central in GL(2,C ) and X is faithful, the inverse image H of T in G is abelian. Hence the restriction of X to H is the sum of two one dimensional representations, x and x’. If x = x’,Clifford’s theory implies that x extends to G in two different ways (differing by the sign character on G I H ) . But X is irreducible two-dimensional, hence x’ # x,x’ = gx for any g E G - H , and X = Indg x. 0

2. COROLLARY. Suppose Im(Sym2 OX) is dihedral, where X : Gal(F/F) 4 GL(2,C) is two-dimensional. Then n(A) exists as a cuspidal representation of GL(2, AF). PROOF. By Proposition 1 there is a quadratic extension E of F and a character x of Gal(F/E) such that X = Indgx, x # “x for all c E Gal(F/F) - Gal(F/E). The existence of 7i.(Indgx) is proven in [JL], [LL], ~31. 0 The irreducible representations of the symmetric group S, are parametrized by the partitions of n , and the associated Young tableaux. The representation A’ associated to the dual Young tableaux is X . sgn, where X is associated with the original Young tableaux, and sgn is the nontrivial character of S,/A,. The representation X of S, becomes reducible when restricted to A , precisely when the Young tableaux is selfdual. The dimension of X is the number of removal chains, by which we means a chain of operations of deleting a spot of a Young diagram at the right end of a row under which there is no spot. For example, S 4 has the representations listed in the table on the next page. There the partitions (4) and (1111111) are dual. They parametrize the trivial and sgn one dimensional representations of S,. The partitions (3,l) and (2,1,1) are dual (obtained from each other by transposition), and parametrize 3-dimensional representations whose restrictions X3 to A4 remain irreducible and equal to one another. The selfdual partition (2,2) parametrizes the 2-dimensional irreducible representation of S 4 whose re-

II. O n Artin’s Conjecture

305

striction to A4 is reducible, equal to the sum of the two quadratic characters of A4 (trivial on the 3-Sylow subgroup).

partition

Young T

chains

dim

(4)

2x22

(xxx,2 2 , x)

1

2

1

The 3-dimensional representation X3 of A4 is induced. Indeed, consider the 2-Sylow subgroup A> of A d . It is generated by (12)(34), (13)(24), (14)(23), and is isomorphic to 2 / 2 @ 2/2. The quotient A4/A& is 2 / 3 . The restriction to the abelian A&of the irreducible 3-dimensional representation A3 of A4 is the sum of 3 characters permuted by the quotient 2 / 3 of Aq, hence A3 is induced Ind;; x, x2 = 1 # x. ofGL(2, AF) GL(2, C) is an irreducible representation such that

3. THEOREM. There exists a cuspidal representation .(A) where X : Gal(F/F) -+ Im(Sym2 OX) = A4.

We record Langlands’ proof ([L6]). 4. LEMMA.There exists a cuspidal representation r(Sym2OX) of GL(3, A F ) .

PROOF.The composition of Sym2OX with the projection A4 4 213 is a surjective map Gal(F/F) 4 2 / 3 . Its kernel has the form Gal(F/E), where E / F is a cubic extension. As noted before the lemma, Sym2OX = Ind; xI

306

11. O n Artin's Conjecture

where x : Gal(F/E) { f l } , and ax # x for 0 # 1 in G a l ( E / F ) = 2/3. The existence of 7r(Indg x ) now follows from the theory of (cubic) basechange for GL(3) [ A C ]or the endoscopic lifting for SL(3) of [ K l ]and [Fl;I]. 0 --ff

Put XE for XI Gal(F/E).

5. LEMMA.There exists a cuspidal representation 7 r ( X ~ )ofGL(2, AE). PROOF.We claim that XE is irreducible. If not, it would be the direct sum of two characters, permuted by Gal(F/F)/ Gal(F/E) = Gal(E/F) = 2/3. This action would then be trivial and X be reducible. But X is irreducible, hence so is XE. Now Sym20XE has as image the order 4 dihedral group, hence A(.), exists. 6 . PROPOSITION. Suppose T is a cuspidal representation of GL(2,AF) whose basechange bcEIF(7r) t o E is ~ ( X E ) ' whose central character w, is det A, and such that its symmetric square lift Sym2(7r) is 7r(Sym2OX). T h e n 7r = 7r(X).

PROOF.Denote by [a,b] the conjugacy class of diag(a, b ) in GL(2, C). At any place v where 7r, and A, = AID, are unramified (0, 1: Gal(F,/F,) is the decomposition group of w in G a l ( F / F ) ) , put t(7rv) = [a,b] and X(Fr,) = [.,PI. If v splits in E then bcEIF(7r) = T ( X E ) implies that [a,b] = [ a ,PI. We need t o show this also when E, = E @F F, is a field, to conclude that 7r = .(A) by rigidity theorem for GL(2). When E, is a field, from bcEIF(7r) = 7 r ( X ~ )we conclude that [ a 3 b3] , = [a3,p3],and from w, = det X that ab = ap. Hence a =

Automorphic forms and Shimura varieties of PGSp (2)

Automorphic forms and representations

Families of automorphic forms