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 401402, Hackensack, NJ 07601 UK ofice: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library CataloguinginPublicationData 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 9812564039
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 quasipackets 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 HeckeGalois bimodule, 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 nonexperts 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, TelAviv, 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 MPIBonn Arbeitstagung 2003, ERAAMS 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 writeup 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 MaxPlanckInstitut 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 ,
GALOIS REPRESENTATIONS Tempered Case Nontempered Case
1. 2.
xi
25 1 252 254 255 256 258 258 262
PART 3. BACKGROUND
267
I. ON AUTOMORPHIC FORMS 1. Class Field Theory 2. Reductive Groups 3. Functoriality 4. Unramified Case 5. Automorphic Representations 6. Residual Case 7. Endoscopy 8. Basechange 11. ON ARTIN’S CONJECTURE
269 269 2 75 280 285 288 290 295 300
REFERENCES
311
INDEX
321
303
I. PRELIMINARIES 1. Introduction According to the “principle of functoriality” , “Galois” representations p : LF + LG of the hypothetical Langlands group LF of a global field F into the complex dual group LG of a reductive group G over F should parametrize “packets” of automorphic representations of the adde group G(A). Thus a map X : L H +LG of complex dual groups should give rise t o lifting of automorphic representations ~ T Hof H(A) to those 7r of G(A). Here we prove the existence of the expected lifting of automorphic representations of the projective symplectic group of similitudes H = PGSp(2) to those on G = PGL(4). The image is the set of the selfcontragredient representations of PGL(4) which are not lifts of representations of the rank two split orthogonal group SO(4). The global lifting is defined by means of local lifting. We define the local lifting in terms of character relations. This permits us to introduce a definition of packets and quasipackets of representations of PGSp(2) as the sets of representations that occur in these relations. Our main local result is that packets exist and partition the set of tempered representations. We give a detailed description of the structure of packets. Our global results include a detailed description of the structure of the global packets and quasipackets (the latter are almost everywhere nontempered). We obtain a multiplicity one theorem for the discrete spectrum of PGSp(2), a rigidity theorem f o r packets and quasipackets, determine all counterexamples to the naive Ramanujan conjecture, compute the multiplicity of each member in a packet or quasipacket in the discrete spectrum, conclude that in each local tempered packet there is precisely one generic representation, and that in each global packet which lifts to a generic representation of PGL(4) there is precisely one representation which is generic everywhere. The latter representation is generic if it lifts to a properly induced representation of PGL(4, A). 3
I. Preliminaries
4
We also prove the lifting from SO(4) to PGL(4). This amounts to establishing a product of two representations of GL(2) with central characters whose product is 1. Our rigidity theorem for SO(4) amounts to a strong rigidity statement for a pair of representations of GL(2, A). Our method is based on an interplay of global and local tools, e.g. the trace formula and the fundamental lemma. We deal with all, not only generic or tempered, representations.
2. Statement of Results 2a. Homomorphisms of Dual Groups
Let G be the projective general linear group PGL(4) = PSL(4) over a number field F . Our initial purpose is to determine the automorphic representations 7r (BorelJacquet [BJ],Langlands [L4])of G(A), A is the ring of adkles of F , which are selfcontragredient: 7r E 5,equivalently (BernsteinZelevinski [BZl]), 8invariant: 7r 2 %. Here 8, 8(g) = J  l t g  l J , is the involution defined by
where t g denotes the transpose of g E G , and On(g) = 7r(O(g)). According to the principle of functoriality (Bore1 [Boll, Arthur [A2]) these automorphic representations are essentially described by representations of the Weil group WF of F into the dual group 2= SL(4, C) of G which of are &invariant, namely representations of W F into centralizers
z,(.+@
e.
Int(.+)iin Here 8 is the dual involution e(4) = J  l t 6  l J , and d is a semisimple element in g . These centralizers are the duals of the twisted (by .+6) endoscopic groups (KottwitzShelstad [KS]). In fact these are the connected components of the identity of the duals of the twisted endoscopic groups Zc(i8) x W F . But in our case the endoscopic groups are split so the product of Z ~ ( i 8with ) the Weil group WF is direct. Hence it suffices for us to work here with the connected component of the identity. A twisted endoscopic group is called elliptic if its dual is not contained in a proper parabolic subgroup of G. Representations of nonelliptic endoscopic groups can be reduced by parabolic induction to known ones of A
2a. Homomorphisms of Dual Groups
5
h
smaller rank groups. For our G, up to conjugacy the elliptic twisted endoscopic groups have as duals the symplectic group H = Z,(e) = Sp(2,C) 4
and the special orthogonal group C = Z,(O@ = “ SO(4, C)”
=so((w!!I;),C) = { g E S L ( 4 , C ) ; g O J t g = d J = which consists of all A @ B =
(A=
(z :) ,
(z:
I!,(
;)}
7
ii), where
B ) E [GL(2,C) x GL(2,C)]/CX
satisfy det A . det B = 1. Here z E C x embeds as the central element
( z ,z  ’ ) , B = diag(1, 1,1,1) and w =
( Y ;).
The group i? is the dual group of the simple Fgroup H = PSp(2) = PGSp(2), the projective group of symplectic similitudes, which can also be denoted by the shorter symbol PGp(2). It is the quotient of GSp(2) = {(g,X) E GL(4) x 6,; tgJg = XJ} by its center {(X,X2)} Y G,. Since X is uniquely determined by g (we write A = X(g)), we view GSp(2) as a subgroup of GL(4) and PGSp(2) of PGL(4). The group 6 is the dual group of the special orthogonal group (“SO(4)”)
Here z E 6, embeds as the central element ( z ,z ) . Also we write [GL(2) x GL(2)]’/ GL(1) for C , where the prime indicates that the two factors in GL(2) have equal determinants. The principle of functoriality suggests that automorphic discrete spectrum representations of H(A) and C(A) parametrize (or lift to) the 0invariant automorphic discrete spectrum representations of the group of Avalued points, G(A), of G . Our main purpose is to describe this lifting, or parametrization. In particular we define tensor products of two
1. Preliminaries
6
automorphic forms of GL(2, A) the product of whose central characters is 1. Moreover we describe the automorphic representations of the projective symplectic group of similitudes of rank two, PGSp(2,A), in terms of &invariant representations of PGL(4, A). Motivation for the theory of automorphic forms is attractively explained in some articles by S. Gelbart, see, e.g. [GI. For a more technical introduction see part 3, “Background”, of this volume. It is based on a course I gave at the Ohio State University in 2003. It gives most definitions used in this work, from adbles to Weil and Lgroups, to twisted endoscopy, and a proof of (Emil) Artin’s conjecture for two dimensional Galois representations with image Aq, S, in PGL(2, C). 2b. Unramified Liftings We proceed to explain how the liftings are defined, first for unramified represent at ions. An irreducible admissible representation 7r of an adkle group G(A) is the restricted tensor product @7rw of irreducible admissible ([BZl]) representations T , of the groups G(F,) of F,points of G I where F, is the completion of F at the place w of F . Almost all the local components 7rw are unramified, that is contain a (necessarily unique up to a scalar multiple) nonzero K,fixed vector. Here K , is the standard maximal compact subgroup of G(F,), namely the group G(R,) of R,points, R, being the ring of integers of the nonarchimedean local field F,; G is defined over R, at almost all nonarchimedean places w. For such w, an irreducible unramified G(F,)module 7rw is the unique unramified irreducible constituent in an unramified principal series representation l(qw),normalizedly induced (thus induced in the normalized way of [BZ2]) from an unramified character q, of the maximal torus T(F,) of a Bore1 subgroup B ( F , ) of G(F,) (extended trivially to the unipotent radical N(F,) of B(F,)). The space of I(q,) consists of the smooth functions q5 : G(F,) 4 C with
&,(a) = det[Ad(a)I LieN(F,)], and the G(F,)action is (g.q5)(h)= q5(hg),
9, h E G(Fw).
The character q, is unramified, thus it factors as q, : T(F,)/T(R,)
+
2c. The Lifiing from SO(4) to PGL(4)
C x . As X,(T) = Hom(G,,T)
N
7
T(F,)/T(R,), q, lies in
Hom(X,(T),Cx) = Hom(X*(?),CX),
e,
both fixed in where ? is the maximal torus in the Borel subgroup B^ of the definition of the (complex) dual group (? (Borel [Boll, Kottwitz [K o ~ ] ) . Now Hom(X*(?), C”)= X*(?) @ C x = ? c
e.
Thus the unramified irreducible G (F,)module 7rw determines a conjugacy class t(7rw)= t ( l ( q , ) ) in (? represented by the image of 17, in ?. This class t(7r,) is called the Langlands parameter of the unramified T,. In the case of G = GL(n), take B to be the group of upper triangular matrices, T the diagonal subgroup, and q v ( a l , . .. ,a,) = q,(ai) (1 5 i 5 n ) . If T, is a generator of the maximal ideal of R, then t(I(q,)) is the class of diag(ql(n,), . . . , q , ( ~ , ) ) in G = GL(n,C). If G = PGL(n) then q 1 . . . qn = 1 and t(I(q,)) is a class in G = SL(n, C ) . We make the following notational conventions: If the components of q are q,,772,,. . . , we write I(ql,, 7 7 2 , , . . . ) for I(q,). For a representation 7r and a character x we write x 7 r for g H X ( g ) 7 r ( g ) , and not x @ 7 r , reserving the notation n1 Bn2, or n1 x n2, for products on different groups: ( h ,g ) H 7rl(h) 8 7 r 2 ( g ) (for example, if ( h , g ) ranges over a Levi subgroup, the representation normalizedly induced from the representation 7r1 @ 7r2 on the Levi will be denoted by I ( 7 r l , 7 r 2 ) or 7r1 x 7 r 2 , depending on the context). We prefer the notation 7rl x 7r2 for a representation of a group which is a product of two groups, such as our C = SO(4, F ) . By a representation we mean an irreducible one, unless otherwise is specified.
n
A
2c. The Lifting from SO(4) to PGL(4)
We next describe our results on our secondary lifting XI, from C = SO(4) to G = PGL(4). We now return to G = PGL(4), 8 and C = [GL(2) x GL(2)]’/GL(l). Note that an irreducible unramified GL(2, F,)module TI, is parametrized by a conjugacy class t(7rlw)in GL(2,C) (the Langlands parameter of the representation; its eigenvalues are called the Hecke eigenvalues of the representation). An unramified irreducible representation 7r1, x 7r2, of C(F,) is parametrized by a class t(7r1,) x t ( 7 r 2 , ) in [GL(2,C)x GL(2, C)]”/Cx
2:
SO
((,01 :)
, C ) = (? c
G.
I. Preliminaries
8
(Double prime means detgl . detg2 = 1). If xi, is the unramified constituent of
I(71iw),
t(.i?J) = dkdrlil, 7122),
71ij = 71ijw(7rw),
7111711271217122 =
1,
we define the “lift” 7rlV [XI 7r2, = XI(TI,x 7r2,) of TI, x 7rzv with respect to the dual group homomorphism A 1 : 6 = SO(4, C)~f G = SL(4, C)(the natural embedding) to be the unramified irreducible constituent 7rv of the PGL(4, F,)module I(qv)parametrized by the class
t(.rrw) = diag(71117121, 71117722,71127121,71127122) in
= SL(4,C). In different notations,
Xl(I(a1,az) x provided that
I(bl,b2)) = I(albl,albz,azbl,a2bz)
ala2blb2
I(alb1, alb2, bla2, a2b2)
xI(a1,.2)
(%bi
E
ex),
= 1. Note that the inverse image under X1 of consists only of
x xlI(b1, b2)
and X W l ,62) x
xlI(a1, (32)
where x is any character of F,“. Thus, A 1 is twotoone unless T I , = ir2, (the contragredient of 7r2,), where A 1 is injective on the set of orbits of multiplication by x in Hom(F,X,Cx). The rigidity theorem for the discrete spectrum automorphic representations of GL(n, A) asserts that discrete spectrum automorphic representations 7r1 = @7r1, and 7r2 = 87r2, which have 7r1, 21 7rzv for almost all places w of F are equivalent (JacquetShalika [JS], MoeglinWaldspurger [MWl]). Moreover they are even equal, by the multiplicity one theorem for GL(n) (Shalika [Shall). Representations of PGL(n,A) (or PGL(n, F,)) are simply representations of GL(n, A) (or GL(n, F,)) with trivial central Gm)= {0}), and the rigidity theorem applies then character (since H1(F, to PGL(n). Both multiplicity one theorem, and the rigidity theorem for packets (the latter asserts that 7r = @IT, and 7r‘ = @IT; must lie in the same packet if 7rw N 7rh for almost all w) hold for SL(2) ([F3]) and fail for SL(n), n L 3 (Blasius [Bla]). The rigidity theorem holds for C = SO(4); this is the content of the assertion that the lifting X I is injective, made in the second paragraph of the following theorem. The first paragraph asserts that the lifting exists. By an elliptic representation we mean one whose character (HarishChandra [HI) is not identically zero on the set of elliptic elements.
2c. The Lifting f r o m SO(4) to PGL(4)
9
2.1 THEOREM (SO(4) TO PGL(4)). Let T I = @7r1,, 7r2 = 8 7 r 2 , be discrete spectrum automorphic representations of GL(2, A) whose central characters w 1 , w 2 are equal, and whose components at two places v 1 , 212 are elliptic. Then there exists an automorphic representation 7r = A 1 (7r1 x i r z ) of P G L ( 4 ,A) with K , = A 1 ( r l u x ii2,) f o r almost all v. W e have A 1 ( x l 7 r l x x 2 7 r 2 ) = x l x 2 A 1 ( 7 r 1 x 7 r z ) f o r xi : AX/,' + Cx with ( ~ 1 x 2 = ) ~1. If 7r1 = ~ ~ ( p l7r2) = , ~ ~ ( p are 2 ) cuspidal monomial representations of GL(2,A) associated with characters p 1 , p 2 of A 2 / E X where E is a quadratic extension of F such that the restriction of p 1 p 2 to A X is 1, then Al(TE(p1)
x
rE(P2))
= 1(2,2)(7rE(p1772)1 r E ( p l p 2 ) ) .
are cuspidal but not of the f o n n { ~ ~ ( p 7lr )~ (, p 2 ) } and , 7r1 # x 7 r 2 for any quadratic character x of A X / F X then , 7r1 1xI T Z is cuspidal. If 7r1 is the trivial representation 1 2 and 1r2 is a cuspidal representation ofPGL(2,A), then A l ( 1 2 x n  2 ) is the discrete spectrum noncuspidal PGL(4,A)module J ( ~ ~ / ~ 7~ r 2' / ,~ 7 r 2 ) .Here v ( z ) = 1x1, and J is the quotient of the representation I ( ~ l / ~ ~r z' /, ~ 7 r 2 )nonnalizedly induced from the parabolic subgroup of type (2,2) of PGL(4). The global map A 1 is injective o n the set of pairs T Ix ir2 with w 1 = w 2 up to the equivalence 7r1 x ir2 21 X K ~x x  l i r 2 , x a character of A X / F X , and 7r1 x 772 N ir2 x T I . If
{ T I ,7 r 2 }
The injectivity means that if 7r1, 7r2, 7ryl 7ri are discrete spectrum automorphic representations of GL(2, A) with central characters w 1 , w 2 , wy, w: satisfying w l w 2 = 1 = wywi, each of which has elliptic components at least at the three places q , 212, v3, and if for each v outside a fixed finite set of places of F there is a character xu of F," such that the set { 7 r 1 , x u , 7r2,x;') is equal to the set { 7 r ~ , , 7 r i U } (up to equivalence of representations), then there is a character x of A XI F Xsuch that the set { 7 r l x , 7 r 2 x P 1 } is equal to the set {ry,~ i }In. particular, starting with a pair 7r1, 7 ~ of 2 automorphic discrete spectrum representations of GL(2,A) with w l w 2 = 1, we cannot get another such pair by interchanging a set of their components 7 r l V , 7r2, and multiplying 7 r l u by a local character and 7r2, by its inverse, unless we interchange T I ,7r2 and multiply 7r1 by a global character and 7r2 by its inverse. A considerably weaker result, where the notion of equivalence is generated only by 7 r l v x j i 2 , N *av x 7 r l v but not by 7rlv x ir2, 2: x v 7 r l v x x;lirzvl
I. Preliminaries
10
follows also on using the JacquetShalika [JS] theory of Lfunctions, comparing the poles at s = 1 of the partial, product Lfunctions LV(S,7rY
x iil)Lv(s,7r; x
iil)
= LV(S,7r1x
iil)LV(S,7r2 x
iil).
Our global results are complemented and strengthened by very precise local results. If 7r N % there is an intertwining operator A with A7r(g) = .rr(B(g))Afor all g . By Schur’s lemma we may assume that A2 = 1. Then A is unique up to a sign. We put r ( 0 ) = A and 7r(f x 0) = n(f)A. We define XIlifting locally by means of character relations: Xl(7rl x
ji2)
= 7r
if
tr7r(f x 6) = tr(n1 x iiZ)(fc)
for all matching functions f , fc (and a suitable choice of A ) . This definition is compatible with the one given above for purely induced 7r1 and .rr2 and unramified representations. We have X1(12(p, p’)xiiz) = Id(pLjT2, p‘frz) (the central character of the GL(2,F)module 7r2 is p p ’ ) . The local and global results are closely analogous. 2d. Special Cases of the Lifting from SO(4)
Let us describe some special cases of the lifting XI. When 7r2 = irl is the contragredient of “1, A 1 (7r1 x el) is the PGL(4, A)module normalizedly induced from the maximal parabolic of type (3,l) and the PGL(3,A)module Sym2(7rl)on the GL(3)factor of the Levi subgroup (extended trivially to the GL(1)factor of the Levi, and to the unipotent radical). Here Sym2(7rl) is the symmetric square lifting from GL(2) to PGL(3) ([F3]). Indeed, if the local component 7rlW of 7r1 at v is unramified then t(7rlV)= diag(a,b) (thus 7rlw is a constituent of I z ( a , b ) ) , 7rw = A1(7rlV x + l V ) has t(7rV)= diag(a/b, 1,1,b / a ) (thus 7rw is a constituent of 14(13(u/b,1,b / a ) , l ) , and I3(u/b, l , b / a ) is the symmetric square lifting of Iz(u,b)). We write I, to emphasize that the representation is of the group GL(n), and e.g. I(3,1)(7rs,nl) to indicate the representation of GL(4) induced from its maximal parabolic subgroup of type (3,l). However, the results of [F3] are stronger, in lifting representations of SL(2, A) to PGL(3, A) and consequently providing new results such as multiplicity one for SL(2). Although we do not obtain here a new proof of the existence of the symmetric square lift of discrete spectrum representations of PGL(2, A),
2d. Special Cases of the Lifting from SO(4)
11
we do obtain new character identities, relating the &twisted character of 1(3,1)(Sym2 7r2, 1) with that of 7r2 x 5 2 . Clearly in this case the lift A1 is injective: if
then 7r1 = 7r2 = 7rox for some character x of A x / F X . In particular, if 7r1 is a one dimensional representation g H X(det g) of GL(2,A),then X l ( 7 r 1 x51) = 1(3,1)(13, 1)is the representation of PGL(4, A) normalizedly induced from the trivial representation of the maximal parabolic subgroup of type ( 3 , l ) . An alternative purely local computation of this twisted character is developed in [FZ]. Let 7r1 = ~ ( pbe ) a cuspidal monomial representation of GL(2,A) associated with a character p of A g / E X where E is a quadratic extension of F (denote by (T the nontrivial element of Gal(E/F)). Then
where X E f F is the quadratic character of A X / F X N E l F A z(NE/F is the norm map from E to F). Moreover,
is an induced representation from the parabolic subgroup of type (2,1,1) of PGL(4). Note that the central character of the GL(2,A)module n(p) is XE/F .plAX, for any character p of A g / E X . If ~ ( pis)a PGL(2, A)module we have that the restriction of p to A g / F X is X E / F , nontrivial but trivial On FXNE/Fl$g. If 7r1 = 7 r ~ ( p 1 )7r2 , = 7rE(p2), cuspidal monomial representations of GL(2,A) associated with characters p1, p2 of A g / E X where E is a quadratic extension of F such that the restriction of p1p2 to A X is 1, then
Indeed
12
I. Preliminaries
where CE = A g / E X (globally, and E X locally), and the representation corresponds to
where p1p2(02) = 1 and & ( z ) = p i @ ) , p17Z1p27i2 = 1 and p i ( z ) are abbreviated to pi in the line of z . When p1 = pY1 we have 7r(p1p2)= 7 r ( p 1 / p l ) and 7r(p1pz) = I ( x E / F1). , Thus
Note that if p : : A 4 C x has ( P / ~ Z ) ~= 1 # jZ/p then there are quadratic extensions E2, E3 and characters pi : Agi/E: 4 C" with T E i (pi)= T E ( p ) * Another interesting special case is when 7r1 is taken to be the trivial representation 1 2 of PGL(2, A) while 7r2 is a cuspidal representation of PGL(2,A). Then X l ( l 2 x 7r2) is the discrete spectrum noncuspidal representation J(v1l27r2,v  ' I 2 7 r 2 ) of PGL(4, A), the quotient of the nor2 ) from , the parabolic of type (2,2) of malizedly induced I ( V ~ / ~~ ~~/ ' ~7 r Z Indeed, 1 2 is the quotient of the induced PGL(4). Here v(x) = 1x1. I(v1l2,Y  ~ / ~ Hence ) . Then
is the quotient J ( ~ v 1 / ' 7 r 2 ~v11/27r2v) , of the induced I ( ~ v 1 / ~ 7 rvV1/27r2v) 2~, for all Y where 7rzv is unramified. Hence it is J ( V ' / ~ ~ ~ ~ , globally V~/ by the rigidity theorem for this noncuspidal discrete spectrum ([MWl]). On the set of pairs 7r1 x 7 r ~such that at least one of 7r1 or 7r2 is one dimensional, the lifting XI is injective. Indeed, a discrete spectrum representation of GL(2, A) with a onedimensional component is necessarily
2e. The Lifting from PGSp(2) to PGL(4) onedimensional. If
13
is not cuspidal but rather trivial, then the quotient J ( v 1 l 2 1 2 , ~ ~ ' 1 ~ of 1 21 4)( v 1 / ' 1 2 , ~  l / ~ 1 2is) not discrete spectrum, but the induced 14(13) from the trivial representation of the (3,1)parabolic; this is X l ( l 2 x 1 2 ) . 7r2
2e. The Lifting from PGSp(2) to PGL(4)
We now turn to the study of our main lifting A, and of the automorphic representations of the Fgroup H = (PSp(2) =) PGSp(2) = GSp(2)/Gm, where the center (6,m of GSp(2) = { g E GL(4); t g J g = XJ,
3X = X ( g ) E G m } h
consists of the scalar matrices. Its dual group is H = Sp(2,C) = Z,(O) C G = SL(4,C), where $(g) = J  l t g  ' J . It has a single elliptic endoscopic group C Odifferent than H itself. Thus aOOb
 a,
c O O d

where s0 = d i a g (  l , l , l , 1), and C O= PGL(2) x PGL(2). Write XO for 6. the embedding e o and X for the embedding = Sp(2, C) defines The embedding XO : & = SL(2,C) x SL(2,C) the "endoscopic" lifting A0
: 7r2(p11 pT1)
n 2 b 2 1pT1)
TPGSp(2)(pl, p 2 ) ~
Here 7r2(piI p:') is the unramified irreducible constituent of the normalizedly induced representation I(pi,p:') of PGL(2, F,) (pi are unramified characters of F,X, i = 1, 2); 7 r p G S p ( 2 ) ( p 1 , p 2 ) is the unramified irreducible constituent of the PGSp(2, F,)module 1 p G S p ( 2 ) ( p 1 , p 2 ) normalizedly induced from the character n . diag(a,P,y, 6) ++ pl(a/r)pz(a/P) of the upper triangular subgroup of PGSp(2, F,) (n is in the unipotent radical, a6 = Py). The embedding X : H = Sp(2,C) SL(4,C) = G defines the lifting X which maps the unramified irreducible representation 7rITpGsp(2)(pl,p 2 ) of PGSp(2, F,) to 7 r 4 ( p l I 112, p z l ,p L 1 ) , an unramified irreducible representation of PGL(4, F,).

I. Preliminaries
14
The composition X o A0 : 60= SL(2,C) x SL(2,C) takes 7r2(P11p ; ' ) x 7r2(p2, p;l) to
+
G = SL(4,C)
namely the unramified irreducible PGL(2, F,,) x PGL(2, F,)module 7r2 x 7rh to the unramified irreducible constituent ~ 4 ( ~T2; ), of the PGL(4, I?,,)module I4(7r2, 7rL) normalizedly induced from the representation 7r2 8 7rh of the parabolic of type (2,2) of PGL(4, F,,) (extended trivially on the unipotent radical). For example XoAo takes the trivial PGL(2, F,,) xPGL(2, F,,)module 1 2 x 1 2 to the unramified irreducible constituent 7 r 4 ( 1 2 , 1 2 ) of I 4 ( 1 2 , 1 2 ) , and 1 2 x 7r2 to 7r4(12,7r2) = 7r4(v1/2X21 ~  ' / ~ 7 r 2 ) .Note that this last 7r4 is traditionally denoted by J . The definition of lifting is extended from the case of unramified representations to that of any admissible representations. For this purpose we define below norm maps from the set of &stable ®ular conjugacy classes in G = G ( F ) to the set of stable conjugacy classes in H = H(F), and from this to the set of conjugacy classes in C o ( F ) ,extending the norm maps on the split tori in these groups which are dual to the dual groups homomorphisms X and XO. This is used to define a relation of matching functions f, fH and fc, (they have suitably defined matching orbital integrals) and a dual relation of liftings of representations. To express the lifting results we use the following notations for induced representations of H = PGSp(2, F ) . For characters p1, p 2 , g of F X with p 1 p 2 a 2 = 1 we write p1 x p 2 A (T for the Hmodule normalizedly induced from the character p =mu t+
pl(u)p2(b)(~(X),
m = diag(u,b,X/b,X/a),
a , b, X 6 F x of the upper triangular minimal parabolic of H . For a GL(2,F)module 7r2 and character p we write 7r2
u E U,
A
p for the
PGSp(2,F)module normalizedly induced from the representation p =mu H 7r2(g)p(X), m = diag(g,Xwtglw),
u E U(2),
X
E FX
(here the product of the central character w of 7r2 with p2 is 1)of the Siege1 parabolic subgroup (whose unipotent radical U(2) is abelian).
2f. Elliptic Representations
15
We write p x 7rzl if w p = 1, for the representation of PGSp(2,F) normalizedly induced from the representation
p = mu H p(a)m(g),
m = diag(a,g,X(g)/a),
uE
U(1),
X(g) = det g, of the Heisenberg parabolic subgroup (whose unipotent rad
ical U(1) is a Heisenberg group). These inductions are normalized by multiplying the inducing representation by the character p H 1 det(Ad(p))lLie U1’/2,as usual. For example, I H ( P l , P 2 ) = PlP2 x
p1IcLz
pT1.
Note that 7r x c !x ir >a w a and p(7r x a ) = T >a pa. Complete results describing reducibility of these induced representations, stated in SallyTadic [ST] following earlier work of Rodier [ R o ~ ] , Shahidi [Sh2,3], Waldspurger [Wl], are recorded in chapter V, section 1, Propositions 2.12.3, below. For notations see chapter 11, section 4. For properly induced representations, defining A and Aoliftings by character relations (A(7r~)= 7r4 if tr7r4(f x 0) = trrH(fH) for all matching f, f H ,and Ao(7rl x ~ 2 = ) T H if t r 7 r ~ ( f ~=) tr(7rl x n2)(fco) for all matching f ~fc,), , our preliminary results (obtained by local character evaluations) , are that w  l x 7r2 Alifts to 7r4 = I ~ ( ~ z , i r that z ) , p7r2 >a p’ (here w = 1) Alifts to 7r4 = I c ( p ,T Z ,p  ’ ) , and that I z ( p , p  ’ ) x 7rz Aolifts from COto p7r2 x p’ on H = PGSp(2, F ) . Let x be a character of F X / F x 2 . It defines a onedimensional representation x ~ ( h = ) x(X(h))of H , which Alifts to the onedimensional representation x ( g ) = X(detg) of G (if h = Ng then X(h) = detg; on diagonal matrices N(diag(a, b, c, d ) ) = diag(ab, ac, db, d c ) ) . The Steinberg representation of H Alifts to the Steinberg representation of G, and for any character x of F X with x2 = 1 we have A(XHStH) = x StG. 2f. Elliptic Representations Our finer local lifting results concern elliptic representations (whose characters are nonzero on the elliptic set). They follow on using global techniques. Elliptic representations include the cuspidal ones (terminology of [BZ]. These are called “~upercuspidal~~ by HarishChandra, who used the word “cuspidal” for what is currently named “discrete series” or “square integrable” representations).
I. Preliminaries
16
2.2 LOCALTHEOREM (PGSp(2) TO PGL(4)). (1) For any unordered pair 7r1, 7r2 of square integrable irreducible representations of PGL(2, F ) there exists a unique pair T;, 7rE of tempered (square integrable if T I # ~ 2 cuspidal if 7r1 # 7r2 are cuspidal) representations of H with
f o r all matching functions f f H , fc,. If 7r1 = 1r2 is cuspidal, 7r;H and 7rG are the two inequivalent constituents of 1 x 7r1. If 7r1 = 7r2 = asp, where a is a character of F X with a2 = 1, then T; and 7rH are the two tempered inequivalent constituents 7(v1I2 sp2,av'I2), 7(v1/212,av1/2) of 1 x usp,. If 7r1 = csp2, a2 = 1, and 7r2 is cuspidal, then 7rL is the square inte2 , l / of ~ the ) induced uv1/27r2x U U  ' / ~ ; n; grable constituent 6 ( ~ v ~ / ~ 7ar ~ is cuspidal, denoted here by
If 7r1 = asp2 and 7r2 = [asp2, E (# 1 = J2) and a (a2 = 1) are characters of F X ,then 7r; is the square integrable constituent
of the induced
JU'/~
sp2 ~ a 112. v , rH  is cuspidal, denoted here by
(2) For every character a of F X / F X 2and square integrable exists a nontempered representation 7rG of H such that
f o r all matching f ,
fHl
fc,. Here
7rH  = 7r'H(asp2X7r2)]
7r;
= L(av1/27r2,av'/2).
7r2
there
,
2f. Elliptic Representations
17
(3) For any characters 5, a of F X / F X and 2 matching f, f H , fc, we have that tr(a512 x a l 2 ) ( fc,) is
and tr I G ( a 5 1 2 , a l 2 ; f x 0) is
HereX=6 i f t f l a n d X = L i f [ = l . (4) Any 0invariant irreducible square integrable representation 7r of G which is not a XIlift is a Xlift of a n irreducible square integrable representation TH of H , thus tr7r(f x 0) = trrH(fH) f o r all matching f , f H . I n particular, the square integrable (resp. nontempered) constituent b([v,~  ~ / ' 7 r z )(resp. L(Jv,v'/'7r2))2)) of the induced representation Jv x u1/27r2 of H , where 7r2 is a cuspidal (irreducible) representation of GL(2, F ) with central character E # 1 = E2 and <7r2 = 7r2, Xlifts to the square integrable (resp. nontempered) constituent
of the induced representation
IG(V1/27r2, v1/27r~)of
G = PGL(4, F ) .
These character relations permit us to introduce the notion of a packet of an irreducible representation, and of a quasipacket, over a local field. Thus we say that the packet of a representation T H of H consists of TH alone unless it is tempered of the form 7r& or 7rh for some pair T I , 7r2 of (irreducible) square integrable representations of PGL(2, F ) , in which case the packet { T H } is defined to be {7r&, 7 r i } , and we write Xo(7rl x 7r2) = { T ; , 7rG} and A({7rG,7rG}) =IG(T~,T Further, ~ ) . we define a quasipacket only for the nontempered (irreducible) representations 7rg and L = L(v5,E x ~ v  l / ~ to consist of {7rG17r,} and { L , X } , X = X ( E ~ ~ / ~ s p ~ , E a v  ~ /We ' ) . say that 0 1 2 x 7r2 Xolifts to the quasipacket X o ( a l 2 x 7r2) = {T;, T ; } , which in turn Xlifts to I ~ ( a l 2 , 7 r z ) , and similarly, a512 x 0 1 2 Xolifts to X o ( a J l 2 x 0 1 2 ) = { L ,X } which Xlifts t o I ~ ( a 5 1 2 , 0 1 2 ) . Conjecturally our packets and quasipackets coincide with the Lpackets and Apackets conjectured to exist by Langlands and Arthur [A231. Using the notations of section V . l l below, we state the analogue of these results in the real case: F = R. For clarity, denote 7r1 and 7r2
I. Preliminaries
18
above by 7r1 and 7r2. In (I), 7r1 = x k , and 7r2 = T k Z , kl 2 k2 > 0 and Icl, k 2 are odd, are discrete series representations of PGL(2, R), and Wh~ , 7rG~ is the 7rL is the generic ~ k holomorphic z n,":,',, , which are discrete series representations of PGSp(2,R) when k l > k 2 . When kl = k 2 , 7rH + is the generic and 7rE is the nongeneric (tempered) constituents of the induced 1 x 7 r 2 k l + l . There is no special or Steinberg representation of GL(2,R); the analogue is the lowest discrete series T I . The 7rk are self invariant under twist with sgn. In (2) with 7r2 = K 2 k + 3 ( k 2 0), 7r; is ~ ( a v ~ / ~ 7 r 2 k + 3 ~ a 7~ r ~is ~ / 7rk&3,1, ~ ) , 7 r i is 7 r g $ 3 , 1 . In (3), if = sgn then X is the tempered 7rG c 1 >a 7r1, if = 1 then X is L ( V ~ / ~ ~ ~ ~ , O Both of these X , as well as L(vJ,<x a ~  l / ~ are) not , cohomological. In (4), r2 is 7 r 2 k + 2 , L(
<
hol "2k+3,2k+l
<
Wh
@ *2k+3,2k+l'
2g. Automorphic Representations
With these local definitions we can state our global results. These global results are partial, since we work with test functions whose components are elliptic at least at three places, and consequently we cannot detect automorphic representations which do not have at least three components whose (8) characters are nonzero on the (8) elliptic set. Thus we fix three places { V I , u 2 , 213) and discuss only TI x 7 r 2 , TH and 7r = ITG whose components there are (19)elliptic. Let us explain the reason for this restriction. The (noninvariant) trace formula, as developed by Arthur, involves weighted orbital integrals and logarithmic derivatives of induced representations. Arthur's splitting formula shows that these can be expressed as products of local distributions, which are all invariant (orbital integrals or traces of induced representations) except at most at rank(H) places. Working with test functions fH = @fHv with rank(H)+l components f H v with tr7rHw(fHw)= 0 for every tempered properly induced representation h7h rof H,, ~ ~(equivalently: fH,,whose orbital integrals vanish on the regular nonelliptic set of H,,), all non elliptic terms vanish. We call such f H w elliptic. At an additional place we use a regular Iwahori biinvariant component (see [FKl], [FK2], [F2]or [FS;VI]) to annihilate the singular orbital integrals. For the twisted trace formula we use the twisted rank, which is equal to rank(H), to obtain the same vanishing. This removes all complicated terms in the trace formulae
29. Automorphic Representations
19
comparison. Here rank means the Fdimension of a maximal split torus in the derived group, or in the derived group of the group of fixed points of the involution in the twisted case. For very little effort we can reduce the number of restrictions to two, rather than three. Using elliptic components f H v l , f H u z , implies that the local factors at each w # wl, w2, in the terms in the trace formula, are invariant. We then use at a third, nonarchimedean, place v3 a regularIwahori function (as in [FKl], [FK2], [F2], [FS;VI]).Similar choice is made for the twisted formula. The geometric sides of the trace formulae consist now of elliptic terms only. As the distributions at v 3 which occur in the trace formula are invariant, such f H U 3 can also be taken to be a spherical function with the same orbital integrals as the Iwahoriregular component. The resulting equality of discrete and continuous measures (the continuous measure comes from the spectral sides), which are invariant distributions in f H v 3 , implies their vanishing by the (standard) argument of “generalized linear independence of characters” (using the StoneWeierstrass theorem) employed in this context in [FKl], [FK2], [F2], [F3]. To simplify our exposition we do not record this argument here, but our global results can safely be used with two restriction, at q ,v2, rather than three. One can do better, and require that only one component, f H v , be elliptic, at a single real place w. This argument, explained in Laumon [Lau], requires very extensive referencing to much of Arthur’s deep analysis of the distributions appearing in the trace formula. Inclusion of these arguments here would have made this work more complicated than the relatively elementary exposition I wish to present. However, our results are provable for global representations with a single elliptic component at a real place. This suffices for all purposes of studying the decomposition of the ladic cohomology with compact supports of the Shimura variety associated with our group, and any coefficients, as a GaloisHecke module ([F7]). These constraints will be removed once the trace formulae identity is established for general test functions. This is being developed by Arthur. A simpler method, based on regular functions, has been introduced when the rank is one (see [F2;I], [F3;VI], [F4;III]) to establish unconditional comparison of trace formulae. But it has not yet been extended to the higher rank cases.
W i t h this reservation, emphasized by a *superscript in the following
20
I. Preliminaries
Global Theorem, the discrete spectrum representations of PGSp(2, A), i.e. H(A), can now be described by means of the liftings. They consist of two types, stable and unstable. Global packets and quasipackets define a partition of the spectrum. To define a (global) [quasi] packet { T H } , fix a local [quasi] packet { T H ~ }at each place w of F , such that {TH,,} contains an unramified member 7rkv (and then { 7 r ~ , , } consists only of 7r&,, in case it is a packet) for almost all w. The [quasi] packet { T H } is then defined to consist of all products @&, , Hv with 7rhv in { T H ~ for } all w, and &,, = 7rgvfor almost all v. The [quasi] packet { T H } of an automorphic representation T H is defined by the local [quasi] packets {TH,,} of the ~ T H at almost all places. components T H of The discrete spectrum of PGSp(2,A) will be described by means of the Ao and Aliftings. We say that the discrete spectrum 7r1 x 7r2 Aolifts to a packet { T H } (or to a member thereof) if {TH,,} = A0(7rlv x ~ 2v 2 for almost all w, and that a packet { T H } (or a member of it) Aliftfts to an irreducible selfcontragredient automorphic representation 7r if A( { TH,,})= 7rV for almost all w. The unstable spectrum of PGSp(2,A) is the set of discrete spectrum representations which are Aolifts; its complement is the stable spectrum. A [quasi] packet whose automorphic members lie in the (un)stable spectrum is called a(n un)stable [quasi] packet. (PGSp(2) TO PGL(4)). The packets and 2.3 GLOBALTHEOREM* quasipackets partition the discrete spectrum of the group PGSp(2, A), thus they satisfy the rigidity theorem: if T H and 7rh are discrete spectrum representations locally equivalent at almost all places then their packets or quasipackets are equal. The Alifting is a bijection between the set of packets (resp. quasipackets) of discrete spectrum representations in the stable spectrum (of PGSp(2, A)) and the set of self contragredient discrete spectrum representations ofPGL(4, A) which are one dimensional, or cuspidal and not a A1 lift f r o m C(A) ( o r residual J ( V ~ / ~ ~ Q , V where  ~ / ~7r2 T ~is )a cuspidal representation of GL(2,A) with central character [ # 1 = E2 and [m = m ) . The Aolifting is a bijection between the set of pairs of discrete spectrum representations
and the set of packets and quasipackets in the unstable spectrum of the
2g. Automorphic Representations
21
group PGSp(2,A). The Alifting is a bijection f r o m this last set to the set of automorphic representations I ~ ( 7 r 1 , 7 r 2 ) of PGL(4, A), normalizedly induced from discrete spectrum 7r1 x 7r2 (7r1 # 7r2) on the parabolic subgroup with Levi factor of type (2,2). If 7r1 x 7r2 is cuspidal, its Aolift is a packet, otherwise: quasipacket. Each member of a stable packet occurs in the discrete spectrum of the group PGSp(2, A) with multiplicity one. The multiplicity m ( 7 r H ) of a member T H = @ T H ~ of an unstable [quasi]packetX ~ ( 7 r lx 7r2) ( T I # 7r2) is not ("stable", or) constant over the [quasi]packet. If 7r1 x 7r2 is cuspidal, it is m(7rH) =
1 (1+ (  l ) n ( " H ) ) 2
( E (0,l)).
Here n ( 7 r H ) is the number of components 7rzv of ITH (it is bounded b y the number of places v where both 7 r l u and 7raU are square integrable). Each ITH with m ( 7 r H ) = 1 is cuspidal. The multiplicity m ( 7 r H ) (in the discrete spectrum of PGSp(2,A)) of ITH = @ V T H ~ f r o m a quasipacket X o ( a l 2 x 7 r 2 ) , where 7r2 is a cuspidal representation of PGL(2,A) and CT is a character of A X / F X A X 2is,
where n ( 7 r H ) is the number of components 7rEu of T H , and ~ ( ~ 2 , is s ) the usual &factor which appears in the functional equation of the Lfunction L(7r2,s ) . I n particular 7rff = @7rGv ( n ( 7 r H ) = 0 ) is in the discrete spectrum i f and only if c ( m 2 , $) = 1. Finally we havern(.rrH) = for7rH = @ T H ~in X o ( a J l 2 ~ ) 7 r =~X,.~ Here T H = @Lu ( n ( 7 r H ) = 0 ) is 0 1 2 ) with ~ ( T H components residual.
i(l+(l)n("H))
I. Preliminaries
22
2h. Unstable Spectrum Note that the quasipacket packets
Xo(a[l2
x
al2)
is defined by the local quasi
c2
for every u,where E (f l),a are characters of A X / F Xwith = 1 = a2 and EVl aV are their components. When E,, a, are unramified, this quasipacket contains the unramified representation 7rLV = L,. Members of this quasipacket have been studied by means of the theta correspondence by Howe and PiatetskiShapiro, see, e.g., [PSl], Theorem 2.5. They attracted interest since they violate the naive generalization of the Ramanujan conjecture, which expects the components of a cuspidal representation to be tempered. (The form of the Ramanujan conjecture which is expected to be true asserts that the components of a cuspidal representation of PGSp(2, A) which Alifts to a representation of PGL(4, A) induced from a cuspidal representation of a Levi subgroup, are tempered.) Members of this quasipacket are equivalent at almost all places to the quotient of the 2 properly induced representation v[ x [ x o v  ' f 2 . Let 7r2 be a cuspidal representation of PGL(2,A), a a character of A x / F X A X 2 . The packet A o ( a l 2 x 7 r 2 ) contains the constituent 7r; = v1/2 18~nG of ~the representation av1/27r2 XI ~ v  ' pv / ~avlf27r2 XI properly induced from an automorphic representation, hence it is automorphic by [L4]. It is known that 7rG is residual precisely when L(a7r2,$) # 0; hence ~ ( 0 ~$)2 = , 1 in this case. Let n(7r2)denote the number of square integrable components of 7 r 2 . The quasipacket X o ( a l 2 x 7 r 2 ) thus consists of 2n("2) (irreducible) representations. If n(7r2)2 1, half of them in the discrete spectrum, all cuspidal if L(a7r2,?j) = 0, all but one: T; = @ v 7 r ~ v lare cuspidal if L(a7r2,f) # 0. If n(712) 2 1 and L(a7r2,?j) = 0, the automorphic nonresidual 7rG is cuspidal when ~ ( a 7 r 2 , = 1. If 7r2 has no square integrable components (n(7r2) = 0), the packet A o ( a l 2 x 7r2) consists only of 7rG. This 7rG is residual if L(a7r2,;) # 0; cuspidal (by [PSl],Theorem 2.6 and [PS2], Theorem A.2) if L(a7r2, = 0 and ~(a7r2, = 1; or (automorphic but) not in the discrete spectrum otherwise: L ( m 2 , f) = 0 and 4a7r2, = 1. In this last case the Ao
i)
i)
i)
i)
2h. Unstable Spectrum
23
lift of 0 1 2 x 7r2 is not in the discrete spectrum, and there is no discrete specrtum representation Alifting to I ~ ( 0 1 2~ ,2 ) . At a place w where 7r2, is induced I ( p v ,p ; ' ) , the packet
x is the irreducible induced pVa,12 p ; ' , which Alifts t o the induced I G ( ~ g, ,V 1 2 ,p ; ' ) , and not the irreducible induced
) which Alifts t o the reducible induced I,(pv, ~ , I ( v t / v,"~), ~, p ; ' ) , which has the constituent I G ( ~ 0 ~, , 1 2 ,p;'). Members of the quasipacket A o ( a l 2 x 7 r 2 ) were studied numerically by H. Saito and N. Kurokawa, and using the theta correspondence by PiatetskiShapiro and others, see [PSl], Theorem 2.6. They attracted interest since they violate the naive generalization of the Ramanujan conjecture. They are equivalent at almost all places to the quotient of the properly induced representation av1I27r2x a ~  ' / ~ . A discrete specrtum representation T H with a local component
(whose packet consists of itself), where 7rzv is a cuspidal representation with central character tv # 1 = t: and t v 7 r 2 v = 7 r ~ ~isz ~in , the packet of L ( V ( , V  ' / ~ T ~Here ) . 7r2 is cuspidal with central character [ # 1 = t2, hence E7r2 = 7r2, whose components a t u are 7r2v and tv. It Alifts t o J ~ ( v ' / ~ 7 r ~ , v  ' / ~At 7 ~v2 with ). tv = 1 the component 7rzv is induced. If 7rav = I ( p v , p v t v )[," , = 1 and p: = 1 (in particular whenever Ev # 1 and 7rav is not cuspidal), then L(v,[,, vv ' I 2 7 r 2 , ) is L, = L(vvEv,tvx p, vv 1 / 2 ) , which Alifts to I c ( p v 1 2 ,pvtv12),and its packet contains also X , = X ( V ; / ~ <S,P ~ , Evpvvv , ' I 2 ) .Thus the packet of T H is determined by {L,,X,} at all v where 7rzv = I(pv,pvtv), p: = 1 = E,", and by the singleton { L , = L(v,~v,v~1127r2v)} a t all other w , where 7rzv is cuspidal, or [, = 1 and 7rav = I ( p , , p ; ' ) , p: # 1. Each member of this infinite packet occurs in the discrete specrtum with multiplicity one, and is cuspi~ V ' / ~ 7 r 2 ~which ), is dal, with the exception of L ( u t ,v'l27r2) = @vL(vv&,,
24
I. Preliminaries
residual ([Kim],Theorem 7.2). Members of the packet L ( I / [v,  l I 2 7 r 2 ) are considered in the Appendix of [PSl] and its corrigendum. If 7r1 and 7r2 are cuspidal but there is no place u where both are square integrable, Xo(7rl x 7r2) consists of a single irreducible cuspidal representation. This instance of the lifting A0  where i ~ are ~ i cuspidal can also be studied ([Rb]) using the theta correspondence for suitable dual reductive pairs (S0(4), PGSp(2)) for the isotropic and anisotropic forms of the orthogonal group, to describe further properties of the packets, such as their periods. 2i. Generic Representations Our proof of the existence of the lifting X uses only the trace formula, orbital integrals and character relations. However, for cuspidal representations 7r1, 7r2 of PGL(2,F), F local, we get only the character relation
Here f on G = PGL(4,F) and f H on H = PGSp(2,F) are any matching functions, and m = m(7rl,7rz)is a nonnegative integer. To prove multiplicity one theorem for PGSp(2,A) we need the fact that m = 0. Our proof is global. It uses the following results from the theory of the theta correspondence, Whittaker models and Eisenstein series. (1) GinzburgRallisSoudry [GRS], Theorem A: Each representation I(7r1,7r2) of PGL(4,A) normalizedly induced from a cuspidal representation 7r1 x 7r2 of its (2,2)parabolic, where 7rl # 7r2 are cuspidal representations of PGL(2,A), is a Xlift of a unique generic cuspidal representation TH of SO(5, A) = PGSp(2,A). (2) KudlaRallisSoudry [KRS], Theorem 8.1: If 7ro is a locally generic cuspidal representation of Sp(2,A) and the partial degree 5 Lfunction L(S,7r0,ids1s) is # 0 at s = 1 then TO is (globally) generic. (3) Shahidi [Shl], Theorem 5.1: If 7r0 is a generic cuspidal representation of Sp(2,A), then L(S,ro,ids,s) is # 0 at s = 1. See chapter V, section 7, and the final remark in section 6, for further comments. We do not use the assertion (attributed to “a yet to be published result of Jacquet and Shalika”) in the Remark following the statement of Theorem 8.1 in [KRS],p. 535 (that a cuspidal representation of GSp(2) is generic iff it lifts to GL(4)), which contradicts  at least as stated  our result that all
2i. Generic Representations
25
representations but one in a packet of PGSp(2) are nongeneric, yet they all lift to PGL(4). Our global proof resembles (but is strictly different from) the second proof of [F4;II],Proposition 3.5, p. 48, which is also based on the theory of generic representations. This Proposition claims the multiplicity one theorem for the discrete spectrum of U ( 3 , E I F ) . 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 l”,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 T in L& have multiplicity one only in the subspace L& of the discrete spectrum. This claim does not exclude the possibility of having a cuspidal T‘ perpendicular and equivalent to T C Li,l. Multiplicity one for the generic spectrum would follow via this global argument from the statement that a locally generic cuspidal representation is globally generic (multiplicity one implies this statement too). In our case of PGSp(2) we deduce from [KRS], [GRS], [Shl]that a locally generic cuspidal representation which is equivalent at almost all places to a generic cuspidal representation is globally generic. A proof for U(3) still needs to be written down. The usage of the theory of generic representations in the proof described 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 representation near the origin. Details of this proof are given in [F4;IV] in odd residual characteristic in the case of basechange for U(3). It implies that in a tempered packet of representations of U(3, E / F ) there is precisely one generic representation, and that each generic packet of discrete spectrum representations of U(3, AE/AF) where a generic packet means one which lifts to a generic representation of G L ( ~ , A E ) would contain precisely one generic member. Moreover, a locally generic cuspidal representation of U(3, A,/A,) F is generic. This type of a local argument was introduced in [FKl] in the proof of the metaplectic correspondence and the multiplicity one theorem for the discrete spectrum of the metaplectic group of GL(n,A). We have not
26
I. Preliminaries
carried out this local proof in the case of PGSp(2) as yet. In the case of PGSp(2) our global proof implies that a local tempered packet contains precisely one generic representation, and that a global packet which lifts to a generic representation of PGL(4, A) contains precisely one everywhere generic representation. The latter is generic if the packet is unstable (in the image of the lifting XO). We do not show that a locally generic cuspidal representation of PGSp(2, A) which is stable (Xlifts to a cuspidal representation of PGL(4, A)) is generic. There is some overlap between our results on the existence of the Xlifting and the work of [GRS] which asserts that the weak (i.e., in terms of almost all places) lifting establishes a bijection from the set of equivalence classes of (irreducible automorphic) cuspidal generic representations of the split group S0(2n+1, A), t o the set of representations of PGL(2n, A) of the form 7r = I(7rl1.. . , T,), normalized induction from the standard parabolic subgroup of type (2n1,. . . , 2n,), n = nl . . . n,, where 7ri are cuspidal representations of GL(2ni,A) such that L(S, 7 r i , A', s ) has a pole at s = 1 and 7ri # 7rj for all i # j , and the partial Lfunction is defined as a product outside a finite set S where all 7ri are unramified. Of course we are concerned only with the case n = 2, where PGSp(2) N SO(5). Our characterization of the lifting X is (as in [GRS]) that I ( 7 r 1 , 7 r ~ ) , cuspidal representations 7r1 # 7rz of PGL(2,A), are in the image; and that self contragredient cuspidal representations 7r of PGL(4, A) are in the image of the lifting X from PGSp(2, A) (= SO(5,A)) precisely if they are not in the image of the lifting X1 from SO(4, A). The cuspidal 7r = X ( 7 r ~ ) ,generic T H , are characterized in [GRS] as the 7r N ir such that L(S, 7rl A', s)' is 0 at s = 1. Thus the characterization of the cuspidal image of X here is complementary to but different than that of [GRS].
+ +
However, the methods of [GRS] apply only to generic representations, while our methods apply to all representations of PGSp(2). In particular, we can define packets, describe their structure, establish multiplicity one theorem and rigidity theorem for packets of PGSp(2), specify which member in a packet or a quasipacket is in the discrete spectrum, and we can also Alift the nongeneric nontempered (at almost all places) packets to residual selfcontragredient representations of PGL(4, A). Our liftings are proven in terms of all places, not only almost all places. In addition we establish the lifting X I from SO(4) to PGL(4), determine its fibers (that
2j. Orientation
27
is, prove multiplicity one theorem for SO(4) and rigidity in the sense explained above) , and show that any selfcontragredient discrete spectrum representation of PGL(4,A) which is not a Alift from PGSp(2,A) is a XIlift from SO(4, A). 2j. Orientation
This work is an analogue for (S0(4),PGSp(2),PGL(4)) of [F3], which dealt with (PGL(2),SL(2),PGL(3)), thus with the symmetric square lifting, and of [F4], which dealt with quadratic basechange for the unitary group U(3, E / F ) , thus with (U(2,E / F ) , U(3, E / F ) , GL(3, E ) ) . These works use the twisted  by transposeinverse (and the Galois action in the unitary groups case) trace formulae on PGL(4), PGL(3), GL(3, E ) . They are based on the fundamental lemma: [F5] in our case, [F3;V]and [F4;I] in the other cases. The technique employed in these last works benefited from work of Weissauer [W] and Kazhdan [Kl]. The present work, which deals with the applications of the fundamental lemma and the trace formula to character relations, liftings and the definition of packets, is analogous to [F3;IV] and [F4;II]. The trace formula identity is proven in [F3;VI] and [F4;III] for all test functions. Here we deal only with test functions which have at least three elliptic components. The trace formulae identity for a general test function has not yet been proven in our case. Perhaps the method of [AC] could be used for that, as it has been applied in a general rank case. It would be interesting to pursue the elementary techniques of [F3;VI]and [F4;III], and [F2;I],which establish the trace formulae identity for basechange for GL(2) by elementary means, based on the usage of regular, Iwahori test functions. In particular the present work does not develop the trace formula. It only uses a form of it. Our approach uses the trace formula, developed by Arthur (see [All), as envisaged by Langlands e.g. in his work on basechange for GL(2). Of course Siegel modular forms have been extensively studied by many authors (e.g., Siegel, Maass, Shimura, Andrianov, Freitag, Klingen ...) over a long period of time, and several textbooks are available. As noted above, an important representation theoretic approach alternative to the trace formula, based on the theta correspondence, Weil representation, Howe’s dual reductive pairs, Lfunctions and converse theo
28
I. Preliminaries
rems, has been fruitfully developed in our context of the symplectic group by PiatetskiShapiro, Howe, Kudla, Rallis, Ginzburg, Roberts, Schmidt, Soudry, and others, see, e.g., [PSI, [KRS], [GRS], [Rb], [Sch]. A purely local approach to character computations is developed in [FZ]. Our results are used by P.S. Chan [Ch] to determine the representations of GSp(2) which are invariant under twisting by a quadratic character. The classification of the automorphic representations of PGSp(2) has applications t o the decomposition of the &ale cohomology with compact supports and twisted coefficients of the Shimura varieties associated with GSp(2), see [F7]. Our techniques extend to deal with admissible and automorphic representations of GSp(2), but this we do not do here. The present part is divided into five chapters: I. Introduction, 11. Basic Facts, 111. Trace Formulae, IV. The Lifting X I , V. The Lifting A. Each is divided into sections. Definitions or propositions are numbered together in each section.
3. Conjectural Compatibility Our local results are analogous to those of Arthur [A2], who verified them in the real case, and are consistent with his conjectures. We shall assume in this section, not to be used anywhere else in this work, familiarity with [A2], [A3], and briefly highlight some of the definitions and conjectures of [A21 in our context, in our notations ( H , C o in place of Arthur’s G , H ) . For brevity we write WF for the Weil group of the local field, but as in [A2],2.1, this group has to be the motivic Galois group of the conjecturally Tannakian category of tempered representations of all GL(n)’s in the global case, a complex proreductive group, or an extension of W Fby a connected compact group (WFx SU(2,IR) in the padic case). Thus @ ( H I P )denotes the set of ficonjugacy classes of admissible (in particular, pr, 04 = idWF) maps
4 : WF+ L H = fi x WF
(fi= L H o ) .
It contains the subset @temp(H/F) defined using the 4 with bounded , .2 is simply Im(pr, 04). Note that for a split adjoint group H over F,connected, and for any semisimple s in H , the centralizer CO= Zg(s) of s in 2 specifies the endoscopic group H uniquely (up to isomorphism). Write S, = S,f = Z E ( ~ ( W F(centralizer )) in the connected group H of h
h
3. Conjectural Compatibility
29
the image of 4), 2 = Z S ( ~ H c ) g,and note that S, = SQ/S$Z^is a finite abelian group, conjecturally in duality with the packet II, to be associated with 4 E I I t e m p ( H / F )(this is the case when F = R, see [A2]). Arthur [A21 defines a further set 9 ( H / F ) of gconjugacy classes of maps $ : W F x SL(2,C) 4 L H such that $(WFE atemp(H/F), and a map
which embeds 9 ( H / F ) in @(H / F ) . Each 11, can be viewed as a pair
( 4 , ~E) (@temp(H/F)x Hom(SL(2, C S,))/Int(S,) (quotient by S#conjugacy). Then Q t e m p ( H / F )embeds in 9 ( H / F ) as the (4>1).Put S$ = s; = ZS($(WF x SL(2,C))). It is equal to ZS4,
(P(SL(2,el)>,
a subgroup of S,+, and there is a surjection S, = S,/S$ZZ n S,, . The group S, is in duality with the quasipacket II$ conjecturally associated with 111. Globally, the quasipacket II, contains no discrete spectrum representations of H unless S$ is finite. /. Let us review the examples of [A2], where H = Sp(2,C) II Co = h
SL(2, C ) x SL(2, C ) =
{ (i i i) }.
h
The parameter $ can be described
by the maps
(4 = 41 x If
4i : WF
42,
p = p1 x pz) : W F x SL(2,C) + SL(2,C) x SL(2,C).
SL(2,C) are irreducible and inequivalent, p = 1,
ZsL(2,c)(Im4i)= {*I),
s,, = 2 / 2
x 2/27
s,,
= 2/2,
S, = 2 / 2 x 2 / 2 , S, = 2/2. This is a “classical” tempered case, as Im4i are bounded. If 41 = 4 2 is irreducible, p = 1, S,, = O(2, C ) = S, (this group consists
of the diag(g,g*),g* = wtgP1w, w =
( y i), which commute with (: ),
thus gtg = I ) , S$ = SO(2,C) and S,, = S, = 2 / 2 (= (diag(w,w))).
I. Preliminaries
30
These cases correspond to Xo(7rl x xz), where 7r1, 7r2 are in the discrete spectrum; a local packet consists of 2 = [2/2] elements. A global packet in the second case consists of no discrete spectrum representations since S+ = O(2,C) is not finite. In the first case, where 7r2 # 7r1, the packet consists of 2n irreducibles, where n is the number of places where both ~1 and 7r2 are square integrable; half of the members in the packet are in the discrete spectrum (one, if n = 0). If 41 is irreducible and Im(4z) C { & I } , and p = 1 x id, we have S,, = 212 x C x (= {diag(L,z,z',L); z E C x , L E {&1}}),S, = {l}, S, = 2 / 2 x 2/ 2, S+ = 2/ 2. This is the case of Xo(7rl x Cpzlz), where Cpz is a character, If Imq5i c { & I } but 41 # $2, and pi = id, 5'4, = C x x Cx (= {diag(z, t , t  l , z'); z , t E C " } ) , S$$ = {l},S+= 212 x 2/2, S+ = 2/2. This is the case of X o ( $ l l z x 4 2 1 2 ) , where $1 # 4 2 are characters of FX/Fx2 or A x / F X A X 2 . If d1 = 42 with image in { & I } , and pi = id, ,S , = GL(2,C) (= {diag(g, g*); g E GL(2, C)}), S4, = {l},S+ = O(2, C), S+ = 2/2. This is the case of Xo(q51lz x q5112), whose packet contains no discrete spectrum representations, and indeed S+ = O(2, C ) is not finite. In addition we determine that the multiplicity d+ of [A2],p. 28, is one.
4. Conjectural Rigidity This section explains the rigidity theorem for SO(4) via the principle of functoriality. It is based on conversations with J.P. Serre at Singapore. 4.1 PROPOSITION. Let
qi, 77;: W F 4 GL(2,C) be (irreducible continuous) representations of the Wed group WF of F which are unramified at almost all places v ( s o they depend there only on the Frobe~ 71 @ T]~IWF, for almost all Y and nius element) with 71 @ r 7 2 l W ~ Y with det ql . det 7 2 = det 7: . det 7;. Then there exists a homomorphism x : W F C x such that 7 ; = xql and q; = ~ ~ ~ 7 or 7 2qh, = xqz and 771, 772,
f
q; = x
h.
Since the subgroup of WF generated by the Frobenii is dense, we may consider instead a group r (instead of W F ) ,and two representations pi
4.
Conjectural Rigidity
31
(instead of 771 €4 772) which are locally conjugate, which means that p l ( y ) is conjugate to p2(y) for each y in I?, or alternatively that the restrictions of p1, p2 t o any cyclic subgroup are conjugate. We wish to know whether they are conjugate as representations. We say that a group G over C has the rigidityproperty if for any group I?, any two locally conjugate representations p1 , pz : r + G(C) are conjugate. Variants are naturally defined (for special r and p). For example, if r is finite and G = GL(n), character theory asserts that locally conjugate p 1 , p 2 : I? + GL(n,C) are conjugate. The group G = GL(n) has the rigidityproperty for any semisimple continuous representations p1, p2 of the Weil group. On the other hand, the group PGL(n,C) does not have the rigidityproperty since it is the dual group of SL(n), for which rigidity does not hold. In our case we wish to know whether locally conjugate p 1 , p2 into SO(4,C)are conjugate. They are not, but almost are: they are conjugate in O(4, C),which is the semidirect product of SO(4, C)with an element which maps 771 @ 772 to 772 €4 71. We proceed to explain this via the group theoretical notion of fusion control. 4.2 DEFINITION.Given groups G 3 H' 2 H we say that H' controls t h e fusion of H in G if for any sets A , B in H and g in G with g A g  l = B there is h in H' with hah' = gag' for every a in A , namely h'g g lies in the centralizer C G ( A )of A in G. 4.3 EXAMPLE. Let S be an abelian pSylow subgroup in a finite group G, and N = NG(S)the normalizer of S in G. Then S C N c G and N controls the fusion of S in G.
PROOF.Since S is abelian and A is a subset of S we have that S is contained in the centralizer C G ( A )of A in G. Hence S is a pSylow subgroup of CG(A).Now the abelian S commutes with any subset B of S , hence g  l S gg commutes with g  l B gg = A , and so g  l S g h i sis a pSylow subgroup of C G ( A )for any g in G. Since pSylow subgroups are conjugate, Then there is u in &(A) with g  l S gg = u5'ul; take h = g u E NG(S). 1 1 hah' = guaulg' = gag' for any a in A. 0 4.4 EXAMPLE. Let G be an algebraic reductive group, T a maximal torus and N = NG(T) the normalizer of T in G. Then T c N c G and N controls the fusion of T in G.
32
I. Preliminaries
PROOF. If A is any subset of the abelian T , we have that T lies in the centralizer CG(A)of A in G. Hence T is a maximal torus in C c ( A ) . Now T commutes with any of its subsets B , hence g  l T g commutes with g  ' B gg = A , and so g  l T g is a maximal torus in C c ( A ) . Since maximal tori of a reductive group are conjugate, there exists u in C c ( A )such that g'Tg g = uTu'. Hence h = g u lies in NG(T)and satisfies huh'1 s = g u m  ' 1 = gag' for any a in A. 0 4.5 PROPOSITION. Let S = t S be a symmetric matrix in GL(n, C). Put g* = S t g P 1 S  l . T h e n the orthogonal group O ( S ,C) = { g E GL(n, C ) ;g = g * } controls its o w n fusion in GL(n, C).
PROOF. Suppose that A , B are subsets of O ( S , C ) and g E GL(n,C) 1 satisfies gAg' 1 = B. For each a in A we have a* = a , hence g*ag*' = ( g a g p 1 ) * = gag' (as b = b* for b = gag'). Then c = g'g* commutes with each a in A, and c*' 1 = StcS' 1 = Stg*tgl,'l  1 = glStg'S' 1 = g'g* = c. Let d be a square root of c, thus c = hd 2 . Using the binomial expansion u ' / ~= C,"=,( i ) ( u  l)nfor a unipotent matrix u and ( r e i 0 ) 1 / 2 = r 1 / 2 e i 0 / 2 (0 < 9 < 27r, r > 0), the Jordan decomposition c = su = u s and diagonalization, we express d as a function f(c) in c, where f satisfies f(syz') = zf(y)z' and f(tz)= 'f(z). Then
and h = (gd)' = g*d* = gcd' 1 = gd satisfies (gd)a(gd)' = gag', for all a in A. 0
4.6 COROLLARY. If p1, p2 : r 4 O(S,C) are representations of a group r into the orthogonal group O ( S ,C ) , and there is g in GL(n, C) with 0 pz = gp1g', then there is h in O ( S , C ) with p2 = h p l h  l . REMARK.The last Proposition and its Corollary hold (with the same proof) for the symplectic group Sp(S, C), defined using S =  tS.
4.7 PROPOSITION. Let 771, 772, q i , 7;: r +h GL(2, C)be representations of a group r with 771 @I 772 !x 77; @I 77; in GL(4,C) and det q1 . det q 2 = det 77; . det 77;. T h e n there exists a homomorphism : r + C x such that q; = xql and 7; = x'772h2 oor 77; = x77z and 77; = x'r]l.
x
4.
Conjectural Rigidity
33
PROOF.The tensor products p = 71 @ 772 and p' = @ 77; have images in SO(S,C) c O(S,C) where S = 6J = antidiag(l,1,1, l),
d = diag(1, 1,Hence p and p' are equivalent in O ( S ,C ) = SO(S,C)x (L ) , where L = diag(l,w,l) acts on a @ b in SO(S,C) (a,b in GL(2,C), detab = 1) by L : a @ b H b@a. So p is equivalent under SO(S,C ) to p' or to 'p' = q i @ ~ i , and ( q l , q 2 ) is equivalent to (xq;,x'q;) or to ( x q ; , x  ' q ; ) . The map x : r + C" is a homomorphism since so are the qi, q:, i = 1,2. o We also note the following analogue for the group of similitudes.
I f the representations p, p' : r + GO(S,@.)( o f a 4.8 PROPOSITION. group l' into the group of orthogonal similitudes) are conjugate in GL(n, C) (3 S = t S ) and have the same factor X of similitudes, then they are conjugate in O ( S ,@.).
r
PROOF. Replacing I' by the 2fold cover = T x c x,$"C (fiber product of X : r 4 C x with C" 4C",0 : z I) z 2 ) ,there is a character p : f' + @." with X = p 2 :
F 2 @.x 1 10 x
r

cx
Then p  l p , plp' : p + O ( S , C ) are conjugate in GL(n,C) hence also in O ( S , C ) , and so p, p' : T 4 O(S,C) are conjugate in O ( S , C ) and they 0 factorize via pr:F 4 r. We can now return to our initial Proposition 4.1. If the irreducible continuous representations 7 7 1 , 772, qi, q; : WF + GL(2,C) are unramified and satisfy ql @ q2(F'rv) 2~ 7: 8 vh(Fr,) for almost all places w , then p = 771 @ 772 and p' = 77: @ qk : WF * SO(S,C)c O (S ,C)c GL(4, C )
are conjugate in GL(4,C) (since the Frobenii are dense in WF and p, p' are semisimple). Hence they are conjugate in O ( S , C ) and there is a homomorphism x : WF + C x with qi = x q 1 , ~ = ; x'772, or qi = ~ 7 7 2 , q; = Xlql.
34
I. Preliminaries
Had we known the Principle of Functoriality, namely that discrete spectrum representations 7rz of GL(2, A) are parametrized by two dimensional representations qz : I? + GL(2,C) of a suitable Weil group r(=W F ) , we could conclude the rigidity theorem part of our global theorem about the lifting XI from C = SO(4) to PGL(4). However, this Principle is known W E , J F )induced , only for monomial representations qz = Ind(pz;WE,JE,, from characters p2 of WE,JE,= A;,/E:, where E, is a quadratic extension of F . Thus we get an alternative proof  based only on class field theory and the basic group theoretic consideration above  of the special case for monomial representations 7rz = 7r(pz)stated after that theorem. Note that the rigidity property, that any locally conjugate p, p’ : r + G(@) are conjugate, holds for G = GL(n), O ( n ) ,Sp(n) and Ga, and for any connected, simply connected, complex Lie group precisely if it has no direct factors of type B,(n 2 4), D,(n 2 4), E, or F4. For this and related results see Larsen ([Lar]).
11. BASIC FACTS
1. NormMaps The norm maps are formally defined by the dual group maps, as we proceed the diagonal torus in &, and by the diagonal to explain. Denote by torus in fi, T G in Co and T> in H. Then
PH
PO
h
X*(?o) = X*(TH)= { ( a ,b, b, u); U ,b E Z} is the lattice of 1parameter subgroups, while the lattices of characters are
X*(?O) = X*(?H) = {(x,y,a,t)mod(n,m,m,n);x,y,z,t E Z}; here (x,y, z , t ) takes diag(u, b, bl, u  ' ) in f ? or~diag(u,u  ' ) x diag(b, b  l ) in ?o to aztbYz. Further we have X*(?o) = X*(Tg),while the isomorphism X*(?,) 2 X,(T*,)
is given by
Y,t, t ) H .(x
(2,
+ Y,x +
2,y
+ t ,z + t ) ,
with inverse (u:,P,Y,6 )
(u: Y,a  P , O , O ) .
In particular the map is
and we make 1.1 DEFINITION. The norm m a p N : T*, 2 T;I is defined by
diagf 35
II. Basic Facts
36
The elements ( a ,b, b, a) of X*(?O)= X*(TE)can be viewed as characters of T::
Under the isomorphism N : Th 7TE, diag h
the elements (a,b, b, a) of X*(TH)1: X*(TT,)can be viewed as characters of T&:

( a ,b, b, a) : diag(a, P, 776) H (,./r)"(a/P)b. A
Hence corresponding to the "endoscopic" lifting A0
A0
:
TO1?H induced by XO :
EO
H
we have
: r 2 ( p 1 , p y 1 ) x rZ(p2,/'~;~)I+ r P G S p ( 2 ) ( p l r p Z ) .
Here x z ( p i , pi') is the unramified irreducible constituent of the normalizedly induced representation I ( p i , pi') of PGL(2, F,) (pi are unramified characters of F,", i = 1 , 2 ) ; r p G S p ( 2 ) ( p 1p, 2 ) is the unramified irreducible p ~ normalizedly ) inconstituent of the PGSp(2, F,)module I p G S p ( 2 ) (PI, duced from the character n . diag(a,P,y,6) H p l ( a / y ) p 2 ( a / P )of the upper triangular subgroup of PGSp(2, F,) ( n is in the unipotent radical, a6 = p y ) .

Corresponding to the embedding A : 2 = Sp(2,C) have the natural embedding
SL(4, C)=
we
z}
x*(TL) X*(?H)= ((2,Y,9, .); z,y E tX,(?) = { ( ~ , y , z , t )€ Z 4 ; ~ + y + z + t = 0 } = X * ( T * ) . The torus TT, consists of diag(a, P, y,6) mod(zI4), a6 = Py, and the character ,.( y, y, ).maps this element to ( a / ~ ) ~ ( a / P (*). ) g The torus T* consists of diag(a, P, y, 6) in PGL(4). Dual to the embedding X : ?H = {diag(a, b, b  l , u  ' ) }
? = {diag(a, b, c, d ) ; abcd = 1)
~1
1. Norm Maps
37
there is the map of the character lattices
The isomorphism
leads us to make the
1.2 DEFINITION. The norm map N : T* 4 TL is given by N(diag(a, b, c,d)) = diag(ab,ac,bd,cd). The dual map of Ccharacters
is given by X(x)(diag(a, b, c, d ) ) = X(N(diag(a, b, c, d ) ) = X(diag(ab, ac, bd, cd)). If
x = (z,9, y,
z) then
X(X)(diag(a, b, c, d)) = (ab/bd)z(ab/ac)Y = a”bYcYdz (by ( 0 ) 18 lines above) as expected. In other words the lifting X maps the unramified irreducible PGSp(2, F,)module T I T ~ G S(PI, ~ ( ~p2) ) to the unramified irreducible PGL(4, F,)module 7r4(pl, p2, p;’, p ; ’ ) . Note that the norm map extends to the Levi M(2,2)of PGL(4) of type
(2,2)by N
(f
i) = (
detA E B E A 0 det B
),where
E
=
(; !).
It takes 8
conjugacy classes in M(2,2)to conjugacy classes in the Levi of type (1,2,1) in PGSp(2). Indeed,
0
(cd:tAx
cd det B
)
= cd
(
detA C  ’ E B E A G 0
det B
II. Basic Facts
38
where c = det C , d = det D , and
X = E W ~ C W B D E W ~ D= WcdCleBEAC AC is conjugate to EBEAtimes cd. Moreover, it extends to the Levi of PGL(4) of type (1,2,1) by
0 dsAs
)
It takes
to IJvdetB
(
aB'AB
0
0
deBlABe
The composition
x 0 A0 : eo = SL(2,C) x SL(2,C) + (3 = SL(4,C) takes
7r2(Pll PI1)
x
7r2(P2,
pi1)
T4(p1,p2r&1,pT1)
to
= 7r4(p1,pT1ik2ipF1)i
namely the unramified irreducible PGL(2, F,) x PGL(2, F,)module 7r2 x 7rh to the unramified irreducible constituent ~ 4 ( ~ 7rL) 2 , of the PGL(4, F,)module 14(7r2,7rL) normalizedly induced from the representation 7r2 @ T ; of the parabolic of type (2,2) of PGL(4, F,) (extended trivially on the unipotent radical). For example XoXo takes the trivial PGL(2, F,) x PGL(2, F,)module 1 2 x 1 2 to the unramified irreducible constituent 7 r 4 ( 1 2 , 1 2 ) of 1 4 ( 1 2 , 1 2 ) , and 1 2 x 7r2 to 7r4(12,7r2) = 7 r 4 ( d / 2 7 r 2 , ~  ~ / ~ 7 r 2Note ). that this last ~4 is traditionally denoted by J. The embedding A1 :
e = [GL(2,C)x GL(2, C)]'/Cx 1SO
i))+ 6 = SL(4, C )
defines an embedding of diagonal subgroups
% = {( (; a ; ) , ( 2 ;)) = ~ ~ ~ ~ ( ~ 1 ~ 1 , ~ 1 ~ 2 , ~ 1 ~ 2 , a z = b zI}, ~ ; T^ = {diag(a, b, c, d);abcd = l}, ~f
2. Induced Representations
39
and lattices
(X*(TZ)=)
X,(Tc)
Q
X,(T)
(= X*(T*)),
(51,52;~1,~2 H ) ( ~ i + y i , ~ i + y a ; ~ i f ~ z , ~ z +51+52+y1+yz yz), = 0. Here TZ and T* are the diagonal subgroups of C = [GL(2)xGL(2)]'/ GL( 1)and G = PGL(4). The dual map N : X,(T*) = X*(?) X*(?C) = X*(T&), or f
satisfies
((:
a:1a;2by1by = ( 2 1 , ~ z ; Y l 1 Y 2 ) = (51
I ( 2
;))
=x(N(a,P,y16))
+ y l , x 1 + YZ;YI + 52,xz + y2)(diag(oIPI y,6))
 (pl+Yl

aoz)
P
Zl+YZ
r
Yl+ZZ(p+YZ
= (QP)"l(y6)Q (ay)Y'(P6)YZ
for all
x = ( ~ 1 ~ xy1,yz) 2; in X*(Tb),hence we are led to make the
1.3 DEFINITION. The norm map N : T* + TZ is defined to be
2, Induced Representations Let us recall the computation of the character of a representation w = I(q) of G = G ( F , ) normalizedly induced from the character q of the Bore1 subgroup B = A N , N the unipotent radical and A the maximal torus in B. If K is the maximal compact subgroup with G = B K = N A K , the space of w consists of the smooth 4 : G + C with d ( n a k ) = (61/2q)(u)@(k)l where S ( a ) = I det(Ada1 LieN)I and w acts by right translation; of course a E A , k E K , n E N . In Lemma 2.1 G can be any quasisplit connected reductive group. Recall that the character ([HI) of an admissible representation w is a conjugacy invariant locally integrable function xs satisfying tr w(fdg) = S G x T ( g ) f ( g ) d g for any test function f E CF(G). It characterizes the representation up to isomorphism.
40
II. Basic Facts
2.1 LEMMA.The character xa of the induced representation is supported on the split set and we have for regular a E A
IT
= I(7)
WEW
PROOF.There is a measure decomposition dg = 6'(a)dndadk corresponding to g = n a k , G = N A K . For a test function f E CF(G) the convolution operator IT( f d g ) = JG .rr(g)f(g)dg maps q5 E .rr to
The change of variables n1 ct n, where n is defined by nlana' has the Jacobian I det(1  Ada)[Lie NI.
= nl,
The trace of T(f d g ) is obtained on integrating the kernel of the convolution operator  viewed as a trivial vector bundle over K  on the diagonal h = Ic E K . Hence, writing
A ( a ) = 61/2(a)ldet(1  Ada)l LieNI, we have
where w(A)is the cardinality of the Weyl group W . Here W is the quotient of the normalizer of A by the centralizer of A in G. To conclude the proof of the lemma we now use the Weyl integration formula
2. Induced Representations
41
Here T ranges over the conjugacy classes of tori, x(g) is a conjugacy class function, A(t)z is the Jacobian Idet(1  Ad(t))l(LieN$LieN)l (over an algebraic closure F of F the torus T splits). N is the unipotent radical of a Bore1 subgroup containing T and N  is the opposite unipotent group: Lie(G/T) = Lie N Lie N and (det(1 Ad(t))(LieN( = b'(t)(det(l Ad(t))lLieNI.
0 Similar analysis applies in the twisted case, where A(t8) is defined in the course of the following proof. 2 . 2 LEMMA. The twisted character x,,(tO) of the induced 0invariant representation n = I(q) with q = q o 8 vanishes outside the 8split set (the set of 8conjugacy classes of A ) , and is given by
on the 0regular a E A .
PROOF. Let 8 be an involution of G preserving B and K , for example 8(g) = J  l t g  l J where G = GL(n,F) (or PGL(n,F), etc.) and J an antidiagonal matrix. Then tr(r(f8)) is zero unless r is equivalent to 'n(:g H n(8(g))),in which case, for r = I(q), we have
hence tr n(8f dg) =
///
f(8(k)1nlak)(61/2q)(a)dnldadk.
II. Basic Facts
42
We change variables n1 H n,where O(n)lanal = n l r which has the same Jacobian as if naO(n)lal = nl,which is (det(1Ad(aO))ILieN(, to get
Here we put
l  Ad(a0))l LieN(, A(a0) = ~ 5  ' / ~ ( a )det(1
A'
= { a E A ; a = O(a)}l
Al0
= {ae(a)'; a E A } .
We may choose a set of representatives T for the 0conjugacy classes of tori in G with T = B(T) ([KS]),such that on the regular set G=U
U
T
u e(gl)tg.
gETa\G
The corresponding Weyl integration formula is
sG X ( g ) f(g)dg
where A(t0)2 = I det(1  Ad(tI9))l Lie(G/T)I and d ( T )is the cardinality of the group W O ( Tof) 0fixed elements in the 0 Weyl group W ( T )of T . The lemma follows. 2 . 3 LEMMA.For t = diag(a, b, c, d ) we have A(t0) =
I
( a c  bd)2(ab  cd)2(a  d)2(b  c)2 (~bcd)~
PROOF.Note that Lie(G/T) = LieN invariant. We have,
I det(1  Ad(t0))l Lie"
@
LieN
= 1 n(1
I
and N , N are 19
a(t))l
2. Induced Representations
43
where the product ranges over the 0orbits 0 of the positive roots cr. > 0, and the sum over the roots in the &orbit. Thus for t = diag(a, b, c, d ) we cbtain I(1
if) ); (1
(1 
;)
(1 
;) I.
Further,
I det(1  Ad(t0))lLieNI
= 6(t0)ll det(1  Ad(t0))l Lie"
where 6(tB) is
The lemma follows. Recall now that the map A; : G = PGL(4)
+ C =
[GL(2) x GL(2)]'/GL(l)
dual to A1 :
6 = [GL(2,C ) x GL(2, C)]'/C"
maps diag(a, b, c, d ) to
((?
:d)
~t
6 = SL(4, C)
(?bod))'
2.4 DEFINITION. Let F be a local field. We say that f E C F ( G ( F ) ) weakly matches fc E C F ( C ( F ) )if
and
(t E T ( F ) ) are related by F f ( t 0 ) = F f c ( A T ( t ) )for t E A(F)'"g. This is a temporary definition, sufficient for the study of induced representations; it will be completed below.
II. Basic Facts
44
2.5 DEFINITION. We say that the induced C(F)module 7r1 x 7r2 lifts to the induced G(F)module 7r if tr(.rrl x 7 r z ) ( f ~ ) = trr(t9f) for all weakly matching f and f C .
Note that the characters of the induced C(F)modules and the twisted characters of the induced G(F)modules are supported on the split set, hence our temporary definition of weakly matching is sufficient. We conclude 2.6 PROPOSITION. The induced representation "C = k4Pl1P'1) x ~ Z ( P 2 l P 1 )
of C(F) X1lifts to the induced representation 7r
= 14(PIP2,PlPL PZPL PL:P1)
of G(F). Here p,,pi : FX3 C X are any characters with plpip2p; = 1.
PROOF. It suffices to observe that (PlP2, PIP11 P2PL Pu:P1)(diag(ajb, C? 4 ) = (PlP2)( a )(P;P;)(d)(PlP;)(b) (P2Pu:)(C)
on the Gside is equal to ~ l ( u b ) ~ : ( c d ) ~ 2 ( . c ) ~ on ~ ( bthe d ) Cside, and use the computation of the character of the induced and 0induced representations. 0 Similarly the map A* : G = PGL(4) 4 H = PGSp(2) dual to
x : il=
Sp(2,C) L+ 2= SL(4,C)
maps diag(a, b, c, d ) to diag(ab, ac, bd, cd). 2.7 DEFINITION. Let F be a local field. We say that f E CP(G(F)) weakly matches f H E CF(H(F))if Ff(tr3) and
are related by Ff(t0) = FfH(X*(t))for t E A(F)'"g. We say that the induced H(F)module T H lifts to the induced G(F)module 7r if for all weakly matching f and f H we have tr 7 r ~f H ( ) = tr 7 r ( O f ) .
2. Induced Representations

45
2.8 PROPOSITION. The induced H ( F )  m o d u l e I ~ ( p 1 , p z lifts )  via X to the induced G ( F )  m o d u l e 7r = 14(pu1, p2, pT1,11.1'). PROOF. It suffices to observe that (pil/*21pL21,p11)(diag(ul b,c,d)) = ~ i ( u / d ) ~ 2 ( b / c ) ;
and that ~ ~ ( pp2) 1 is , induced from the character d i a d a , P,r,6)
Pl(a/r)P2(m
of the diagonal subgroup of PGSp(2,F), and that the value of this last character at X*(diag(a,b, c, d ) ) = diag(ub, uc, bd, cd) is p1(ub/bd)p2(ub/uc).
0 Finally note that the map

A; : H = PGSp(2) dual to
A0
:
EO
4
Co = PGL(2) x PGL(2)
takes
We define 2.9 DEFINITION. The functions f H E CF(H(F))and fo E C?(Co(F)) are weakly matching if F f H ( t )= F f o ( t ) on t f A(F)reg. The induced Co(F)module 7r0 = T I x 7r2 lifts to the induced H(F)module X H if tr n ~ ( f =~tr)T O ( fo) for all weakly matching f H and fo. 2.10 PROPOSITION. The induced representation
12(P11PT1) x I2(P2, P A of Co(F) XOliftsto the induced representation I ~ ( p 1 , p of ~ )H(F).
PROOF. On the Hside, we induce from diag(a, Pl 71 6)
p 1 ( a / r ) p 2(alp).
This matrix is mapped by XY, to
((: ;) , (: ;))
1
and the Comodule is induced from the character whose value at this last 0 pair of matrices is p1(ayIy)p2(a/p).
II. Basic Facts
46
3. Satake Isomorphism Our liftings are summarized in the following diagram (X = GL(2, C))
e = SO(4,C)
eo = SL(2, C)x SL(2, C) Xo
\
ii = Sp(2,C)
x L)
J
N
[X x XI1/@'
A1
E = SL(4,C)
The dual group homomorphisms define liftings of unramified (local) representation. These representations are uniquely determined by the semisimple conjugacy classes that they define in the dual group. Thus the Xo, A, A 1 define liftings as follows. A0 : TZ(p1, pT1)
x
XZ(pZ,
pT1)
XPGSp(2)(pl,p2)
E
x
x PdPZ x P
A
T4(P1,p2,p21ip11)E JH(I~(p1,p21p21~p11)),
A : rPGSp(2)(pl,p2) A 1 : TZ(pl, pi)
JH(PlP2
7r2(P27
&)
T4(plpZ,p1pki p2p:,pipk), plp:l*.Zpk
= 1.
We write JH(7r) for the set of irreducible constituents of a representation 7 r , for example 7r = 7r1 x . . . x 7rT x D on GSp(n, F ) or 7r = 7r1 x . . . x 7rT = I(7rl,. . . , T ~ on ) GL(ln1,F). The subscript indicates that 7r2 is a representation of GL(2, F,) and 7r4 of PGL(4, F,). The p l , p2 are unramified characters of the local nonarchimedean field F,X; write pr for their values p i ( 7 ~ )at a uniformizer. Then the class t(xz(p.1,p i ) ) associated to the unramified irreducible 7rz(p1,p i ) is that of diag(p7, p i * ) in GL(2, C ) , t(TPGSp(2)(Pl,p2)) is the class of
in Sp(2, C), t(K4(plp27Plpk, p2P;i
pi&?))
is that of diag(P;P;,
I e
1.
PlPZ
7
PuXPCl'l', Pl
I .
P2
)
in SL(4,C). Note that the homomorphisms A, XO, A1 define dual homomorphisms of Hecke algebras, e.g., with G = G(F,), K = G(R,), . . . , A* : WG = C,"(K\G/K)
4
WH = C,"(KH\H/KH),
4. Induced Representations of PGSp(2,F)
47
by A* : f H f H , f g ( t ( ~ = ~ f"(t(n) )) x e), where the Satake isomorphism f H fV,fromWGto@[(Ao(@)x6)W] isgivenbyf"(tx0) = trn(t)(fxO), and f$(tH) = trnH(tH)(fH). In particular, by definition of corresponding functions f H A*(f) = f H , we have that
= trl4(f x O ) , where the traces of the full induced representation
at a spherical function f is equal to that at its unramified constituent.
4. Induced Representations of PGSp(2,F) We use results recorded in SallyTadic [ST]  using those of Rodier [ R o ~ Shahidi [Sh2,3] and Waldspurger [Wl]  on reducibility of induced representations of H(F) = PGSp(2,F), and unitarizability. Let us recall some notations. Denote by GSp(n) (or GSp(n)) the group of symplectic similitudes
Here w = w, = (&,n.j+l)in GL(n). Its standard parabolic subgroups are the upper triangular subgroups P, = PE with Levi subgroups
M, = ME = { m = diag(g1,. . . , g r , h , X ( h ) T g ; l , . . . , X ( h ) ' g ~ l ) } ; gi E GL(n,), h
E
GSp(n

Inl). Here n = (n1,.. . , n T ) , ni 3 1, r 3 0,
Put GSp(0) = G, = {X(h)}. These groups are in bijection with the set of
subsets of the set of simple roots of GSp(n); to a subset we associate the Levi subgroup generated by the root subgroups of the simple roots in the subset and their negatives. For GSp(2) the standard parabolic subgroups
II. Basic Facts
48
are P(o)= P{a,o} = GSp(2), the Siege1 parabolic P(2)= P{al which has Levi M(2) = Mia} = {diag(g, X'g');
the Heisenberg parabolic
P(1)= Pip}
9 E GL(2),
E Gm},
which has Levi M(1) = M{P}
= {diag(a, h , X ( h ) / a ) ;a E Gm, h E GSp(1) = GL(2), X(h) = det h } ,
and P(1,,) = Po is the minimal standard parabolic subgroup with Levi subgroup M(I,l)= M0 that we usually denote by Ao, consisting of {diag(a,b, X / b , X / a ) ; a , b, X E Gm}. If T I , . . . , 7rT are representations of GL(ni, F ) , and a of GSp(n  In/,F ) , F a local field, as in [ST] denote by 7r1 x . ' x 7rT x1 (T the representation I(7r1,. . . , 7 r T , a ) of GSp(n, F ) normalizedly induced from the representation +
P = mu
H rl(g1) 8
. * * 8 XT(gT) 8 ~ ( h ) of P n = MnUn
Here U, denotes the unipotent radical of P,. Note that a is a character if In1 = n (thus h E GSp(0, F ) = 3''). The induction is normalized by multiplying the inducing representation by the character SA"((p), where 6,(p) = I det(Ad(p)I Lie(U,))l. Normalized induction takes unitarizable representations to unitarizable representations. NOTATION.As in [BZ2],4.2, we write v(x) = 1x1 for x E F X .
EXAMPLE. The simplest example is where GSp(l)=GL(2). Here Mt,) is the diagonal subgroup, G(diag(a, b ) ) = Ia/bl, and p x1 CT is the representation usually denoted by I ( p a ,a ) , normalizedly induced from the character diag(a,b) . u ++ ( p a ) ( a ) a ( b )(if b = X/a, this is = ( p a ) ( a ) a ( X / a )= p ( u ) a ( X ) ) . The trivial representation 1 2 of GL(2, F ) is a subrepresentation of I(v'I2, vl/') = v' XI vl/' and a quotient of I(vl/', vd1I2) = v >a vl/'. 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, 2.12.3 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 Ftorus 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 TateNakayama 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 Ftori in H injects naturally in the image of ker[H1(F, N ) 4H1(F,H)]in H 1 ( F ,W ) . When 0 H is quasisplit 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 Ftori follows that of loc. cit. The list of Ftori 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 Hl 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) 0conjugacy classes of a strongly ®ular element t in G. We fix a &invariant Ftorus T*, to wit: the diagonal subgroup. The stable 8conjugacy 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 Ftorus Tt: it is ZG (ZG ( t o ) )
= hlT*h = Tt. The torus
Tt is Ot = Int(t) o &invariant:
We have zG(t6) = TP: if tl E Z ~ ( t 8=) hlT*'h C h'T*h = Tt then . t e . tl = te, thus e t ( t l )= te(tl)tl = tl. The 8conjugacy classes within the stable 8conjugacy 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 0conjugacy classes within the stable 8conjugacy class oft. The Galois action on Ti: = a(hlt*e(h))= hI
. h a ( h )  l . a ( t * ) e@(h)hl)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*').
5. Twisted Conjugacy Classes
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
hlt*o(h) = t = a(t)=
at *.~ ( ~= ha(h)let*e(ela(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 8conjugacy 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 Gmodules placed in degrees 0 and 1 is the group of 1hypercocycles, quotient by the subgroup of 1hypercoboundaries. A 1hypercocycle is a pair ( a , @with ) a being a 1cocycle of G in A and p E B such that f ( a ) = a@;ap is the 1cocycle 0 H @'a(@) of G in B. A 1hypercoboundaryis 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 (1Ot)r(c(gl)g;')
= a(b)6l
where 6 = O ( Z ) Z  ~ = ( 1  O t ) ( t  ' )

E
V,;
(18tb
a(g1)gT1,b) defines an element inv(t,tl) of H1(F,TY
Vt). This element parametrizes the 8conjugacy classes under Gsc within the ( ~ 7H
stable 8conjugacy 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 8conjugacy classes within the stable 8conjugacy class of t , D ( F ,0, t )
is the image under
of (18t)"
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 0conjugacy classes of strongly ®ular elements t in G = PGL(4) as in [F5]. Thus we describe the Ftori T, as ZG(t8) = T> and T = Tt = Z G ( Z G ( t 0 ) ) . The conjugacy classes of Ftori 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 .
5. Twisted Conjugacy Classes
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 8conjugacy classes within a stable semi simple 8conjugacy 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 8conjugacy classes of type (I) in GL(4,F ) which split over E can be found in a torus T = T ( F ) , T = hlT*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 8conjugacy classes within a stable 8conjugacy class can be chosen in T . Indeed, if t = hlt'h and tl = hltTh 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*)pl. Since t is 8regular, p lies in the 8normalizer of T * ( F )in G ( F ) . Since the group W e ( T *G , ) = Ne(T*G , ) / T * ,quotient by T * ( F )of the 8normalizer of
5. Twisted Conjugacy Classes
57
T*(F)in G ( F ) ,is represented by the group W e ( T * , G )= N ' ( T * , G ) / T * , quotient by T* of the 8normalizer 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 0conjugacy classes within the stable 8conjugacy 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 0conjugacy classes within a stable semi simple 8conjugacy 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 0classes within a stable class are parametrized only by r, or equivalently only by s. A set of representatives for the 8conjugacy classes within a stable semi simple 0conjugacy 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 8conjugacy classes can be taken in the torus T = hlT*h, consisting of
+
t=
(z 'f)
= hlt*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 8conjugacy classes within the stable 8conjugacy class of t can be taken in T . In fact if t' = gtQ(g)' lies in T and g = hlph, 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 8conjugacy classes within the stable 8conjugacy class of t is given by t . diag(r, r ) , as T ranges over E: / N E / EE~X . Representatives for the stable regular 8conjugacy 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),
= hlt*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 <, in which case the Galois group is the dihedral group D4. In both cases the Galois group contains an element a with a& = u a= a 2 a= In the 0 4 case Gal(E'/F) contains also T with r<= <, we may choose D = r D = D and a f i = In any case, t is 8regular when a # u2a. We write a = a b a E E X, u = a1 +a2& E E:, b = b l + b 2 f i E E:, ua = a a + u b G , a2a = a  b a . Also
+
a. a,
al ml
Representatives for all 8conjugacy classes within the stable 8conjugacy class of t can be taken in T . In fact if t' = gtO(g)l lies in T and g = hlph, p E T*(F),then p 8 ( p )  l = diag(u,au,a3u,a2u) has u = u2u, thus u E E:. If g E T , thus p E T * ,then p = diag(v, uv, a3v, u2v) and
pO(p)' = diag(va2v,u(va2v),u(va2v),vu2v),
5. Twisted Conjugacy Classes
59
with wov E N E I E ~ E It~ follows . that a set of representatives for the 8conjugacy classes within the stable 8conjugacy class of
t = hlt*h = (i'f)
where
t* = diag(a,a;, 03a102a),
is given by multiplying a by T , that is t* by tz = diag(r,or,a3r,02r), where r = 02r ranges over a set of representatives for E l I N E , E 3 E X . NOWto = h't,*h =
(0' .)"I
Hence a set of representatives is given by
t .diag(r,r), r E E , X I N E I E ~ E ~ .
111. TRACE FORMULAE 1. Twisted Trace Formula: Geometric Side The comparison of representations is based on comparison of trace formulae, which are equalities of geometric and spectral sides. In this section we first state the spectral side of the 0twisted trace formula on the discrete spectrum, and then we record the geometric side of the 0twisted trace formula for G , in fact only its &elliptic strongly ®ular part, stabilized according to its &elliptic endoscopic groups, as in [KS].This geometric side is a linear form, with complex values, on the space Cr(G(A)) of smooth compactly supported complex valued functions on G(A). This space is spanned by products @f,, where fv E C,"(G(F,)) for all w and fv = f," is the characteristic function of K , = G(R,) for almost all w . In fact we need the measure f d g where dg = @dg, is a Haar measure on G ( A ) ,but we suppress the d g from the notations. We later compare trace formulae for test measures f d g , f H d h , fc,,dco, etc., with matching orbital integrals. The dependence on measures is implicit. The trace formula is obtained on integrating over the diagonal g = h in G(F)\G(A) the kernel K f ( h ,g ) of the convolution operator r ( f ) r ( O )on L2 = L2(G(F)\G(A)),defined by
for 4 E L2. The discrete part Ld of L2 splits as a direct sum @,L, of subspaces transforming according to inequivalent irreducible representations 7r of G ( A ) .Thus L , = rn(.rr)7r is a multiple of an irreducible 7 r , occurring with finite multiplicity rn(.rr)in L 2 , and the sum is over inequivalent 7r. If {@} is an orthonormal basis of L, then the kernel of r(f)r(O) on Ld is
60
1. Twisted Truce Formula: Geometric Side
61
Indeed,
The trace of r(fx )=r(f)r( ) over the discrete spectrum is the integral of Kd over the diagonal k = of Kd over the diagonal k=g in G(A)
7r
6;
7r
where ~ ( fand ) ~ ( 0 denote ) the restriction of r ( f ) and r ( 0 ) to 7rl and ~ ( xf19) = 7r(f)n(O). One can see that the sum C,rn(7r)lt r r ( f x 19)l is convergent. The 7r in L d which contribute a nonzero term to this sum are those which are &invariant: ' 7 r N T . The contribution to the trace formula from the complement of Ld in L2 is described using Eisenstein series; we describe this spectral side below. This side will be used to study the representations T whose traces occur in the sum. We now turn to the geometric side of the trace formula. The geometric side of the trace formula is obtained on integrating over the diagonal g = h E G(F)\G(A) the kernel of the convolution operator r ( f ) r ( O )on L2: here (r(f)r(O)4)(h) is
III. Trace Formulae
62
and we consider only the subsum
9) =
~ e ( h i
C
f(h'be(g))
6 E G ( F )e
over the set G(F)e of Osemisimple, strongly 8regular and &elliptic elements 6 in G(F). An element 6 of G(F) is called 0semisimpleif the automorphism Int(6)o 8 = Int(6O) is quasisemisimple, by which we mean that its restriction to the derived group is semisimple (thus there is a pair (B,T) in G fixed by the automorphism). As for 8regularity, denote by 16 = z G ( 6 6 ) the centralizer of 68 in G (this is the group { g E G;6O(g)6' = g } of fixed points of Int(6). 0). A 8semisimple 6 in G is called 8regular if ZG (68)' is a torus, and strongly 0regular if Z ~ ( 6 6is) abelian. If 6 is strongly 0regular (centralizer in G of Z ~ ( 6 0 ) ' is ) a maximal torus then T6 = z~(z~(6O)'))) in G fixed under Int (SO), and ZG (60) = A 8semisimple element 6 of G(F) is called &elliptic if ( 2 G ( 6 0 ) / Z ( G ) e )iso anisotropic over F . The integral Te(f,G, 0) over h = g in G(F)\G(A) of Ke(g,g ) is the sum over a set of representatives 6 for the 0conjugacy classes in G(F)eof orbital integrals:
TF(6e).
It is rewritten in [KS], (7.4.2) as a sum over a set of representatives (H, N ,s, I ) for the isomorphism classes of elliptic endoscopic data for (G, 0) ([KS], (2.1)) and over a set of representatives for the H(F)conjugacy classes of elliptic strongly Gregular y in H ( F ) (y E H is called strongly Gregular if the image under the norm map AH,G ([KS], (3.3)) of the conjugacy class of y consists of (strongly) 6'regular elements): aG (H,7l,s,E)
'
IOut(H,x, S, e)l'
@;(f). Y
Here Out(H,N, s, I ) is the group defined in [KS], (2.1.8); CLGis the number defined in [KS], (6.4.B);and the twisted &orbital integral @,Yn(f) is defined in [KS], 3 lines above (6.4.10) and 3 lines above (6.4.16).
1. Twisted Trace Formula: Geometric Side
63
If f H = @ f v ~ ,f v H E c,"(H(F,)) has matching orbital integrals with f, for all w ([KS],( 5 . 5 ) ) ,then @;(f) can be replaced by the stable orbital integral @:(f~), and the stabilized trace formula takes the form ([KS], (7.4.4))
L(GI~IH S T)e ( f H ) , (H,'H,%€)
where
and
In our case G = PGL(4), 8 ( g ) = J  l t g  l J , there are two elliptic 8endoscopic groups H = PGSp(2) and C = [GL(2) x GL(2)]'/GAl with = Sp(2,C) and e = { ( A , B ) E [GL(2,C) xGL(2,C)]/CX;d e t A d e t B = l } ,
Z ( c ) = p4, Z ( g ) = Z(@ = p2 = { & I } , the Galois group l? = Gal(F/F) acts trivially as G I H and C are split, 2 1 = Z(@ n (?)' of [KS],Lemma 6.4.B, is {*I}, hence 7 there (= Z(G)/Zl) is p ~ 21 , n (Z(G)')O is trivial, kerl(FIZ(@) = 1, hence UG = 2. For H and C there is no 8, Z1 = 1 Z ( g ) and = 1, ker ( F , p z ) = 1, hence U H = 2 = a c . In particular L(G,O,H)= IOut(H,'H,s,E)Il is 1 and L(G,e,C) =
z
a.
Similarly we consider the elliptic regular part of the geometric side of the trace formula of H = PGSp(2) and stabilize it, to obtain (here 8 is trivial and is omitted from the notations)
where fc,, is a function on c o ( A ) matching f H . Here CO = PGL(2) x PGL(2) is the only elliptic endoscopic group of H other than HI =
e0
III. Trace Formulae
64
SL(2,C) x SL(2,C) has center {*I} x (41)of order 4, hence a c 0 = 4. Also Out(C0,. . . ) has order 2 and U H = 2, hence L(H,1,Co) = The &endoscopic group C of G has a proper endoscopic subgroup CE for each quadratic extension E of F. Its connected dual group EE = ZZ(;E)', ,?E = (diag(1, l), diag(1, l)),is
i.
{(diag(al,u2),diag(bl,b z ) ) modCX; ala~blb2= l}, and Gal(E/F) acts via
(( _", i) , (_",i)). Thus
Gal( E / F ) and CE(F) = ((z1,zz) E ( E X x E X ) / F X ; z l Z l= z222}, and CE is 2 / 2 , generated by (1,I). Since Out(CE,. . . ) has order 2,acE = 4 and a c = 2, we get L ( C , ~ , C = E ) and
This identity can be used to associate to any pair p1, pz of characters of Ag/EX whose restriction t o A X / F X is x E / F the pair ~ ( p l x) 7 4 ~ 2 of representations of C(A) = [GL(2,A) x GL(2,A)]'/AX. This lifting is wellknown. Whenever possible we shall work with fc = @fc,v whose component at a relevant place has orbital integrals which are stable, so that there won't be a contribution from T E ( fc,) = STE(fc,). In summary, the &elliptic 8semisimple strongly ®ular part of the geometric side of the &twisted trace formula for G, Te(f,G ,O), takes the form
Here fH = @ f H v and fc = @fcvhave orbital integrals matching the (stable and unstable) &twisted orbital integrals of f = @f v for each place v, those of fc, = @fcov match those of fH and those of fc, = @fCEv match those of fc. Of course by this we mean that the measures fdg, fHdh, fcdc, are matching, and fCodco and fHdh are matching, and so are
)
1. Twisted Dace Formula: Geometric Side
65
fcEdCE and f c d c . The identities of trace formulae hold for such matching measures. We suppress the measures from the notations. Complete analysis of the geometric sides of the trace formulae would include terms related to singular and to nonelliptic orbital integrals. In order to not deal with these in this work, we take a component of all global functions at a fixed place 210 of F to vanish on the singular set, and then the integrals over the singular classes vanish apriori and need not be computed. This mild restriction does not restrict the uses for lifting applications of the identity of trace formulae. For example we may take these functions to be biinvariant under the Iwahori subgroup, and supported on double cosets of elements in the maximal split torus (diagonal, in our case) on which the absolute values of the roots are big (the eigenvalues have distinct absolute values, in our case). To avoid dealing with the non(&)elliptic conjugacy classes, we observe that using the process of truncation, integration over these orbits leads to (0) orbital integrals weighted by a factor which can be expressed as a sum of local products involving number of factors bounded by the (twisted) rank. Thus these weighted (8) orbital integrals are sums of products of local factors which are all  except for at most rank(G,0) factors orbital integrals on the non&elliptic class. In our case the &twisted rank of G is two, and the ranks of HI C and C Oare two too. The restrictive assumption that we make is that we fix three places: 211, vz, 213, of F , and work with functions f whose components fv at 21 = vi (i = 1 , 2 , 3 ) have &orbital integrals equal to 0 on the strongly ®ular orbits which are not &elliptic. In this case the geometric side of the twisted trace formula is equal to the 0elliptic part Te(f,G ,0). The matching functions on H, C and CO can also be chosen now to have components at 211, 212, 213 whose orbital integrals vanish on the regular nonelliptic sets of these groups, and the component at 210 vanishes on the non regular set. The geometric sides of the trace formulae are then concentrated on the elliptic regular sets, and are equal for such test functions to T e ( f ~ l H ) , Te(fco,Co), Te(fc,C). The requirement that the orbital integrals of f V i ( i = 1 , 2 , 3 ) be zero on the strongly ®ular non &elliptic set is weaker than an assumption that the functions themselves be zero there. The requirement that we make permits applying the trace formula with coefficients of elliptic representations
66
III. Trace Formulae
at the places 'ui (i = 1 , 2 , 3 ) . We compare the geometric sides of the trace formulae with the spectral sides, which include, in addition to the contribution C , m(n)tr T (f x 0) (in the case of the &twisted trace formula for f on G) from the discrete spectrum Ld, also contributions from the continuous spectrum. These contributions are described in terms of Eisenstein series, and lead to a sum of discrete terms and integrals of continuous series of representations, involving logarithmic derivatives. The weight factor splits as sum of local products whose number of terms is bounded by the (&)rank. In our case the rank is 2, and assuming as we do the vanishing of the orbital integrals on the regular nonelliptic set at 3 places leads to the vanishing of all continuous sums, or integrals, of traces of representations which contribute to the (&)trace formula. We proceed to describe only the discrete sums contributions to the spectral sides of the trace formulae.
2. Twisted Trace Formula: Analytic Side We now record the analytic side of the twisted trace formula; it involves twisted traces of representations. The expression is taken from [CLL],XV, p. 15. Fix a minimal &invariant Fparabolic subgroup P o of G, and its Levi subgroup Mo. Denote by P any standard (containing Po) Fparabolic subgroup of G, by M its Levi subgroup which contains Mo, and by AM the split component of the center of M. Then AM C A0 = AM^. Let X*(AM)be the lattice of rational characters of AM,dM the vector space X,(AM) 8 JR = Hom(X*(AM),R),and Ah the vector space dual to AM. Let Wo = W ( A 0 ,G) be the Weyl group of A0 in G. Both B and every s in WOact on do.The truncation and the general expression to be recorded depend on a vector T in A0 = AM^. In our specific case of G = PGL(4) we shall use only the constant term, or value at T = 0, and in fact only the discrete part, of terms where A* = {0}, below.
2.1 PROPOSITION ([CLL]). The spectral, or analytic side of the trace formula is equal to a sum ouer (1) the set of all Levi subgroups M containing Mo of the Fparabolic subgroups of G ;
2, Twisted Trace Formula: Analytic Side
67
(2) the set of subspaces A of do such that for some s in WOwe have A = A;', where Age is the space of s x 6invariant elements in the space AM associated with a 0invariant Fparabolic subgroup P of G;
(3) the set W A ( A M ) of distinct maps on A M obtained as restrictions of the maps s x 0 (s in WO) on A0 whose space of fixed vectors is precisely A; and (4) the set of discrete spectrum representations I of M(A) with ( s x 6 ) r N I  , s x 0 as in (3). The terms in the sum are equal to the product of
and
1,.
t r [MAT(P,~)MP,e(P)(s,O)Ip,r(~, f x 6 ) ) ldi'l.
Here [W,"] is the cardinality of the Weyl group W," = W(A0,M ) of A0 in M . Also P is an Fparabolic subgroup of G with Levi component M; Mp~e(p)is an intertwining operator; M;(P, A) is a logarithmic derivative of intertwining operators, and I P ,(C) ~ is the G(A)module normalizedly in standard notations. induced from the M(A)module m H I(m)e(C,H(m)) The sum of the terms corresponding to M = G in the formula is equal to the sum I = C tr T (f x 0) over all discrete spectrum representations 71 of G(A). We proceed to describe, in our case of G = PGL(4) and the involution 6 , the terms corresponding to M # G and A = ( 0 ) in the formula. Let Mo be the diagonal subgroup of G. There are [ W o ] / [ W f=] 4 Levi subgroups M 2 A0 of maximal parabolic subgroups P of G (of type (3,l)) isomorphic to GL(3), that is to the image of GL(3) x GL(1) in PGL(4). The space dM = { ( a , a , a , b ) * ; a , b E R} (the superscript * means image in R4/R, where R is embedded diagonally), has A = A;' = ( 0 ) for any s E W (for which s x 0 maps dM back to AM), and the contribution is
III,
68
Dace
Formulae
Here PI denotes the upper triangular parabolic subgroup of G of type (3,l). We write a1 = (12), a 2 = (23), a3 = (34), J = (14)(23) for the transpositions in the Weyl group WO.In the last sum, x ranges over the characters of A x / F x of order at most two, while r ranges over the discrete spectrum representations of GL(3, A) whose central character is x and re Y r . There are [W0]/2[W,"] = 3 Levi subgroups M 3 A0 of maximal parabolic subgroups P of G (of type (2,2)) isomorphic to the image of GL(2) x GL(2) in PGL(4). The space d M = { ( a ,a , b, b)*; a, b E
R2}
has d = dgeequal to {(O,O,a,a)*} for s E W,", and d = (0) for all s # 1 in W/W,". Consider only the case of d = {0}, and choose s = J t o be a representative. Then 1 J x I9 : ( a ,a, b, b)* H (a, a, b, b)*
has determinant 2 on the one dimensional space d M , and the contribution is
Here P2 denotes the upper triangular parabolic subgroup of G of type (2,2). The last sum ranges over the ordered pairs ( T I , 7 2 ) of discrete spectrum representations 11, 7 2 of GL(2,A) with central characters w,, and wr2 with w,,w,, = 1 and I," E ri, thus 11 and 12 are discrete spectrum representations of PGL(2, A) (then we write x = 1) or ri = pi), pi characters of A 2 / E X A x , and E / F is the quadratic extension determined by x = w,, = wr2. Thus the sum over x ranges over all characters of A X / F X of order at most two. There are [Wo]/2[WoM]= 6 Levi subgroups M 3 A0 of parabolic subgroups P of G (of type (2,1,1)) isomorphic to the image of GL(2) x GL(1) x GL(1) in PGL(4). If s E WOis such that s x I9 maps d M = {(a,a,b,c)*} to itself, then (up to multiplication by (a1 = (12)) = W,"), s can be (1) s = (14)(23), in which case s x
I9 : ( a ,a , b, c)*
H
(a, a,
4, c)*,
2. Twisted Trace Formula: Analytic Side
A = (0) and det(1  s x
d)ld,
69
= 4, or (2) s = (13)(24), then
s x 0 : ( a ,a, b, c)*
H
(a, a, c, b)*
and A # ( 0 ) . The term with A = (0) is
is the upper triangular parabolic subgroup of type (2,1, l),and equivalent to (T~,x;',x;').If x denotes the central character of TI then x = ~ 1 x 2 and , TI N rf = r1x. We can write the induced representation as x21(71,X I , 1). If x = X I # 1 then TI = r ( p 1 ) where p1 is a character of A g / E X ,where E / F is determined by x, The central character of ~ ( p 1is) x . p l I A x ; if this is equal to x, then p l l A x = 1, hence there is a character po of A g / E X with p 1 ( z ) = po(z/F). Put p o ( z ) = PO@), z E A;, then r1 = 7r(po/po).If on the contrary x = x1 = 1 then 71 is a discrete spectrum representation of PGL(2, A). We then obtain from the terms with d = ( 0 ) the sum Here
T
=
P3
( T I , X I , X ~ )is
1

8
tr M((14)(23),o ) ( X I P 3 ( T 1 ,1,l))(fX 0) X>Tl
c
1
+4
trM((14)(23),O)(XlP,(T(pO/iZ,),Xl, l))(f x 6 )
X1#13PO,X
where x is any quadratic character, x1 is a quadratic character # 1, 71 ranges over the discrete spectrum of PGL(2,A) and p1 = po/jT0 over the characters of A2/AxE X ,or po over the characters of Ak.E1, where Ah = { z E A;; z? = 1). Note that I ( T ~ , x1), and 1(r1,1,x ) contribute two equivalent contributions when x # 1. Let 7r be an irreducible &invariant representation of G = PGL(4), which is properly induced from a parabolic subgroup. We proceed to list these. If 7r is induced from the (standard) parabolic of type (3,l) the T = I ( T , x ) , where T is a representation of GL(3) and x is a character (of GL(l)), and w x = 1 where w = w, is the central character of T . From 7r' N IT we conclude that T0 N T
( T B ( g )= 7 ( J t g  l J ) , J = antidiagonal(1, 1,l))
70
III. Trace Formulae
and x2 = 1. Then 7r = x1(7x,l), where 7%is a representation of PGL(3) ~ hence the image of the symmetric square lifting from with ( 7 ~ cv) rx, GL(2) (or rather SL(2), see [F3]) to PGL(3). Globally we have that the lifting A1 : SO(4) = [GL(2) x GL(2)]'/ GL(1) + PGL(4) takes 71 x r1x to I(XSym2(71),X). If 7r is induced from the (standard) parabolic of type (2,2) then it is
If 7 ; cv 7 2 , then 7r lies in a continuous family I(T~v',T~v'),v(g) = I det g ( , of 0invariant representations. Otherwise 71"N 71 and 72"cv 7 2 , thus w;~= 1, and w,,wTZ= 1. If wTi= 1 then ~i is a representation of PGL(2), and if wT,# 1 then ~i = n(pi),where pi is a character of CE(=E X in the local case, A g i / E i in the global case) which is trivial on C F ,where E I F is the quadratic extension determined by w,, = wTz.
Interlude about GL(2): if E / F is the quadratic extension determined by a quadratic character w of F X ( F local), and p is a complex valued character of E X ,there is a two dimensional representation p ( p ) of the extension
of Gal(E/F) = (a)by W E / , = C E ,given by
Then det p ( p ) ( z ) = ~ ( z Z )det , p ( p ) ( o ) = p(a2). The corresponding admissible (globally: automorphic) representation of GL(2) is denoted by ~ ( p )and , its central character is W(Z)= X E / F ( Z ) ~ ( Z ) . In the case of the parabolic of type (2,2) above, wTi# 1then implies that pilFX = 1, hence there is a character pi : E X +C x with p i ( z ) = p i ( z / Z ) so that T~ = 7r(pL/pL). Choose square roots of
2. Twisted Trace Formula: Analytic Side
71
then
(" a1
1
and
If
7r
is induced from the standard parabolic of type (2,1,1)then
and 7r = x 2 1 ( 7 ~ 2 xlxz, , 1). Further r8 N 7 (it is a representation of GL(2)), and TO = T X ~N and xo = ~ 1 x 2has order two. If xo = 1 then 70 is a representation of PGL(2),while if xo # 1 then 70 = 7r(po/po), where po is a character of the quadratic extension E of F determined by XO. In this case
TO",
If
7r
is induced from the minimal parabolic, of type (l,l,l,l), and
is not in a continuous family of &invariant representations, then xp = 1. If two xis are equal then 7r is xoI(x, 1 , l ) . Otherwise 7r is the twist by xo of I(x1,x 2 , x1x2,l). Denote by E2 the extension of F determined by x2, put p ( z ) = x1(zZ) ( Z is the image of z E E; under the Galois action over F ) . Then p = ji ( F ( z ) is p ( Z ) ) and p = pl since x: = 1. Then there is p1 on E$ with p ( z ) = pl(z/F)(= pl(F/z) $ l), and
x',
We now take the Levi subgroup A0 and list the different types of maps s x 8. The involution 8 maps an element ( a , b , c , d ) * of do to (d, c, b, a)*, and it is convenient to write s as s J ( J = (14)(23)). In these notations, there are 1 (resp. 8, 6, 6, 3) distinct maps sJ x 8 where s is 1 (resp. has order 3, is a transposition, has order 4, is a product
72
III. Trace Formulae
of two transpositions of disjoint support). Representatives are given by s = 1 (resp. (321), (12), (4321), (12)(34)). The subspace A of vectors in do fixed by sJ x 8 is {0} (resp. {0}, {(u,a,O,O)*}, { ( a , b , a , b ) * } , { ( u , b , c , a + b  c ) * } ) a n d d e t ( l  s J x O ) i s 8 ( r e s p . 2,2, 1, 1). Werecord only the discrete part, where A = (0).
The T in the first sum are the characters ( x ~ , x Z , X ~ , X ~ ) of A0 fixed by J x 8, that is x: = 1. There are 4! such ordered 4tuples of distinct x:s, 3! . 2 ordered 4tuples where {xi} has 3 distinct elements, 3 x 2 ordered 4tuples wh,ere each xi occurs twice in ( X I , x2, x3, x4), 4 ordered 4tuples where exactly 3 of the 4 xis are equal. The first sum becomes
xi
x;', xi'), the fixed T have = 1 Since (321)J x 8 maps T to (xB1,xY1, and ~ 1 x 3= x1x2= ~ 2 x 3= 1, thus x1 = x2 = x3 has xs = 1. Since ~ 1 x 2 ~ = 3 1, ~ we 4 get x4 = x. The contribution is then
3. Truce Formula of H: Spectral Side
73
3. Trace Formula of H: Spectral Side The spectral side of the trace formula for H = PGSp(2) can be written out too as the case where 8 is trivial. We proceed to specify the objects involved. As usual, a superscript * indicates image in the projective group. We choose Po to be the upper triangular subgroup in H. Its fixed Levi subgroup is chosen to be Ao = { t = diag(a, b , X / b , X / a ) * } . A basis of Ao) = { a ,P } , a ( t ) = a/b, P ( t ) = b2/A, the root system is A = A(H, PO, and the root system is R = Rf U  R f , where R+ = R+(H,Po,Ao) = { a ,p, a+pl 2a+p} is the set of distinct homomorphisms in the action (Int) of Ao on Lie(Po/Ao). The group X,(Ao) = Hom(G,,Ao) is a lattice in the vector space d o = X,(Ao) @R which we identify with R2 via the map log : X,(Ao) 4 d o = R2,
The roots, characters of Ao(F) = X,(Ao)@FX,lie in the group X*(Ao) = Hom(Ao, G,), which is a lattice in the dual space A,* = X*(Ao) 8 R to d o = Hom(X*(Ao),R). Identifying AT, with R2 with the usual inner product: A; x d o + R,( ( z , ~ )(,u ,u ) ) = xu+yw, the roots can be identified with the vectors cy = (1,l), p = (0,2), a p = (1,1),2 a ,8 = ('40) in A; = R2. The coroots aV = 2a/(a,a ) ,. . . are in d o identified with av = (1,l), pv = (0, l),( a p)" = (1,l),pa P)" = ( L O ) . The Weyl group Wo = W ( H , A o )of Ao in H , which is the quotient by (the centralizer in H of) Ao of the normalizer of Ao in H , viewed as a group of permutations in the symmetric group S4 on 4 letters, is generated by the reflections s, = (12)(34),sp = (23), s,+p = (13)(24), s2,+p = (14) in S4. Put a = sps, = s,sz,+p(= (23)(12)(34) = (12)(34)(14)). Then
+
+
+
+
.
.
Wo = (g,s p ; a4 = s; = 1,s p a s p = (71) = {sbaJ;i = 0 , l ; j = 0,1,2,3} is the dihedral group D4. Note that s p a = ,s, spa2 = s2,+pl spn3 = sa+p. Under the identification of X,(Ao) with a lattice in do = R2, the Weyl group can be identified with a group of automorphisms of do: s,(x, y) = (Y, XI,
III. Trace Formulae
74
o(z,y) = (y,x), o2 = 1. Note that for each root y in { a , p , a + P , 2 a +
P } and for 6
E dg perpendicular to y, we have sry = y and s,6 = 6. Then the sy are reflections, and o is a rotation of 7r/2, clockwise. The Levi (components of parabolic) subgroups of H containing A0 other than H and A0 are M', = s p M a s p , Mb = saMps,,
M,
= M ( 2 , 2 ) = {diag(A, X W ~ A  ~ WA) *in; GL(2),X in
Gm},
Mp = {diag(a, A, det A/a)*; A in GL(2), a in GL(1)).
We determine the subspaces of do associated with these. The (split component of the) center A, = AM, of M, consists of t = ( a ,a , A / a , A / a ) * , thus X,(A,) = Hom(G,,A,) is Z ( 1 , l ) and A, = X,(A,) 8 R is R(a p)" in d o . Since X,(AM;) = s ~ X * ( A M , )we have d M ; = Ra". From X,(AM') = Z(1,O) we obtain d M o = R(2a P)" in d o , and since AM;) = s , x , ( A ~ ' ) we have d M b = RP". Hence
+
+
Here is the diagram (where a1 = ~1  ~
2 ) :
To list the contributions to the trace formula, note that the w in WO with = {0} are 0,0 2 ,03.Recall that o ( a ,b, X / b , X / a ) = (b,X/a, a, X / b ) and the character ( ~ 1 ~ from ~ 2 which ) J ( p l , p 2 ) = p1p2 x pz/p1 M pF1 is induced takes at t = ( a ,b, X / b , A/a) the value
dr
Then
Ol(p1, pz)(g) =
( P I , pz)(og) is the character
3. Trace Formula of H: Spectral Side
75
We have u(p1,p2) = (pirp2) if p1 = p2 = p q l . Since
is equal to (pIlp2)(t) if pf = 1 = p i . Note also that 1  0 : (z, y) H (z, y)  (y, x) = (x  y, y + z) has determinant 2, while det(1  a') = 4. Since [WO] = 8 and W t o= {l},the contribution to the trace formula from M = A0 and A = {0}, thus W{O)(Ao)= { o , o ~ , o ~is} ,
Note that the representations Ipo(p1,p2) = p1p2 x p1/p2 >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}, (ls,)(O, y) = (0,2y) hence det(1s,))dp = 2 and det(1sp)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 Fpoints 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 $elliptic
4. Trace Formula Identity
79
Osemisimple strongly ®ular part of the geometric side of the &twisted trace formula, Te(f,G, O ) , is
The assumption that at the place uo of F the components f v o , f ~ , ,. .~. , vanish on the (0) singular set of G(Fv,),H ( F v o ) ,. . . , and the assumptions that the components of f , fH, . . . at v = 211, 212, q have (8.) orbital integrals which vanish on the strongly(&) regular non(8) elliptic sets, imply that the geometric sides of the (&twisted) trace formulae are equal to the ( 0  ) elliptic parts. The geometric sides are equal to the spectral sides  these are the trace formulae for each of the groups under consideration. The spectral side Tsp(f,G , 0 ) of the 0trace formula for G and f will be equal to the (weighted) sum of the spectral sides of the trace formulae:
Here is a summarized expression of the form of the spectral side of the &twisted trace formula for f on G(A) = PGL(4,A): Tsp(f, G ,0)
=I
1
1
1
+ T1(3,1) + 5 1 ( 2 , 2 ) k zI(2,1,1)+ I l ,
where 71
.rr ranges
over the (equivalence classes of) discrete spectrum representations of G(A) which are &invariant. Note that each of these 7r occurs with multiplicity 1 in the discrete spectrum of L2(G(F)\G(A)). Further, .rr
where T ranges over the discrete spectrum representations of PGL(3, A) which satisfy T 11 T$: here O(g) = Jltg' 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 xlrz 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) KVfixedvector 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 wellknown localglobal 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 0elliptic 0regular 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 0conjugate but not 0conjugate 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 8conjugate but not &conjugate to the 8regular t ; K indicates the nontrivial character on the group of 0conjugacy classes within the stable 8conjugacy 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 Afactor 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},
1. From SO(4) to PGL(4)
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 8conjugacy 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 8conjugacy 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 =
1. From SO(4) to PGL(4)
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 8conjugacy coincides with 8conjugacy 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
IV. Lifting from SO(4) to PGL(4)
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,vl12) 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
+
1. From SO(4) to PGL(4)
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
01
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 transposeinverse 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 transposeinverse 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 transposeinverse 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 83invariant cuspidal 773, 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 8elliptic 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 XIlifting 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 = wtAlw. 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 Kfixed 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(lclnlrnr1)rnornlnlc)dndlc,
Mi
and A ~ ( r n 0 0= ) 61/2(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 wtAlur (or A'). Writing the trace again as
rno = diag(A,I), we use the fact that the 8normalizer 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(tAl, I),
and since X ~ , ( ~ A  '= ) xKZ(detAl. A) = wl(detA)x,,(A), we finally conclude that tr.rr(8f dg) is
On the other hand, using the 2fold 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 8Weyl 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
IV. Lifting from SO(4) to PGL(4)
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) =
.2/1I
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 XIlifts 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 X1lift 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,v1/2) x n2) x 7r2) = J , and since the composition series of I(v112,vl12) 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 XIlift 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 0invariant 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 0invariant
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 0twisted trace formula of G. Ifthere 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 XIlift 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”xl 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 XIlift of some 7rk x 7ri where 7ra of 7ri are not cuspidal, since we already know to XIlift 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 0elliptic 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 Xhas 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= 211 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 4dimensional 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 4dimensional 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 Gmodules that such 7r1 x 7r2 can XIlift 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 XIlift to 7rE = X l ( 7 r z ) . Consequently, each of the nu, pairs 7r&, iib of cuspidal representations of C(A) which XIlift to X1(7r&), basechange from F to E to the pair 7 r z l iig of cuspidal representations of C(AE), which XIlift 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 JacquetShalika [JS] theory of Lfunctions, comparing the poles at s = 1 of the partial, product Lfunctions L V ( S , 7r;
x i r l ) L V ( S , 7r; x
*I)
= L V ( S , n1 x i r l ) L V ( S , 7r2
x
el).
Moreover, such a proof assumes the theory of Lfunctions. 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 8invariant representation of the group PGL(4, F,,). Its 0character is not identically zero on the 8elliptic regular set ( b y the orthonormality relations). Suppose it is not a 0stable function o n the 0elliptic regular set. Then it is a Xllij? 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 pseudocoefficient 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 8stable 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 0trace 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
IV. Lifting from SO(4) to PGL(4)
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 XIlifts to the corresponding component of 7 r , and similar statement holds for the archimedean places. Using the pseudocoefficients 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 squareintegrable 7rZVO x 7rkwo which X1lifts to 7rwo. This has already been treated by our 0 complete description of the XIlifting.
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 inducedfromtheBore1 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 Xoli.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 xwl 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 = nmnlm' Jacobian. Hence with
AM,(^) = d&112(m)ldet(1 Ad(m))lLieNI =
a  b ac ud 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,,(alA) and U x ~ , I ~ ~=(x,(A) ~ ' ) = w,(det(alA))lX,(alA).
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
U1 b1 .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 8Weyl group W e ( M )of M in G (Onormalizer/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 AwtAIw)
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
%),
V. Lifting from PGSp(2) to PGL(4)
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 , Alifts 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 Alijh to the Steinberg representation of G.
PROOF. We use Lemma 3.5 of [ST],which asserts the following decomposition result: v2 x v >a v3/2a is equal to
in the Grothendieck group (Zmodule 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~ ) ~ ( vvla ~ , 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 Alifting to a), as the lifts of two of its term is known:
2. Reducibility
119
Next we apply the Xlifting to b) and note that aI(ulsp2 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 Xlift of a S t G S p ( 2 ) is aSt4. An alternative proof is obtained on Xlifting c) to get ul'(.U112
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 Xlift of the left side of d); the latter has o Stq as the other irreducible in its composition series, hence the Xlift 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 u3/2a 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 ul (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 u1/2x 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 u1/2a = 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 ull2a 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 ull2a) 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 u1/2a = 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 Xlift of v J x J
XI
121
v1/2a is a I ( ~ l V/ ~~ ,/ ~v J ,~ / ~vll2) 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 , v112)
It is invariant under multiplication by 6 ( J 2 = 1). To determine the liftings of the constituents of v J x J XI vll2a 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 4.64.9, 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 v1/2a 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.
V. Lifting from PGSp(2) to PGL(4)
122
(c) The irreducible unitarizable representations of H supported on P(ll which are not tempered are L(vP<, ~~/'7r2), 0 < P I 1, # 1 = t2, and 7r2 a cuspidal unitarizable representation of GL(2, F) with 7r2t N 7r2 and t W T 2 = 1. 0
<
3. Transfer of Distributions In relating characters on the group Co = PGL(2,F) x PGL(2,F) with those on H = PGSp(2,F), F local, we need a transfer Do 4 DH of distributions which is dual to the transfer of orbital integrals fH 4 fo for functions on H and on CO. This transfer is crucial to the orthogonality relations of characters, a main tool in our work. Let us recall (from chapter 11, section 5) some basic definitions. Two regular elements h , h' of HI and two tori TH,Tfi of H, are called stably  conjugate if they are conjugate in H(F);F is a separable algebraic closure of F. Let A(TH/F) be the set of z in H(F) such that Tfi = T; = X~THX is defined over F . The set B(TH/F) = TH(F)\A(TH/F)/Hparametrizes the morphisms of TH into H over F , up t o inner automorphisms by elements of H . If TH is the centralizer of h in H then B(TH/F)parametrizes the set of conjugacy classes within the stable conjugacy class of h in H. The map x H { T ++ 2, = ~ ( 2 ) z  l ; T in Gal(F/F)} defines a bijection
B(TH/F)11 ker[H1(F,TH)4 H1(F1H)]. Since F is nonarchimedean, H1(F,Hsc)= (0). Hence
Consequently it is a group, which is isomorphic theory  to

by the TateNakayama
3. Transfer of Distributions
123
Stable conjugacy for regular elliptic elements of H = PGSp(2, F ) differs from conjugacy only for elements in tori of types I and 11, where the stable conjugacy class consists of two conjugacy classes. ) Weyl group of TH in H , and by ~ ' ( T H the) Weyl Denote by W ( T H the group of TH in A ( T H / F ) . Let dH be a locally integrable conjugacy invariant complex valued function on H . The Weyl integration formula asserts that
The sum ranges over a set of representatives TH for the conjugacy classes of tori in H ; [XI denotes the cardinality of a set X . Suppose t is a regular element of H which lies in T H . Then the number of 6 in B ( T H / F )such that t6 is conjugate to an element of TH is [W'(TH)]/[W(TH Hence ) ] . when the function d H is invariant under stable conjugacy, we have
Here { T H }is~a set of representatives for the stable conjugacy classes of tori in H .
3.1 DEFINITION. Given a distribution DO on CO,let DH = D H ( D o ) be the distribution on H defined by D H ( ~ H=) Do(fo), where fo is the function on COmatching f H on H . Our next aim is to compute DH if DO is represented by a locally integrable function. We first state the result, and explain the notations at the beginning of the proof. 3 . 2 PROPOSITION. Suppose that DO is a distribution on COrepresented by the locally integrable function do. T h e n the corresponding distribution DH = D H ( D o )o n H is given b y a locally integrable function dH defined o n the regular elliptic set of H b y d H ( t ) = 0 if t lies in a torus of type III or IV, and by
124
V. Lafling from PGSp(2) to PGL(4)
if t is of type I or 11, where r ranges over F X/ N E / F E X or E l /NE,E3 E X , and x is the nontrivial character of this group; if r # 1 then tr indicates the element stably conjugate but not conjugate to t . If to = tb x t: E Co = PGL(2, F ) x PGL(2, F ) , then to" indicates t{ x tb.
REMARK. If DOis represented by do and D H by dH = d ~ ( d 0 for ) DH = D H ( D o ) .
dH,
we shall also write
PROOF.We need to recall the description of elements of types I (and later 11) and their properties. A torus TH of type I splits over a quadratic extension E = F ( a ) of F , and we choose explicit representatives for the two tori in the stable conjugacy class:
Here r ranges over a set of representatives for F X/ N E / F E X o , is the nontrivial automorphism of E over F , xi = ai +pi@ E E X are the eigenvalues of tr, and h, are suitable matrices in Sp(2,F), described in [F5],p. 11. The norm map relates the elliptic torus TOof COwhich splits over E to TL, on the level of eigenvalues it is given by
t:
N
= (diag(t1, a t l ) ,diag(t2, at2))++
t* = diag(x1 = t l t 2 , x2 = tlotz,ox2
= otl . t2,ax1 = at1 . at2).
Now the Weyl group of an elliptic torus in PGL(2, F ) is Z/2, hence the Weyl group W(T0)of TOin COis 212 x 212. The Weyl group W ( T H of ) TH in H (of type I) contains 212 x 212: it contains s1 = (12)(34) (acting on the diagonal matrix t*) ,which is represented by diag
(( i) , ( y i)) in H
(acting on t E T H ) and , (14)(23), which is represented by diag(1, 1,1, 1) (and hence W ( T H contains ) also (13)(24)). To use the Weyl integration formula we need to compute ~ ' ( T HThere ). are two cases for TH of type I. In case 11, 1 $ NE/FEX (this happens when E / F is ramified and 1 @ F x 2 ) . Then we can take r # 1 to be 1. Choose i E F with i2 = 1, and put w = diag(1, i, i, 1). It lies in Sp(2,F), and wltw = t'. .Then w represents b # 1 in B ( T H / F ) , and
3. Transfer of Distributions
125
~'(TH = 0) 4 (w acts as (23) on t*,and s2 = (23), s1 = (12)(34) generate 0 4 ) contains W ( T H )= 2 / 2 x 2 / 2 = "(To) as a subgroup of index 2. In case I2 we have 1 E N E / F E X, hence we can write diag( 1,l) as cs, s E SL(2, F ) , c E C G L ( 2 , F ) ( T E ) (= centralizer in GL(2, F ) of an elliptic torus TE which splits over E ) , and w = diag(1, 1,1,1) as ch with h in Sp(2, F ) and c in CGL(~,F)(TH). Then (ch)'tch = t r , t'  h' t '*h, h'* = diag(xl, ox2,x2, 0x1). Hence in case 1 2 we have ~ ' ( T H =W ) ( T H= ) D4, as w E ~ ' ( T H is represented ) by h E Sp(2,F), and it acts as (23) on t*. Note that the action of w in both cases I1 and I 2 is to interchange x2 and 0x2, namely t o = (diag(tl,atl), diag(t2,atz)) with (t")o = (diag(tz,atz), diag(t1, a t l ) ) . Then
and K(tW)
= XE/F((Zl

U Z l ) ( O 5 2  52)/D)
= XE/F(l)K(t).
Let now fH be a function on H such that the orbital integral @(tlf ~ ) is supported on the conjugacy class of a single torus of type I. Then
Note that the norm map N : TO+ TH is an isomorphism. Now in ) XE/F(l)K(t) =  ~ ( t )and , case 11, w represents b # 1, and ~ ( t " = W ( T H= ) W(To),so we get
and since @(t,f ~ is )any function (locally constant) on the regular set of TH,we conclude  by the Weyl integration formula  that
V. Lifting from PGSp(2) to PGL(4)
126
Again, t' is t" when r # 1 in F X / N E / F E Xand d H ( t W ) =  d H ( t ) , t E T H , In case 12 we have that t" is conjugate to t in H , and [ ~ ( T H=) ] 2[W(To)], and @ ( t , f H ) is supported on T& for a single T . Hence D H ( ~ H ) is
Since @ ( t Tfl ~is )any locally constant function on the regular set of T H , we obtain AH( t ) d H (t') = X E / F ( T )Ao ( t o )K(t)2do ( t o )
Tori of type I1 split over a biquadratic extension E = E1E2E3 of F , where El = E' = P ( f i ) , E2 = E"' = F ( m ) , E3 = E" = F(./71); A , D , A D are in F  F 2 , and we write tl = a1 for elements of E l , f p 2 m for elements of E2. The norm map takes tz =
+
tg
= (diag(t1, a t l ) ,diag(t2,7t2))
to
Thus TON E : / F X x E $ / F X (in contrast to case I where El = E2 = E is quadratic over F ) has Weyl group W(T0)= 212 x 212. The tori T& consist of
where X I = a + b f i ; a = a l + a z a , b = b l + b 2 a E E:; ux1 = a  b f i , 7x1 = ra r b . filthus
+
3. Transfer of Distributions
127
Further, r ranges over E,X/NEIE~E~, and if r = r1
( (
T2
la
(i :), and h = hD =
Then h, = h
a), (i "), T1
E
(hA
+ r 2 a we put r = ) ( i  s ) i h A =
see [F5],P. 12.
=
The Weyl group W ' ( T L )of TL in A ( T L / F )is 2 / 2 x 2 / 2 , generated by 5 = (14)(23), which maps the eigenvalue x1 of t* to 0x1, and ? = (12)(34), which maps x1 to 7x1. It is equal to the Weyl group W(T&)of TL in H , since (14)(23) is represented in H by diag(1,  I ) , and (12)(34) is represented by diag(1, 1,1, 1). We shall compare the norm map To 1 TH with that for ?O 2? H , where the tilde indicates that the roles of El and E2 are interchanged. Thus = H
((diag(t~,~t2),diag(t~,at~))
? = diag(x1 = tztl, U T X 1 = t2 . d l , 7 x 1 = Tt2 t i , O X 1 = Tt2 . d l )
and with
'
b' =
( i/A bbz
bl) b2
,
Note that is obtained from t* by the transposition (23). We claim that t and f are stably conjugate. For this choose a in F with a4 = A/4, thus 2 a 2 A / 2 a 2 = 0. Put
+
Then Y E Sp(2,F) satisfies
These t and f are conjugate if 1 $! F x 2 and IAl = 1 (we normalize A and D to lie in R X or ? r R X ) .Indeed in this case we may choose A = 1. Then either 2 E F x 2 or 2 E Fx2,and there is a E F X with a' = 1/2 or = 1/2 (respectively), hence a4 = 1/4 = A/4, and t , f are conjugate in GSP(21F ) . , ( 2 i ~ '= ) ~A has no solution If 1 E Fx2,say 1 = i 2 , i E F X then with a in F X . If IAl = ql then (2a2)2 =  A has no solutions with a in F X ,so f is conjugate to t T ,r # 1.
V. Lifting from PGSp(2) to PGL(4)
128
The transfer factor n(t) is
X E 2 / F ( ( x 1  oXl)(o7xl  7x1)/AD) = X E z / F ( b ' T b ' ) = X E z / F ( b Z  b?/A)
= X E Z / F (  ~ ) X E Z / F ( ~: b;A) = X E z / F (  A ) X E 1 / F ( b ?
 b;A).
For the last equality note that b:  b:A E N E ~ I F E lies ; in N E ~ I F iff E~ it lies in N E l / F E : iff it lies in F x 2 . Note that x E 2 / F (  A ) = 1 if A = 1 @ F x 2 . If 1 E F x 2 and 1Al = 1 then x E 2 / F (  A ) = 1, since then E2IF is ramified, R X n N E 2 / F E T= R x 2 and A $ R x 2so A 6 N E ~ / F E TMoreover . X E ~ / F (  A ) = 1 if IAl = q  l : if E2/F is unramified then A $ N E ~ / F Eand ~ ;if E2IF is ramified we may assume that D = u E R X  R X 2 and , then N E ~ I F E= ; { x 2  y2un} where we write T for A. But if A = A = x 2  y2un is solvable then x E T R and y 2 u = 1 m o d ~ and u be a square in R X . We conclude that n(t) = n(t) if t , are conjugate, and n(t) =  K ( t ) if t , t are not conjugate. Consequently
Ao(to)@(to, fo) = A H ( t ) K ( t )
c
X E / E 3 ( 7  ) @ ( t T fH) ,
TEE:/NEIE~E~
is equal to the expression obtained on replacing t by f, which we denote by tw from now on to be consistent with the case of type I. It follows that each stable conjugacy class of t E TH of type I1 is obtained twice, from to and to (or As W ( T H = ) W(To),we conclude that D H ( f H ) is equal to
tr).
if @(t,fH) is supported on the conjugacy class of T L , and hence
A H ( ~ (t') ) ~ =HX E / E (r)n(t)Ao(to) ~ [do ( t o ) + do((t")o)l. It is clear that tori of types I11 and IV do not contribute to DH(fH),which 0 is equal to Do (fo).
4.
Orthogonality Relations
129
4. Orthogonality Relations We are interested in relating the distributions DH and DO since we need to relate orthogonality relations on H and on Co. 4.1 DEFINITION.(1) Let d H , d h be conjugacy invariant functions on the elliptic set of H . Put
Here { T H } ,(resp. { T H } ~is, a~set ) of representatives for the (resp. stable) conjugacy classes of elliptic tori TH in H . ( 2 ) Let do, db be conjugacy invariant functions on the elliptic set of CO. Put
where {To},is a set of representatives for the conjugacy classes of elliptic tori in Co. (3) Write d r ( t ) for do(t”), where if t = t’ x t” E COthen t” or f is t” x t’. 4.2 PROPOSITION. Let do, db be locally integrable class functions o n the elliptic set of CO,and dH = d H ( d o ) , d h = dH(db) the associated class function o n the elliptic regular set of H . T h e n
PROOF.By definition ( d H , d h ) H is a sum over { T H } ~ For , ~ . tori of type I we have [ W ’ ( T H ) ] = 2 [ W ( T o ) ] so , the contribution is
V. Lifting f r o m P G S p ( 2 ) to PGL(4)
130
where x i l F = 1 and the set of r , F X/ N E I F E X , has cardinality two. For tori of type I1 we have ~ ' ( T H = )W(To),but each t in TH is obtained up to stable conjugacy  twice: once from TOand once from T,". Hence the integral over TH has to be expressed as the sum of integrals over To and T,W,divided by 2. Then the contribution to ( d H , d L ) H will be the sum over { T o } ~ ,of II
+
.[do(to) d o ( t ; ) l p b ( t o ) +Z'(t;;.')ldto, where x i I E 3= 1 and IC' = 1. The cardinality of the r is IE:/NEIE3EXI = 2. We then obtain a sum over all TO,of types I (splitting over a quadratic extension E of F ) and I1 (splitting over a biquadratic extension):
4.3 COROLLARY. Let IT^, IT: (i sentations of PGL(2, F ) . P u t
=
1 , 2 ) denote square integrable repre
d b ( t l l t 2 ) = x7r$l)X7r;(t2)r
d0(t11t2)= X 7 r l ( t l ) X 7 r Z ( t 2 ) , where
xT denotes
the character of
IT.
Then
(do,db)o = ~ ( I T I ~ I T ~ ) ~ ( I T ~ , I T ; )
dy(ti,tz) = do(tz,ti), and
(GIdb)o = S ( n 2 , d)qm1741 where &(IT, IT')is 1 if
( d H l d b ) H=
{
IT
and
IT'are
equivalent and 0 otherwise, so that
$ IT: and
$ IT;;
0,
IT^
2, 4,
ri N IT; and IT^ ?? IT;, or IT^ $ IT: and IT^
where {ilj}= { 1 1 2 } .
Ira
= IT(
IT^
'v
IT;;
cv ITj
0
4.
Orthogonality Relations
131
4.4 PROPOSITION. Let dH be a locally integrable class function o n the elliptic set of H . T h e n dH is stable if and only if (dH,dH(dO))H is 0 f o r every class function do = xxo,where 7ro ranges over the square integrable irreducible representations of CO. PROOF.We have that (dH,dH(dO))H is equal to
where the first sum ranges over a set of representatives for the stable conjugacy classes of tori in H of types I and 11, and r ranges over a set of representatives for the conjugacy classes within the stable classes ( F X/ N E X or E : / N E I E ~ E ~is)0, if d H ( t r ) = d H ( t ) for all r and t. If dH is not stable, note that
A H ( t ) [ C X(r)dH(tr)lK(t) T
is a nonzero class function on the elliptic set of H which is invariant under t ++ t", and introduce a function dH,o on the elliptic set of COby
Then dH,dH(d0) H becomes
But since d a o is a nonzero conjugacy class function on the elliptic set of COthere is a square integrable irreducible representation 7r0 of COsuch that ( d ~ , oxxo)c0 , # 0 , and the proposition follows. 0 Let us review several Alifting facts, used in the study of the character relations below. (1) The representation 1 >a 7r1 of H , where 7r1 is a PGL(2,F)module, Alifts to the &invariant Gmodule I ~ ( 7 r 1"1) , (Proposition V.1.2). If 7r1
132
V. Lifting from PGSp(2) to PGL(4)
is cuspidal then 1 x 7r1 is the direct sum of two irreducible inequivalent tempered representations 7rz = 7rA(7r1) and 7r; = 7 r i ( 7 r 1 ) (Proposition V.2.3(b)). Then
for all matching f,f H . The same assertion holds when 7r1 is ( sp2, E2 = 1, see (3) below. (2) For any PGL(2, F)module 7r2, the Hmodule (7r2v1I2 x (v1/2 Alifts t o the Gmodule I G ( J ~ /7r2, ~C , Y  ~ I (Proposition ~) V.1.2), which has composition series consisting of IG(( sp2,7 r 2 ) and IG(6212,7r2). The Hmodule J7r2v1l2x Jvll2 has a unique irreducible subrepresentation, which we denote by 6 ( ( 7 r 2 ~ ~ / ~ , ( v It  ~is / ~square ). integrable, and a unique quotient L(~~~CV~/~ which , ( Yis nontempered, ~/~) when 7r2 is cuspidal (or is sp2, see (3) below); see Proposition V.2.2. Thus
for all matching f and fH. (3) We have the following decomposition into irreducibles, where the tempered and b is square integrable:
(Proposition V.2.1). Here a and ( are quadratic characters of F X
T
are
5. Character Relations
133
5 . Character Relations Our main local results of character relations are derived from the trace formula identity. 5. PROPOSITION. Let 7r1, 7r2 be two inequivalent cuspidal (resp. cuspidal or special) representations of PGL(2, F ) , F a local padic field. Then there are two cuspidal (resp. square integrable) representations of H = PGSp(2, F ) , 7r; and 7 r i , such that f o r all matching functions f , f H , fo o n G, H , CO,we have
and t11(7rl17r2;f x
e) = t r r & ( f H ) + t r r ; ( f ~ ) .
The same identities hold when 7r1 = 7r2 is squareintegrable, but then 7r& and ITare ; the two irreducible constituents of 1 x TI. They are tempered and 7rh + n; = 1 >a 7rl.
PROOF.This is our main local assertion in this work. The long proof will be cut into a sequence of assertions, most of which we name “Lemmas”. The case of 7r1 = 7r2 is in 6.5. Elsewhere we assume that 7r1 # 7r2, unless otherwise specified. Let ( 7 r , V ) be an admissible representation of a padic reductive group G. As in [BZl], we introduce the
5.1 DEFINITION. (1)Let N denote the unipotent radical of a parabolic subgroup of G with Levi subgroup M . Then the quotient VN of V by the span of the vectors 7 r ( n ) w  w as n ranges over N and 21 over V , is ( 6 ~ ( m=) an admissible Mmodule j i ~ . Its tensor product with ’:6 I det(Ad(m)I Lie N ) I ) is called the normalized Mmodule ( T N , V N )of N coinvariants of 7 r . (2) The representation 7r is called cuspidal when TN = (0) for all N # {l}, that is when t r r N ( 4 ) = 0 for every test measure C#I on M . Analogously, (3) If 8 is an automorphism of G and 7r is Oinvariant (7r N ‘ T ) , we say that 7r is 8cuspidal if for every Oinvariant proper parabolic subgroup of G we have tr 7 r ~ ( $x 0 ) = 0 for every test measure 4.
134
V. Lifting f r o m PGSp(2) to PGL(4)
5.2 LEMMA.The representation and 8cuspidal.
7r = I(7r1,7r2), 7r1
# 7r2, is 8invariant
PROOF.If N is the unipotent radical of a proper parabolic subgroup of PGL(4, F) then T N is zero unless the parabolic is of type (2,2), in which ~ 7r1 x 7r2 + 7r2 x 7r1. case 7 r = However, the irreducible constituents n1 x 7r2 and nz x n1 of this TN are interchanged by 8 so that the trace trTN(q5 x 8) vanishes for any test measure d on M . 0 When 7r1 # 7r2 but one or two of them is square integrable noncuspidal (special) representation of PGL(2, F), the representation I(n1,7r2) is (8invariant and) tempered subquotient of the induced I ( V ~ 7r2, / ~ vP1I2) , (if 7r1 = sp,). This is the Alift of 7rzv1/' x Y  ' / ~ , whose composition series consists of the square integrable 6 ( 7 r 2 ~ ~ v/ ~ l,l 2 ) and the nontempered L ( T ~ v ' /v'/,) ~ , (if 7r2 is cuspidal), or square integrable 6([v1/2 sp2,v'l2) and nontempered L(vl/,[ sp2,Y  ' / ~ )(if 7r2 is the special [sp2, where t2= 1 # [). Since the functor of Ncoinvariants is exact ([BZl]), the central exponents (central characters of constituents) of I(sp2," 2 ) ~correspond to those of 6 ( ~ 2 1 / l /vl/,) ~, ( 7 ~ 2cuspidal) or 6([v1I2 sp2,v'/') (if 7r2 = [sp2), which are decaying. The twisted analogue of the orthogonality relations of Kazhdan [K2], Theorem K , implies in our case where 7r1 # 7r2 are square integrable PGL(2, F)modules:
5.3 LEMMA.There exists a 8pseudocoeficient f 1 of
7r
= I(7r117r2).
A 8pseudocoefficient f 1 is a test measure with the property that trI(7rl17r2;f 1 x 8) = 1 but tr7r'(f1 x 8) = 0 for every irreducible Gmodule 7r' inequivalent to ( i ) I(n1,7r2) if 7r1 , 7r2 are cuspidal, or to (ii) any constituent of I ( a , b ) if 7r1 (or 7 r 2 ) is esp2, E2 = 1, in which case a is [ I ( Y ~ vl/,) / ~ , (or b is such). Note that the 8orbital integral @ ( g , f' x 8 ) of f' is supported on the 8elliptic set, and is equal to the complex conjugate of the 8twisted character xr(gx~)ofI=I(.rr117r2). We shall show below that xI(g x 8) depends only on the stable 8conjugacy class of g, hence @(g,f ' x 8) depends only on the stable 8conjugacy class of g.
5. Character Relations
135
We now pass to global notations. Thus we fix a totally imaginary number field F whose completion at the places v, (0 I i I 3) is our local field, denoted now Fv0.Denote our local representations by 7 r J v O l j = 1, 2. Fix 7rJ,% N 7rJu0 ( j = 1, 2; i = 1, 2, 3) under the isomorphism FU1 rv FUo.
5.4 LEMMA. There exist cuspzdal representation 7r1 and 7r2 which are unramified outside the places v, (1 5 i 5 3) of F whose components at u, are our 7rlvt and TZ,, (respectively).
PROOF.This is done using the nontwisted trace formula for PGL(2), and a test measure f whose components fv, are pseudo coefficients of 7 r j U ,, and whose components fU at all other finite places are spherical. At one of these u # Y, take fv with tr 7r,( f,) = 0 for all onedimensional representations 7r, of PGL(2, F,) (the trivial representation and its twist by a quadratic character). Most of the f, are the unit element of the Hecke algebra, but the remaining finite set can be taken to have the property that the orbital integral of f" = @U
V. Lifting from PGSp(2) to PGL(4)
136
theorem for PGL(4) the only contribution to the &twisted trace formula is I(7r1, n2). The contributions to the trace formula of H are some discrete spectrum representations ITH. Applying generalized linear independence of characters at all places w # wi ( 0 5 i 5 3) where 7rlw x 7rzw is unramified or F, is @, we obtain the identity
"H
W
The products range over w = wi (0 5 i 5 3); m ( 7 r ~are ) the multiplicities of the discrete spectrum representations TH. The identity holds for all triples (fw, fcow, fHw) of matching measures such that at 3 out of the 4 places the orbital integrals vanish on the nonelliptic set. It is clear that
5.5 LEMMA.The distribution fw on
fHv,
H tr1(7r1w,7r2w;fw x 0 ) depends only namely only o n the stable &orbital integrals of f w .
Consequently the &twisted character of 1 ( 7 r l w , ~ 2 is ~a &stable ) function. In particular, the &twisted orbital integral of a 8pseudocoefficient of I(7rlw,7 ~ is not ~ )identically zero on the &elliptic set. This establishes a fact which is used in the derivation of the identity (2) below.
5.6 LEMMA.Fix w E {wi}. The right side of (1) is not identically zero as fHw ranges over the functions whose orbital integrals vanish outside the elliptic set of H . PROOF.Had it been zero we could choose fHw whose stable orbital integrals are zero (and so fw = 0) but with unstable orbital integrals, that ~ , tr(7rlWx 7r2w)(fc0w) # 0. Hence for each w there are T H on ~ is f ~ , ,with the right whose character is nonzero on the elliptic set. Using the &'twisted trace formula and a totally imaginary field F whose completion at wo is the local field of the proposition, we construct a representation 7r as follows.
5. Character Relations
137
5 .7 LEMMA.There exists a cuspidal 8invariant automorphic representation IT with the following properties. Its component at wo is our IT,^ = I ( I T ~ , ~where , ~ ~7rlu0 ~ ,#~7rzvO ) , are square integrable representations of PGL(2, Fvo). At three nonarchimedean places w1, v2, 213 the component is the Steinberg (square integrable) representation St,, . A t all other nonarchimedean places v the component is unramified.
PROOF.We construct IT on using the stable &twisted trace formula and a test function f = @f, whose component fv is unramified at w # wi (i = 0, 1, 2, 3), our pseudo coefficient at VO, and the pseudo coefficient of StUi (i = 1, 2, 3) at vi. Since the &character of St,, is &stable (being the Alift of StH,ui),the &twisted trace formula for f with such a component is &stable, namely its geometric part depends only on the 8stable orbital integrals. As we showed in 5.5 above, the 8stable orbital integral of the pseudocoefficient f,, of 7rv0 does not vanish identically on the 8elliptic set. This determines the nonarchimedean components of f. The geometric side of the stable 8trace formula consists of orbital integrals. We choose the archimedean components to be supported on a small enough neighborhood of a single ®ular stable elliptic 8conjugacy class, such that there will be only one rational stable 8conjugacy class y, which is in the support of the global f and there W(Y,f x 8) # 0. Then the geometric side of the stable 8trace formula reduces to a single nonzero term, namely QSt(y,f x O), and so the spectral side of the 8trace formula is nonzero. By the choice of f there is a representation IT of G(A) which is 8invariant, whose components outside vi ( i = 0 , 1, 2,3) are unramified and at vi are I ( I T ~~2, ~ , ~and , ) Stui (i = 1, 2, 3). In fact, 7r cannot have at vi ( i = 1, 2, 3) components other than StUi because the choice of fui and the fact that tr IT,^ (fui x 8) # 0 imply that IT,^ is a constituent in the composition series of the induced representation I,, (from the Bore1 subgroup) containing StVi. Since IT has the component I ( I T ~ , ~had ,IT it ~ not~been ~ ) ,a discrete series, it could only be induced I ( I T I , Ifrom T ~a) cuspidal representation IT^ x IT^ of the Levi subgroup of type (2,2), and its components at wi (i = 1, 2, 3) would have to be I(vsp2,vlsp,) or I(vl2, vV112), which are not unitarizable. Of course IT,^ ( i = 1, 2, 3) cannot be the trivial representation, since
138
V. Lifting from PGSp(2) to PGL(4)
then T would be trivial, but is has the component I ( T ~ ~ , , I T ~ , , , ) . Now since T has components Stvi it has to be cuspidal. Indeed, having the component I ( T ~ ~7 ,~ , 2 prevents ~ ~ ) T from being a noncuspidal discrete spectrum representation, a complete list of which is given in [MWl]. In 0 particular IT is generic. Having the representation T we can use the trace formulae identity, and a standard argument of generalized linear independence of characters, applied at all places where IT,, is unramified, to obtain an identity x€i
The products range over u = v, (0 5 i 5 3) and the archimedean places. By rigidity and multiplicity one theorem for G = PGL(4) the only contribution on the left side is our cuspidal T . Since it has Steinberg components, the only contribution on the right is of discrete spectrum representations T H of H(A); there can be no contributions from the endoscopic group CO of H, and contributions from the 8endoscopic group C of G have been dealt with already. We now apply generalized linear independence of characters at the archimedean places v of F , where F, = C and the representation T,, is fully induced (from the Bore1 subgroup). At the places u, (i = 1, 2, 3) we use f H v , which is a matrix coefficient of StHw,and f,,, which is a &matrix coefficient of St,, . These functions are matching since StHw, Alifts to St,, @(fv,) = zstvl. Their orbital integrals vanish on and @(fHw,) = zStHv,, the non (8) elliptic set. ~ a~ subOn the side of H we have that tr TH,,, ( f H v , ) is 0 unless T H is quotient of v2 x v >a v  3 / 2 (see Corollary V.1.5). Since a component of an automorphic representation TH has to be unitarizable, the subquotients of v 2 x v >a v  3 / 2which may occur are StHw, and the trivial representation l ~ . But~a discrete , spectrum TH which has a trivial component is trivial (by the weak approximation theorem), and writing v for uo we conclude that 5.8 LEMMA. There are H,modules T H and ~ positive integers such that for any matching functions fw and f H v we have (2)
~'(TH,,)
tII(7Tlv,IT2w;f w x 0) = ~ m ' ( ~ H w ) t r ~ H w ( f H w ) . KHu
5. Character Relations
139
The T H which occur are cuspidal o r square integrable.
PROOF.Let N denote the unipotent radical of any proper parabolic subgroup of H . Let T H , N be the module of Ncoinvariants of T H . Since ( i ) the representation I(7r1, 7r2) is 0cuspidal if 7r1 # 7r2 are cuspidal, and (ii)its ¢ral exponents decay if 7r1 # 7r2 are square integrable, and (iii)each N corresponds to the unipotent radical of a proper &invariant parabolic subgroup of GI the character relation (2) implies that C r n 1 ( 7 r ~ ) x T HisN zero if 7r1 # 7rz are cuspidal, and it decays if 7r1 # 7r2 are square integrable. But the m'(7rH) are positive. Hence all T H , N are zero if 7rl # 7r2 are cuspidal, and decay if 7r1 # 1r2 are square integrable] namely the TH which occur are cuspidal or square integrable] respectively. 0 5.9 LEMMA. The sum (2) with coeficients m' is finite.
PROOF.To see this, write it in the form
where 1 5 b
I 00.
Let
fi
be a pseudo coefficient of the square integrable U
for any finite a over i (1 5 i 5 a ) . Then T H ~ and ,
U
I b put
fu
= Cfi, where
U
U
C indicates the sum
a
Here XI(Ng) = xI(g x 0) is a function on the space of stable conjugacy classes in H , since X I ( g x 0 ) depends only on the stable 0conjugacy class of g in G. The orthogonality relations for twisted characters] which are locally integrable, imply that (XI, XI)H is finite, hence a is bounded, and the sum is finite. 0
V. Lifting from PGSp(2) t o PGL(4)
140
6. Fine Character Relations In this section we conclude the proof of Proposition 5. Using (2), we can rewrite our identity in the form
where the m here are integers, possibly negative. As the left side is nonzero, there are nonzero contributions on the right whose character is nonzero on the elliptic regular set, for each Y. Once again using (2) we write our identity  but on choosing fH, to be a matrix coefficient of I T H , which occurs on the right side, at 3 out of our 4 places, so that trnw,(fH,) is 1 or 0 (or 1) at this places, we get an identity of the form ctr(m,wo x ~ 2 , w o ) ( f c o , w o )

m ( T H , w o ) trnH,w~ (fH,wo) 
m’(TH,wo)
The term following the negative sign is
tr ~ H , w( of H , w o ) .
trI(7rlwo,7r~wo;fwo x 19).
Here
f~,,, fc,,, are arbitrary matching functions, and c # 0 (c =
tr(x1w x
~27J)(fCOW)).
wfvo
Write m for mlc, and note that the 4 places are the same: F,, = FWi (i = 1, 2, 3), and so are the components 7rlw x 7r2,, by our construction. Multiplying the last identity over the 4 places (not only V O , but also ~ 1 , 212, ~ 3 we ) obtain tr(mwx ~ ~ ~ ) ( f c ~ ~ )
n,
Comparing this with the original identity for the left side we deduce that the complex number ~ ( T H , )is an integer, namely ( ~ ( ~ T H , ) / cis) an ~ integer, hence c divides each of the m. We finally get  for all matching functions fH,, and fcoW  the identity
n,
6. Fine Character Relations
141
where the m"(7rHv) are integers, positive or negative. Of course had we used the &trace formula with no restrictions at 3 places the derivation of the last identity from (2) would be easier, but we do not use the unrestricted trace formula identity.
6.1 LEMMA.The sum with coeficients m'' is finite. It consists of 5 2 summands if T I # ~ 2 . PROOF.To see this, write it in the form
where 1 5 b 5
00.
Let
fi
be a pseudo coefficient of the square integrable
T H ~ and , for a finite a 5 b put over i (1 5 i 5 a). Then
fa
&
=
fi,
a
where
5 indicates sum
a
= 2(1+ 6(7rl,R!))U.
The last equality follows from Corollary 4.3. Hence a 5 2 if inequivalent.
7r1, 7r2
are
6.2 LEMMA.The m'' take both positive and negative values. PROOF.To see this, we write x = ~ H ( x ~Then ~ ~x is ~ an ~ unstable ) . conjugacy function on H , thus zero on the elliptic tori of types I11 and IV, and its value at one conjugacy class of type I or I1 is negative its value at the other conjugacy class within the stable class. The ? T H ~(1 5 i 5 u ) which occur in the identity for tr(.rrl x 7r2)(fc0) with mi # 0 are cuspidal or square integrable or constituents of 1 x 7r2, 7r2 square integrable (see Propositions 2.l(d) and 2.3(b)), by Casselman
142
V. Lifting f r o m PGSp(2) to PGL(4)
[C] (compare the central exponents), since 7r1 x 7r2 is cuspidal or square integrable. Hence, choosing F to be totally imaginary, and using pseudocoefficients and the trace formula as usual, we can construct a global discrete spectrum representation TH with (1)a component 7rkvowhich occurs in the trace identity of our local 7r1 x 7r2 at VO, (2) a Steinberg component StHv, at v1, v2, 213, (3) the nonarchimedean components of TH away from vi (0 5 i 5 3) are unramified. This TH contributes to the trace formula identity, where the contribution 7r to the twisted formula of G is necessarily cuspidal, as it has a Steinberg component. We apply as usual generalized linear independence of characters at the unramified components and the archimedean ones, and use coefficients of StHv, at v1, 212, 'us. We deduce that there is a &invariant generic representation IT of G(F,,) with an identity
where f i ( 7 r ~ )2 0 for all ITH and > 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 Alifts to the induced
V. Lifting from PGSp(2) to PGL(4)
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 Oelliptic set, while I(7rz,,, 7rzWi) is not Oelliptic. 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
6. Fine Character Relations
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 Oelliptic set of G,, and it is zero outside the Oelliptic set. As usual fix a totally imaginary field F and create a cuspidal &invariant representation
+
V. Lifting from PGSp(2) to PGL(4)
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 Xolift 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
6. Fine Character Relations
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
V. Lifting from PGSp(2) to PGL(4)
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 Lfunction 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 wellknown isomorphism can be constructed as follows. Let U be the 5dimensional 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 Lfunction L ( S ,T O , id5, s) is nonzero at s = 1 then TO is (globally) generic. Here L is the degree 5 Lfunction associated with the 5dimensional representation ids : SO(5, C)c) GL(5, C)of the dual group SO(5,C) of Sp(2). When TO is generic this Lfunction 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 5dimensional 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) = A17rs’(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
=
V. Lifting f r o m PGSp(2) to PGL(4)
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 8invariant irreducible square integrable representation of G,, over a local field F,,, which is not a XIlift. Then
8. Local Lifting from PGSp(2)
153
there exists a square integrable irreducible representation XH,,, of H,, which Xlifts 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 0character of 7rwo is 0stable, and the character of TH,,, is a stable function o n H,,. PROOF.Since 7rwo is 0invariant, squareintegrable and not a XIlift, its character is 0stable 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 0stable functions (supported on the 0elliptic set), the geometric part of the &trace formula is 0stable: it is the sum of We choose fm = @,f,, w archimedean, 0stable orbital integrals, to vanish on the non 0regular set. Since G(F) is discrete in G(A) and f = @f, is compactly supported, @;t( f) # 0 for only finitely many 0stable 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 0trace formula is nonzero. Consequently the spectral side is nonzero. The choice of f,, (i = 0 , 1 , 2 , 3 ) as a pseudocoefficient can be used now to show the existence of a 0invariant 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 8trace formula, not to 1 ( 2 , 2 ) , etc.). Since the components nut ( i = 1 , 2 , 3 ) and by the same argument also T,,, are our 0stable ones, 7r is not a XIlift, nor it is a Xlift 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 Xlift 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 Hmodule v t x u1/27r2, Alifts to the square integrable submodule S ( L J ~ ~/  ~' / I~ 7~r ~ 2 of ), the Gmodule 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 XIlift 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 u1/27r2 Alifts to I ~ ( ~ ~ / ~ i r 2 , ~  ~and / ~ 7772 r 2=) ,1x2, the decaying central exponents in these fully induced representations correspond, hence S(u<,vP1/'7r2) Alifts to S ( ~ ' / ~ 7 r~ 2 ', / ~ 7 r 2 from ) the proof of the proposition. 0 8 . 3 COROLLARY. The nontempered quotient L ( u ( , in the composition series of the H(F)module uE x v'/'1~2,where 7r2 is a cuspidal GL(2, F)module with central character E # 1 = E2 and E7r2 = 7r2, Alifts to the nontempered quotient J ( u ' / ~ T~~ l, / ~ 7 r 2 in ) the composition series of the induced IG(u1/27r21u'l27r2). PROOF.This follows from
For any irreducible square integrable PGL(2, F)modules have t r ( ~ xi ~ ) ( f c = , )trnj$(fH)  tr7r;(fH), t r I ~ ( ~ 1 , 7 r 2x; f0)
= tr7rj$(fH) +tr7r;(fH),
7r1
and
7r2
we
V. Lifting from PGSp(2) to PGL(4)
156
for all matching functions f , f H , fc,,where rf;,7rs are tempered irreducible (square integrable if 7r1 # 7 r 2 ) representations of H determined by the unordered pair 7 r l , 7 r 2 . If 7r1 = 7r2 is cuspidal, 7rk and 7rk are the two inequivalent constituents of 1 >a 7r1. If 7r1 = 7r2 is [sp,, where E is a character of F X with t2 = 1, IT& and 7rH are the two tempered inequivalent constituents 7(v1/' sp2,[ u  ~ / and ~) 4 2 4 ,J v  1 / 2 ) of 1 >a 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 , ~ v

for all matching f ,
fH,
tr 6  ( 7 r 2 ~ v l / ~ ,
fH),
fc,.
PROOF.This follows from trI(fsp,
x7r2; f
x 6') = trb(fH)
+ tr6(fH),
tr(JSP2 X7r2)(fCo) =trS(fH)  tr6(fH)i and
+
~ 7 r 2 ; f x 0) tr k ( E 1 2 , 7 r 2 ; f x 6') tr I G ( sp2, = tr I ~ ( J v ' ~7r2, ' , ~ v  l / ' ;f x e) = t r ( n 2 ~ v l >a / ~~ = tr(t12 x
v  ~ / ~ ) ( f=Htr)
b(fH)
+ tr L ( f H )
~ ) ( f c= , )tr(J1z x nz)(fc,,)+ tr(lsp2 x7r2)(fcO),
8. Local Lifting from PGSp(2) where
I2
157
= I ( v ' / ~v,  l l 2 ) .
8.5 PROPOSITION. For any characters F X ,for all matching f , f H ,fc, we have
#
1 = t2 and
PROOF. We use the identities displayed above for
and
For the last two identities we use the first two, and
(T
(a2 =
1) of
158
V. Lifting from PGSp(2) to PGL(4)
8.6 PROPOSITION. For all matching fH, fcoland characters with E2 = 1 we have
E
of F X
PROOF.The first equality follows from
and
The second equality follows from this as well as from
Recall (Proposition V.l.2) that for any admissible representation PGL(2, F ) we have that 1 x r Alifts to I G ( T ,T ) , thus t r I G ( r , r ; f x 6’) = tr(1 x T ) ( ~ H ) for all matching f and fH. When n = 7r1
+ 7r2 we get
7r
of
8. Local Lifting from PGSp(2)
159
It follows that the normalization of n(0)on II = I G ( ~7 r ,) , which is unique only up to a sign on any irreducible 0invariant representation of G, as 02 = 1, induces a normalization of II2,1(0), n2,1 = I~(7r2,7rl),which is E different in sign than the normalization of Il1,2(8)~n1,2 = I G ( T I , (T I~(7r2,7r1)when 7r1, 7r2 are irreducible), with the consequence of
for all f . A similar phenomenon is encountered in the following.
8.7 PROPOSITION. For all matching f and
fH
we have
PROOF.The first identity follows from X ( l x "1) = I G ( T ~and , Tthe ~) fact that the composition series of 1 x 7r1 for 7r1 = 1 2 consists of the two irreducible representations L. The second identity is a consequence of the first, as well as
REMARK. On = I G ( 1 2 , 1 2 ) we normalize the intertwining operator II(0), whose square is the identity, by the property that it maps the unramified (Kfixed) vector to itself. This coincides with the normalization , and induces a normalization of 0 on the quotient I(12 x 1 2 ) of I G ( I ~12), of 0 on the subrepresentation IG(sp2,12). On the other hand, we could normalize II'(8) on Il' = 1G(12,sp2)by mapping the Whittaker vector to itself (W H OW). This coincides with the normalization of 0 on the subrepresentation I ~ ( s p , , s p ~of) II', and , of II' which is the induces a normalization of 8 on the quotient I ~ ( 1 2sp2) negative of the normalization of 0 on
160
V. Lifting from PGSp(2) to PGL(4)
viewed as a subrepresentation of lI. Indeed, using
we conclude that
This does not contradict the Proposition, but reinforces it, yet with a 1 2 , sp2). different normalization of the intertwining operator 8 on IG(
9. Local Packets These character relations permit us to define the notion of a packet of tempered representations, and that of a quasipacket, locally. The packet of a nontempered representation T H is defined to consist of TH alone.
9.1 DEFINITION. Let F be a local field. The packet of (an irreducible) tempered Hmodule T H consists of T H alone unless TH is 7r; or 7rk for some pair 7r1, 7r2 of (irreducible) square integrable PGL(2, F)modules, in which case the packet consists of 7r; and ni. For example, if 7r2 is a cuspidal representation of GL(2, F ) with central n z ) of a single character E # 1 = E2, the packet of b ( J ~ , v  ~ / ~consists element. We write tr{.rrH} for the sum of t r & as 7rh ranges over the packet { T H } of TH. 9.2 DEFINITION. The quasipacket of a nontempered (irreducible) Hmodule T H is defined only for such an Hmodule which occurs in the character relation for a12 x 7r2, where 7r2 is a square integrable or one dimensional PGL(2,F)module and (T is a character of F X / F x 2 . It is , which occurs in this character relation defined to be the pair ~ f f 7rg (which also defines 7 r i ) .
10. Global Packets
Thus when
1r2
161
is square integrable the quasipackets are defined to be
{q7r2av1/2, av1/2), 6 (7r2av1/2, av1/2)}, {L(v1/2~sp2,av1/2), 6(tv1/2sp2,av1/2)) and {L(v1/2 sp2, a v  q ,
T(V11212,
av1/2)},
2 cuspidal7r2. Note that the 7r; for any characters E # 1, a of F X / F Xand in the last packet is tempered, but not square integrable. Correspondingly we write Xo(7rl x 7r2) = { T ; , 7 r ; } and X({7r&, T ; } ) = I ~ ( 7 r 1 , 7 r 2 ) when T I , 7r2 are square integrable, X o ( a l 2 x 7 r 2 ) = { 7 r G 1 7 r ; } and X ( { 7 r G 1 7 r ; } ) = I ~ ( a l 2 ~ 7 r 2when ) 7r2 is square integrable and a2 = 1. This notation applies also when 7r2 is sp2, or E 1 2 , in the following sense. ~ 1 = E2, u2 = 1, is defined to The quasipacket X o ( a E 1 2 x ~ 1 2 ) # consist of
We observe that 7r; of X o ( a t l 2 x a14 and of X o ( a t l 2 x asp2) are the same, although the corresponding 7rG are not. Thus it is the 7rff which determines the quasipacket, and not the 7rk. The quasipacket X o ( a l 2 x a l a ) , a2 = 1, consists of
Here we observe our 7rE is not tempered, and is in fact packet X o ( a l 2 x a s p 2 ) .
7r;
in the quasi
10. Global Packets This description of local packets of representations of H will now be used together with the trace formula identity to describe the automorphic representations of H(A). Taking into account the complete results on the lifting XI from C(A) to G(A), the trace formula identity can be phrased as follows: 1 1 I’ + 42 [ 2 , 2 ) = T s p ( f H 1 H)  qTsp(fco, CO)
162
V. Lifting f r o m PGSp(2) to PGL(4)
“2
of PGL(2)
Here I‘ is the subsum of I, namely C , t r r ( f x B), over those discrete spectrum representations 7r N e7r of PGL(4, A) which are not XIlifts (from
W)). Similarly, 1{2,2) is the subsum of I(Z,~) which consists of those induced representations 1(2,2)(7r1,7~2) which are not lifts via A1 from C(A). 10.1 LEMMA.IfI(z,2)(n1,7r2)appears in Iiz,z)then the ni have trivial central characters, namely are representations of PGL(2, A).
PROOF.The 7r1 and
are representations of GL(2,A) with T i N iri. If (hence also of 7r2, since 1 ( 2 , 2 ) ( 7 r l , x 2 ) has trivial central character), then iri N_ W T ~ ,and SO w2 = 1. If w # 1, thus w = x E / F for some quadratic extension E of F , then 7r1 = r ~ ( p L / 1and ) 7r2 = 7 r ~ ( & , ) , where 11: are characters of A g / E X . The central character of such an “ ~ ( p )is X E / F . pIAX. SO for our ~ ~ ( 1  1 : )of central character xE/F, we have p:IAx = 1, which means (since the kernel of z H z/F in A; is Ax)that there are 111, 112, characters of A;, with pI,(z) = p i ( z / F ) . Now 7r2
w denotes the central character of
so
Hence the 1[2,2)(7r1,7r2)with wxi lemma follows.
#
1 are lifts from C(A) via A l . The
0
We shall use  as usual  the form of the trace formula identity where the local component is fixed to be a fixed unramified representation at all places outside a finite set. By the rigidity theorem for PGL(4) at most one of I’ and I{2,z) would have a (single nonzero) contribution. Let 7r1 x 7r2 be a discrete spectrum representation of the group C,(A) = PGL(2, A) x PGL(2, A). It makes a contribution in Tsp(fc,,,CO)as well as in It2,2)
10. Global Packets
10.2 Suppose first that
7r2
163
= T I .Then the contribution to i I [ 2 , 2 )is
1 4
 tr IG(7r2, n2;f x
0).
This is equal to the contribution 1
4tr(I2(1,1) x .2)(fc) to the trace formula of C (see Proposition IV 3.1). Thus these two cancel each other (of course for matching f , fc,and f ~fc, , below). The corresponding contribution (determined by fixing all unramified components) to the trace formula of H is
The corresponding contribution to the trace formula of Co is
At all places v where 7rav is properly induced (and irreducible), R, is the scalar 1, and tr(1 x 7 r ~ , ) ( f ~ , ) = tr(7r2, x 7r2,)(fcO,,), as 7r2, is a representation of PGL(2, F,) (see Proposition V 1.2). If 7r2, is square integrable (or one dimensional), our local results (Propositions V 5 and 2.3(b) for square integrable 7 r 2 , , Propositions V 8.6 and 2.l(d) for one dimensional ~ 2 , ) assert that the two constituents of the composition series of 1 M 7r2, can be labeled 7rLVand 7rG, (= L(v, 1x a ~ and L ( u ~sp2, / ~a v  l f 2 ) when 7r2, is one dimensional alz), such that for matching functions tr(7r2v x
7r2v)(fco,v)
is
t'"j!JfHv)
 tr";,(fHv)
Moreover, R, acts on 7rLVas 1 and on 7 r i , as 1 (this follows for example from the global comparison). Hence these contributions to the formula of H and of Co cancel each other. 10.3 We can then assume that 7r1 # 7r2. Suppose that T I , 7r2 are discrete spectrum representations of PGL(2,A). Note that the pairs ( T I ,7 r 2 ) and
164
V. Lifting from PGSp(2) to PGL(4)
(7r2,7r1) make the same contribution to the formulae of COand of G (in I t z , z ) ) ,hence the coefficient is replaced by When 7r1 and 7rz are cuspidal the corresponding part of the trace formulae identity asserts
a
i.
The products are over the finite set V of places where both 7rlw and 7r2, are square integrable. The sum ranges over all equivalence classes of irreducible discrete spectrum representations T H of H(A) ( T H occurs with multiplicity rn(7rH)2 1 in the discrete spectrum) whose component at each v outside V is Xo(7rlw x ~ 2 , ) . Recall that Xo(I(p1, PT1) x r z ) = Pin2 x PT1
and
X(PlT2 PI1) = IG(P1, XZ,P I 1 ) .
Now at the places u in V the representations TI,, 7r2, are square integrable and the character relations permit us to rewrite the right side of the formula as
f where 7rHw = 7 rfH v ( 7 r l w x ~ 2 , )are the tempered representations of H , determined by 7rlWand r a w . It follows that the discrete spectrum representations 7 r of ~ H(A) with components Xo(7r1, x r a w ) at all v @ V have components 7rgv(mw x~
2
or ~
at all places v E V, and the multiplicity discrete spectrum of H(A) is m(7rH) =
)7 r i w ( m w x m ( 7 r ~of)
m,)
such T H = @TH, in the
1 (1+ (l)+f)), 2
where n ( 7 r ~is) the number of components of T H of the form 7 r i w . The T H with m ( 7 r H ) = 1 are all cuspidal as there are no residual representations with components X0(7rlw x 7r2,) for almost all u.
10. Global Packets
165
In fact since we work with test functions f = @fw with 3 elliptic components we can deduce only a weaker statement, which applies only when the set V has at least 3 members. Namely we cannot exclude the possibility ~ properly induced components that there exist discrete spectrum 7 r with ~ ) all u 4 V ) . So our global at all u E V (and components Xo(7r1, x 7 ~ 2 at results be complete only after removal of the 3places constraint on the test functions o f f , f H . 10.4 Next we deal with the case where 7rz is cuspidal but TI is one dimensional, E l z , E is a character of A x / F x A x 2 . The trace formula identity reduces to
where the product ranges over the set V of places where 7raV is square integrable, and the sum ranges over the discrete spectrum ITHwhose component T H at ~ u 4 V is = XO(I(PlW,P 2 ) x
G12)
= P l W l W l Z >a P;:
if
7r2w
= I(Plv,P 3 .
Note that the involution 8 defined by 8(g) = JltglJ on G(A) and its automorphic forms, induces an involution r ( 8 ) on each automorphic representation. However, abstractly there are two choices of an intertwining operator 7r 2 '7r whose square (7r 2 7r) is 1, and they differ by a sign. We observe that on a generic representation 7 r , the global involution equals the product of the local involutions 7rw(0) which act on the Whittaker functions of nu by 8. This coincides with the choice of the intertwining operator 7rw 2 '7rw, when 7rw is unramified, which maps the Kvfixed vector to itself. Our representation 7r = I ( J 1 2 , 7 r 2 ) is not generic, nor it is everywhere unramified (unless so is 7r2). Hence the global involution r ( 0 ) is the product of the local involutions 7 r u ( 8 ) , and a sign, which we denote by & ( E l 2 x 7r2). The presence of this sign was first noticed in a different context by G. Harder ([Ha], p. 173). Our local character relations express t r ( E w 1 2 x nzw)(fco,v) as the sum of traces at f H of the nontempered constituent
V. Lijling from PGSp(2) to PGL(4)
166
of the indicated induced Hwmodule,and of a cuspidal (if 7raW is), square integrable (if 7r2, = J!, spzw,JL # Jv) or tempered (if 7 ~ =2 Ew~ S P ~ , )representation 7rGv. The trace
is the difference of these two traces. Thus
We conclude that if there is a discrete spectrum T H with components all places where 7r2w is fully induced, then its component at each v in the remaining finite set V lies in the quasipacket {7rGu, T,,}. Its multiplicity is X o ( E W 1 2 x 7 r z W ) at
where ~
( T H is )
the number of components 7rhwin
10.5 LEMMA.For any 4 0 2
x
7r2)
cuspidal7r2
TH.
and quadratic character J we have
=4J7r2,
PROOF.Here ~ ( ~ 2 , iss )the epsilon factor in the functional equation of the Lfunction of 7r2. Note that 4 J 1 2 x " 2 ) is 1 iff ~ f f ,= BW7rGw is discrete spectrum. It is known from the theory of Eisenstein series ([A2], p. 32; [Kim], Theorem 7.1) that this representation is residual, namely discrete spectrum and generated by residues of Eisenstein series, precisely when the Lfunction L(Jn2,s) of 5 x 2 is nonzero at s = The case of E # 1 reduces to that of J = 1 as
i.
To repeat: in this case where L(J7r2,?j) # 0 , ~ ( 5 ' 1 2x 7r2) is 1, as is ~ ( 5 7 r 2 , i ) . To determine, when L(57r2,;) = 0, whether the quotient 7r;f of Jv1/27r2x J u  1 / 2 is cuspidal or not, we appeal to the theory of the
10. Global Packets
167
theta correspondence. As noted above, it suffices to assume J = 1. Then &(J7r2, = 1 implies that 7r; is cuspidal by [W2], Proposition 24, p. 305. The converse follows from [PSl], Theorem 2.2 (1 + 4). To explain this, recall that the theta 0 = 6, and the Waldspurger's Wald = Wald, correspondences depend on a nontrivial additive character 4 : Amod F + C1, which we now fix. These correspondences fit in the chart:
i)
PGSp(2, A) = SO(5, A)
E(2,A)
II E(2,A)
Wald
+
e
c
PGL(2,A) = SO(3,A) T JL
D2
Here E ( 2 , A ) is the metaplectic two fold covering group of SL(2, A), and JL denotes the JacquetLanglands correspondence from the multiplicative group 0: of the quaternion algebra DA. Given 7r1 = @ v 7 r lv on PGL(2,A), Waldl(7rl,) is if 7rlv is principal series, {jiv,gen,jiv,ng} if 7rlv is discrete series. Here jiv,genis the &image of 7rlv while jiw,ngis the &image of 7rf, = JLl(7rlv) at the places u where D ramifies. The product @vjiv,gen defines a representation of E ( 2 , A ) when ~ ( n  1;), = 1, and the &lifting E(2,A) 4 PGSp(2, A) maps @vjiv,gen to
This 7r; is cuspidal when L(7r1, f ) = 0 and &(TI, f ) = 1 by [W2], Proposition 24, p. 305. Now suppose that 7r; is cuspidal. Then L(7r1,f ) = 0. We claim that & ( T I , f ) = 1. By definition, 7rf: is in 0 2 p of [PSl], p. 315. Hence there is a cuspidal irreducible representation 0 of E ( 2 , A ) which &lifts to ;.ff by [PSl], Theorem 2.2 (1 + 4). Moreover Wald(o) = 7r1 by the rigidity theorem for GL(2,A). If & ( T I ,f ) = 1, the representation n in Waldl(r1) it cannot have which 8lifts t o PGSp(2,A) must have a component jiv,ng: the component jiv,genat all places. But the local 8lift takes jiv,ngto a tempered representation of PGSp(2, F u ) , contradicting the assumption that 7rEl with which we started, has no tempered components. As already noted, the case of 5 # 1 follows from this and the equality of J v ' / ~ Tx~[ v  ~ / and ~ J ( J V ~ / x~ ~v Q' / ~ ) . Thus 7r;) is cuspidal iff &([7r2, = 1 and L(J7r1, ); = 0. It is non discrete series iff ~(J7r2,f ) = 1.
a)
V. Lafling from PGSp(2) to PGL(4)
168
In summary: &(El2 x 7r2) = ~(t7r2,;). Further details on Waldspurger’s correspondence can be found in Schmidt [Sch]. 0 10.6 Similarly, for characters E part of the traces identity
# 1, a o f A X / F X A Xwe 2 have the following
1
el.
+&(at12 2 x a 1 2 ) n t r l c ( a u t w l z , ~ w 1 2 ; fxv W
The product ranges over a set V such that uw,tw are unramified for Y $! V . The sum ranges over the discrete spectrum 7 r of ~ H(A) whose component at Y $! V is 7rffw = L(J,v,,J, >a 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 selfcontragredient 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 Xolifting (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 quasipackets 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 Alifts t o a (unique) cuspidal selfcontragredient representation of G(A). The quasipackets 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 quasipacket in the discrete spectrum of H ( A ) with a local component which is onedimensional 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
V. Lifting f r o m PGSp(2) to PGL(4)
170
formula identity as the equality of the sums
The sum on the left, I / , ranges over all selfcontragredient discrete spectrum representations of G(A) which are not XIlifts 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 quasipackets Aolifted 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 Alifts to the residual (discrete spectrum but not cuspidal) selfcontragredient representation J(v1/27rz,u1/27rz)
= @.vJ(u,l/27r~v, v,1/27rzv)
of G ( N . Each irreducible in such a quasipacket occurs in the discrete spectrum of H(A) with multiplicity one, and precisely one irreducible is residual, namely ~ v ~ ( v vv1/27rzv). v ~ v ,
PROOF.If
Cv # 1 and 7rzV is cuspidal, J
( U ~ / ~~ t T’ ” ~7 r~2 ~, )is the Alift
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 Alift 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 > 1pvl2 > ~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 ~ ) Alift . 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 Alifts to ~ ( u , 1 / ~ 7 r 2vi1/27r2v). ,, In summary, the quasipacket of L(vvcv,~ i " ~ 7 r 2 which ~ ) Alifts 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) Alift to the one dimensional representations of G(A); and 2. The discrete spectrum quasipacket { L ( v J ~,  ~ / ~ 7 r 2of) H(A) } Alifts 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,v1/27r2)where 7r2 is a cuspidal representation of PGL(2, A), in which case this J is the XIlift of
172
V. Lifting from PGSp(2) to PGL(4)
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 Alift 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 nzw 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 sixdimensional representation A2 of the dual group Sp(2, C)of PGSp(2) is the direct sum of the irreducible fivedimensional 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, nH , 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
V. Lifting from PGSp(2) to PGL(4)
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 XIlift from SO(4,A). We shall then conclude that 7r is not a Alift 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 Lgroup 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(k1)/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 / 2sgnk1vk/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
V. Lifting from PGSp(2) to PGL(4)
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
<
<
>
+
116. Cohomological Representations
177
given by
z
++
diag((z/lzl)k', (z/14)k2,(14/4kz2, (14/4k')x z
and 0 H
(
0
(:x)
or
w w 0,
x
(odd kl > 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"uk'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(klk2)/2 >a
uk2/2,
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
Quasipackets 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 <2
(t : R x
= J ( ~ Y ~ / ~ , < Y of  ~ ’PGL(2,R) ~)
+
g,
{&I}, ~ ( z=) 1.)
Here J denotes the Langlands quotient of the indicated induced representation, I ( J Y ’ / ~ t, ~  ~ / ~ ) . Similarly the one dimensional representation <4
=J ( ~ v ~ 5u1/’, / ~ < , v~/~,
of PGL(4, R) is parametrized by $ : W, x SL(2, C)
($IWR)(ZO~)
=t(l)j,
$1
+
SL(4, C),
S L ( ~ , C= ) Symi,
thus &(z) = d i a g ( ~ ’ , r , ~  l , r  ~ x )z ,
$+(a)=t(l)14x a.
This parameter factorizes via : WRxSL(2, C) + Sp(2, C ) , which parametrizes the one dimensional representation I H of PGSp(2,R), h H <(X(h)) where X(h) denotes the factor of similitude of h, whose infinitesimal chara . We have acter is ( 2 , l ) = f Ca>O $J
Hij(sp(2,1W),SU(4);[H 8 VO,,) = C
i ( 1+~
for ( i , j ) = ( O , O ) , ( l , l ) ,(2,2), ( 3 , 3 ) . Of course 1~ # sgnH, and sgnH) is the characteristic function of H o in PGSp(2,R). Moreover, the character of f ( 1~ sgnH) vanishes on the regular elliptic set of PGSp(2, R), as ( I H rzf r:?) IHo is a linear combination of properly induced ( “standard”) representations ([Vo]) in the Grothendieck group.
+
+
+ rT;h+ rip: +
l l d . The Nontempered: L ( v sgn, v  1 / 2 7 r 2 k )
179
lld. The Nontempered: L(Ysgn, V  1 / 2 7 r 2 k ) The nontempered nonendoscopic representation L( Y sgn, v  1 / 2 7 r 2 k ) of the group PGSp(2, R) ( k 2 1) is the Langlands quotient of the representation vsgn M v P 1 I 2 7 r 2 k induced from the Heisenberg parabolic subgroup of H . It xlifts to J ( V I / ~ T V ~ 1 / 2~7 r, 2 k ) , the Langlands quotient of the induced representation
Note that the discrete series sgn (f 1). Now
and
($1
SL(2,C))
7r2k
= sgn @3 7r2k = ?2k
has central character
(z i) (zf i:), =
defines
= Sp(2, C)~t SL(4, C)and defines L(vsgn, V  1 / 2 7 r 2 k ) . It factorizes via Note that when 2k is replaced by 2k 1, & ? k + l ( O ) = EW x cr, E = diag(1, l), then
+
thus 41~defines a representation of C(R) (which XIlifts to the representation J(v1/27r2k+l, v1/27r2k+l)
of PGL(4, R)), but not a representation of PGSp(2,R).
V. Lifting from PGSp(2) to PGL(4)
180
As in [Ty] write 7r2k,0 1 for L(sgnu,v1/27r2k+2). We have that 7rik,oN sgn 63 +,k,o] and d j k , o l H o consists of two irreducibles. In the Grothendieck group the induced decomposes as
To compute the infinitesimal character of is a constituent of the induced
u sgn >a
Y1/27r2k, note that it
(using the Weyl group element (12)(34)), whose infinitesimal character is (2k, 1) = ( 2 , l ) ( a ,O), with a = 2k  2 2 0 as k 2 1. For 5 2 1 we have
+
l l e . The Nontempered: L(a from the Siege1 parabolic subgroup of PGSp(2,R). It is the &lift of 7r2k+l x J2 and Alifts to the induced 1 ( 7 r 2 k + l 1 J 2 ) of PGL(4, R). The central character of 7r2k+l is trivial, but that of 7r2k is sgn. Hence 1(7rgk, (2) defines a representation of GL(4, R) but not of PGL(4, R). The endoscopic map
?+b
:
w,x SL(2,C)
f
LCO = SL(2,C) x
defines
which lies in
2 c SL(4, C)since 2k + 1 is odd.
SL(2,C)
3 ill
l l e . The Nontempered:
L(cd/27r2k+l
,[ v  ' / ~ )
181
As in [Ty] we write T i f 1 , k  l for L ( J V ~ / ~ T ~ ~ +k ~2, 0.J ~  ~ / ~ ) , Now [ 7 r 2 i 1 = 7 r 2 , E and 7r2,EIHois irreducible. In the Grothendieck group the induced decomposes as
~ z tis generic, ~ , ~discrete series if k 2 1, tempered if k = 0.
Here Our
5 v 1 / 2 7 r 2 k + l >a
Cvl/'
is a constituent of the induced
which is equivalent to [uk+' x [ v k >a (using the Weyl group element (23)). Its infinitesimal character is ( k + l , k) = ( 2 , l ) + ( k  l , k1). We have
In summary, Hij ( T @ V a , b ) is 0 except in the following four cases, where it is @. (1) One dimensional case: ( a , b ) = (0,O) and 7r is rho' 3,l [H,
T?:,
1
(2) Nontempered unstable case: ( a ,b) = ( k , k ) (k 2 1) and
( 3 ) Nontempered stable case: ( a , b) = ( 2 k ,0) ( k 2 1) and
+
7r
T
is
is
(4) Tempered case: any other (a,b) with a 2 b 2 1, a b even, and 7r is Here k l = a b 3 > 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 2dimensional 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 0module 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 wellknown and easy to check. The quotient R ( m ) / R ( m +1) is a 1dimensional 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 EXorbitson 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+lf& q1 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 ordervaluation 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 Gorbit 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 =wTuR(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'f1l
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 1dimensional 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 Fautomorphism 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 Fanisotropic 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)'lhyhl
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 Fanisotropic (2= center of G and of A X ) .
VI. Fundamental Lemma
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)* = clA* 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 Fvector 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),
qnl(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) 0modules. 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 QLIlattice in L, then M*= {x E LI(x,y) E 0 for all y E M }
VI. Fundamental Lemma
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 0Ltlattice 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 0lattice in L. Put M = O p A , N = 0 ~ i A * . Then M * = {x E L I ~ L , c z A*} is the largest OLIlattice contained in A* and N = DL~/F.largest OLjlattice 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 0Lrsemilinear 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 OLilattices in L and cp : M / N + M / N an 0 LIsemilinear 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 OLIsemilinear, then f H Ifcpis 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 1dimensional 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).
VI. Fundamental Lemma
190
The condition A* = vA means that N, = V,~M: (z = 1 , 2 ) and v~pull = (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 q3m2(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
VI. Fundamental Lemma
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,lurnodnmk.
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.rrmk. 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+2k1 ( 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 q3m2(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 ) ~11
a
becomes
(4
)l
rl O ( ~ I ban2 ) O(bz)bllrU(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 ( b2 k) ) ' ) .
VI. Fundamental Lemma
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 subsums are:
if
a1

a2(7r).
' qM+Nl
(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
2c. L ( F ) is a Quartic Extension
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 certainconditions (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
VI. Fundamental Lemma
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 OLrmodule. 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 rm2 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/)
+
2c. L ( F ) is a Quartic Extension
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.
VI. Fundamental Lemma
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 modsEkOLr.
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~~kl
mk1
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).
2c. L ( F ) is a Quartic Extension
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
VI. Fundamental Lemma
200
+
There are q3"(l q2) 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  a7r2m2' 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; rnodnmkOLI, 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
' blmodnmkOLt
Xb2 = X
and
blblclE1 E F x 2 ,
2c. L ( F ) is a Quartic Extension
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
u1E(3;
Card{cp),
/ o L ( n )x
where we put mnt)L/ (V)IfL,L/
u = ulnL
The following observations can be used to handle the sum over u1. a) If LIL' is unramified, DLI induces a bijection L" / N L I LL~x + 2/22. If L/L' is ramified,
b) Assume L'/F unramified. Then N L I F O= ~ O x if L/L' is unramified, = O X 2if L/L' is ramified.
c ) Assume L / F unramified. Then N L , F : 0 , " / O i 22 OX/Ox2.Moreover, in the case where a p ( t z ) < m+n, we have n > 0, SO O L ( ~ ) 'c O i 2 .
VI. Fundamental Lemma
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 bl?modnL,. One checks casebycase that this is impossible when L J F is not the composite of the three quadratic extensions of F . 0
+
2c. L ( F ) is a Quartic Extension
203
The sums (I)(V) are:
qq2B+2

(L/L’ ramified);
q2  1
1.1)
q2B1
(q2 f
421
q2
+1
q2B+1(q2
42 1 q3B  1 2q421 431
q1 B
qi
111)
+
8

~

}
(L/L‘ ramified);
q31
+ 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 HasseWeil Zeta function of a modular curve as a product of Lfunctions 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 [L351 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 [A241 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 threefold, in the case of F = Q. Laumon [Ln] used the ArthurSelberg 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 GoreskyMacPherson. 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 agadic 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 0group 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 Rhomomorphism 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$, Va,b). For each rational prime e, the representation
np
+
ol
of H’ over F ( S is the set of embeddings of F in R) defines a smooth adic sheaf va)b;$on S K ~We . are concerned with the decomposition of the
I. Preliminaries
210

@QG,
Qeadic vector space H:(SK, Va,b;e) as a C , ( K f \ H ’ ( A Q f ) / K f , Q g ) x Gal(Q/Q)module, or more precisely the virtual bimodule
We fix an isomorphism of fields from Qe to
c. Write H,‘(rHf) for
We are concerned only with a, 2 b, 2 0 with even a ,  b,, and we consider only the part of H,* isotypic under Z ’ ( A Q ~ )Thus . we work with functions in the Hecke convolution algebra of compactly supported modulo the ) center Z’(AQf) of H‘(AQf), Kfbiinvariant functions on H ’ ( A Q ~which . we take our group H to transform trivially under Z ’ ( A Q ~ )Alternatively be the projective symplectic group of similitudes. We make this restriction since this is the case studied in [F6]. The fundamental lemma is established in [F5] for any central character. Thus from now on H’ = R F / Q H , H = PGSp(2). In the next line, C, is C , ( K f \ H ’ ( A Q f ) / K f ) .
THEOREM 1. The irreducible C, x Gal(Q/Q)modules which occur nonK ~ Q Q , V ~ , ~are , ; ~of ) the form r H ;8 H,‘(xHf), where trivially in H , * ( S K@ THf as the finite component @ p < w rof~apdiscrete spectrum automorphic K representation T H of H ‘ ( A Q f ) , and x H i denotes its subspace of Kffixed vectors. The archimedean component T H =~ @ , E S T H , of r, S = { F R} and H’(R) = H ( F @ F , ,R), has components T H , whose infinitesimal character is (a,, b,) ( 2 , l ) . Here ( 2 , l ) is half the s u m of the positive roots. Conversely, any discrete spectrum representation T H of H ‘ ( A Q f )whose archimedean component T H = ~ @ v E ~ ~is ~such , that the infinitesimal (2, l), a, 2 b, 2 0 , even a,  b,, f o r each character of T H , is (a,, b,) K TJ E S ( w e call such representations T H cohomological),and r H $ # { 0 } ,

nvES
+
+
K
occurs in H,*( S K 8~ ~ 0,Va,b;e) with multiplicity one as r H ;@ H,‘(x,yf). The main point here is that the T H which occur in H,* are automorphic, in fact discrete spectrum with the prescribed behavior at 00 and ramification controlled by K f . Each cohomological T H occurs for some K f depending on T H . The first statement here is known for I H by [BC].
2. Statement of Results
211
We proceed to describe the semisimplification of the Galois representaf) to 7 r ~ f . For this purpose we first need to list the tion H , * ( 7 r ~ attached cohomological T H . Note that H ' ( Q ) = H ( F ) and H'(AQ) = H ( A p ) . The T H are described in [F6] in terms of packets and quasipackets, and liftings A : H = Sp(2,C) + G = SL(4,C) (natural embedding), and XO : SL(2,C) x SL(2,C) = 6'0 HI 6'0 is viewed as the centralizer of diag(1, 1, 1,l) in H . A detailed account of the lifting theorems of [F6] is given in the text below, as are the definitions of [F6] of packets and quasipackets. Quasipackets refer to nontempered representations. We distinguish five types of cohomological representations T H of PGSp(2, AF). (1) X H in a (stable) packet which Alifts to a cuspidal representation of G(AF), G = PGL(4); the components 7 r ~ , ( w E S ) are discrete series with infinitesimal characters (avrb,) +(2,1). (2) T H in a (stable) quasipacket of the form ( L ( < V , V  ~ / ~which T~)} Alifts to the residual noncuspidal representation v1/27r2) of PGL(4, AF). Here 7r2 is a cuspidal representation of GL(2, Ap) with quadratic central character [ # 1 with [7r2 = 7r2, and discrete series components 7r," = 7 r 2 k V + 2 , k , 2 0 for all w E S . Here (u,, b,) = (2kv,0). ( 3 ) One dimensional representation 7 r H ( g ) = <(X(g)) of H ( A F ) . Here X(g) is the factor of similitude of g, [ is a character A:/FXAg2 {fl}, and (a,,b,) = (0,O). (4) T H in a packet which is the Aolift of 7r1 x 7r2, where 7r1 and 7r2 are distinct cuspidal representations of PGL(2, AF) such that T,"} = {7rkl, , 7 r k z v } , k l , > k2, > 0 odd integers for all w E S. This packet Alifts 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 quasipacket { L ( [ V ~ /[~v Xl l~2 ,) }which is the Xolift 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 quasipackets

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 Rhomomorphism h : R@/wG, + H' = RF/QH. Over C the oneparameter subgroup p : C x 4 H ' ( C ) , p ( z ) = h ( z , l ) factorizes through any maximal Ctorus TL(C)c H ' ( C ) . The H'(C)conjugacy class of p defines then a Weyl group Wcorbit 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 Wcorbit 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,,
2. Statement of Results
+
+
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 lEprime 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:QldimensionaZ7and 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 quasipacket { L ( E Vv1/27r2)} , 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, v1/27r2k,+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:Q1dimensional 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 ~~), tl t1 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 XOlift of 7r1 x 7r2, where 7r1, 7r2 are cuspidal representations of PGL(2,Ap). It is defined b y means of local packets
+
+ +
2. Statement of Results
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 quasipacket { 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 t1 t1 by t , = diag(tl,,tz,, z U , z U ) , ti, = tUqA/2ziUJ t 2 = ~ E4h/2zlu . The quasipacket {TH} is the Xolifl of 7 r 2 x [ l z , defined using the local quasi~2 [wuw ‘I2) , 7 rH 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
+
+
2. Statement of Results
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 quasipackets 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 Frank 1 (see [F2;I], [F3;VI], [F4;III])but not yet in Frank 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 Kfbiinvariant Qvalued function :f on H(APf),(A is AF and we fix a field isomorphism 2 C ) , and with coefficients in the smooth aesheaf 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 Alift to cuspidal &invariant representations 7r of G(AF), G = PGL(4). The infinitesimal character of each
a,].
nl<m
I. Preliminaries
220
+
archimedean component TH,(U E S ) is (a,, b,) (2, l), determined by (a,b). The components T H , for each place u of F over p are unramified and tempered. In fact the 4 nonzero, namely diagonal, entries of t ( 7 r H u ) are algebraic, all conjugates lie on the complex unit circle. (2) Stable quasipackets ( 7 r ~= L ( ~ V , ~ Y  ~which ~ ~ TAlift ~ ) } to , the quotient J of the induced of PGL(4, AF). Here 7r2 is a cuspidal representation of GL(2, AF) with central character E # 1 = E2, archimedean components T,” = 7r2kVl k , 2 1, v E S , and unramified components 7r:, ulp. The infinitesimal character of T H , is (2k,,l) = ( 2 , l ) + (a,, 0), thus these contributions occur only when all b, are 0. The diagonal entries of t ( 7 r ~ , )are (qA/2zmu)*1, m = 1 , 2 , where the Satake eigenvalues z,, of 7r: are algebraic all of whose conjugates are on the complex unit circle. (3) One dimensional representations T H = 0 A, X denotes the factor of similitude, E a character A g / F X A g 2 , {fl}.This case occurs only when (a,, b,) = (0,O) for all u E S , and we have
<
c, E {il} indicates the value at xu.of the ucomponent of E. The two unstable contributions are parametrized by: (4) Unordered pairs 7r1 x 7r2 of cuspidal representations of PGL(2, AF), 7r1 # 7r2, with discrete series archimedean components 7rA = r k l , , 7r,” = 7rk2v, k l , > k2, > 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 quasipacket 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 (
+
<
i).
H
To express the Zeta function as a product of Lfunctions, recall that 1N
1N
where r is a representation of LH' = H' x Gal@,/Q,) ified. For general E,r = Ind(ro;WQ,W E ) . We now continue with IE = Q,p = ( p ) , Q = p .
and
TH,
is unram
THEOREM 3. The Zeta function is equal to the product over the and the ITH in { T H ) of
{TH}
if T H is stable ( o f type (l),(2) or (3)), or
times
IF
IS UNSTABLE AND TEMPERED) OF TYPE (4), OR
times 1 2
L,(s   dimSK,,
T H , &([7r2
1 )r '2
+r o
IF h IS (UNSTABLE AND NONTEMPERED) OF TYPE (5). hERE
and
(r
S)(tp(THp))
= @ ~ ( p r t(( ~n H u ) ) .
In the case of Shimura varieties associated with subgroups of GL(2), a similar statement is obtained in Langlands [L5]. In general, our result is predicted by Langlands [L351 and more precisely by Kottwitz [ K o ~ ] .
I. Preliminaries
222
4. The Shimura Variety Let G be a connected reductive group over the field Q of rational numbers. Suppose that there exists a homomorphism h : R@/&, + Gw of algebraic groups over the field IR of real numbers which satisfies the conditions (2.1.1.13) of Deligne [D5]. The G(IR)conjugacy class X, = Int(G(IR))(h) of h is isomorphic to G(R)/K,, where Km is the fixer of h in G(IR); it carries a natural structure of a Hermitian symmetric domain. Let K j be , AQf is the ring of adhles of an open compact subgroup of G ( A Q f )where Q without the real component, sufficiently small so that the set
( G ( A Q f ) / K . f= ) lG(Q)\G(AQ)/KmKf is a smooth complex variety (manifold). The group R@/wG, obtained from the multiplicative group G, on restricting scalars from the field C of complex numbers to IR is defined over IR. Its group (R@/RG,)(IR) of real points can be realized as { ( z , z ) z; E e x }in (R@/lwG,)(C) = C" x C x . The G(C)conjugacy class Int(G(C))ph of the homomorphism ph : G,,c + Gc, z H h ( z , I), is acted upon by the Galois group Gal(C/Q). The subgroup which fixes Int(G(C))(ph) has the form Gal(C/IE), where lE is a number field, named the reflex field. There is a smooth variety over E determined by the structure of its special points (see [D5]),named the canonical model S K of ~ the Shimura variety associated ~ displayed above. with (G,h, K j ) , whose set of complex points is S K (C) Let L be a number field, and let p be an absolutely irreducible finite dimensional representation of G on an Lvector space V,. Denote by p the natural projection G(AQ)/K,K~ + SK~(C). The sheaf V : U H x p P 1 U of Lvector spaces over SK,(@) is locally free of rank V,(L) S K f ( @ ) = G(Q)\[xm
P,G(Q)
dimLV,. For any finite place X of L the local system V @ L LA : U ~ E. V,(Lx) x p  l U defines a smooth LAsheaf VA on S K over P,G(Q)
+
The BailyBorelSatake compactification Sk, of S K ~ has a canonical model over IE as does S K ~The . Hecke convolution algebra IHH[K~,L of compactly supported biKfinvariant Lvalued functions on G(Aqf) is generated by the characteristic functions of the double cosets K f . g . K f in G ( A Q ~ )It. acts on the cohomology spaces HZ(SK~(C),V), the cohomology with compact supports H ~ ( S (C),V), K~ and on the intersection cohomology Lspaces I H i ( S k f(C), V). These modules are related
5. Decomposition of Cohomology
223
by maps: H ; + I H i + H i . The action is compatible with the isomorphism H : ( S K ~ ( C ) , V6) 9 LA ~ 21 H,!(SK~8~~ , V A )(same , for Ha and for I H i ( S ’ ) ) ,but the last Qtalecohomology spaces have in addition an action of the absolute Galois group Gal(a/lE), which commutes with the action of the Hecke algebra (X @E abbreviates X x s p e c Speca). ~
a
5 . Decomposition of Cohomology Of interest is the decomposition of the finite dimensional LAvector spaces I H i , H i and H i as I H K ~ , Lx~ Gal(a/E)modules. They vanish unless 0 5 i 5 2 dim S K ~Thus . (1) ~ The (finite) sum ranges over the inequivalent irreducible WKf, J , modules
:.LA.
K
The Hi(7rf,LA)are finite dimensional representations of Gal(a/IE) over L A . Similar decomposition holds for Ha and IHz(S’). In the case of I H , the Zucker conjecture [Zu], proved by Looijenga and SaperStern, asserts that the intersection cohomology of Skf is isomorphic to the L2cohomology of S K ~The . isomorphism commutes with the action of the Hecke algebra. The L2cohomology with coefficients in the sheaf c), a Vc : U H V,(C) x p’(U) of @vector spaces, H { 2 ) ( S ~ f ( C ) , V has PG(Q)
( “MatsushimaMurakami” ) decomposition (see BorelCasselman [BC]) in terms of discrete spectrum automorphic representations. Thus
Here 7r ranges over the equivalence classes of the discrete spectrum (irreducible) automorphic representations of G(AQ) in
and rn(7r) denotes the multiplicity of 7r in L;. Write 7r = 7 r f 69 7rCc as a product of irreducible representations 7 r f of G(AQf) and 7rCc of G(IR), according to AQ = AQfIW, and ’:r7 for the space of Kffixed vectors in
224
I. Preliminaries
7rf. Then 7ry is a finite dimensional complex space on which W K ~= W K ~ , L ~ Lacts @ irreducibly. The representation 7r is viewed as a (g,Km)module, where g denotes the Lie algebra of G(R), and
denotes the Liealgebra cohornology of nTT,twisted by the finite dimensional representation pc of G(R). Then the finite dimensional complex space HZ(g, K,; n I8 pc) vanishes unless the central character wT, and the infinitesimal character inf(n,) are equal to those wec, inf(pC) of the contragredient ,6c of pc; see BorelWallach [BW]. There are only finitely many equivalence classes of 7r in L i with fixed K central and infinitesimal character, and a nonzero Kffixed vector (7rf # 0 ) . The multiplicities rn(7r) are finite. Hence H ~ , ) ( S K (@),Vc) ~ is finite dimensional. The Zucker isomorphism (for a fixed embedding of LA in C ) of W K f , J ,8 L @ = IHIK,rnodules
then implies that the decomposition (1)ranges over the finite set of equivalence classes of irreducible 7r in L; with 7rf”’ # 0 and 7rm with central and infinitesimal characters equal to those of an irreducible HK, ,Lxmodule with spectrum x = 7rf 8 7r,, and
Pc. Further, K
I 8 p ~@~= 7rf
7rEiA
7rELA
of (1) is
for such a discrete
Moreover, each discrete spectrum 7r = 7rf 8 7rm such that the central and infinitesimal characters of 7rm coincide with those of ,6c (where p is an absolutely irreducible representation of G on a finite dimensional vector space over L ) has the property that for some open compact subgroup Kf c G(AQf) for which 7ry # {0}, there is an Lmodel 7rEL of It is also known that the cuspidal cohomology in H:, that is, its part which is indexed by the cuspidal 7 r , makes an orthogonal direct summand in H: 8~~@, and also in IH’ 8~~@ (and Hi I ~ CL) . ~When we study the 7rfisotypic component of H: @ J , @ ~ for the finite component 7rf of
7ry.
6. Galois Representations
225
a cuspidal representation T , we shall then be able to view it as such a component of I H a . Our aim is then to recall the classification of automorphic representations of PGSp(2) given in [F6], in particular list the possible T H = T H f @ T H in~ the cuspidal and discrete spectrum. This means listing the possible T H j , then the T H which ~ make THf C3 T H occur ~ in the cuspidal or discrete spectrum. Further we list the cohomological T H ~ those , for which Hi(lj,K ;T H 8~pc) is nonzero, and describe these spaces. In particular we can then compute the dimensioil of the contribution of T Hf to I H * . Then we describe the trace of Frp acting on the Galois representation H;(?rHf) attached to T H f in terms of the Satake parameters of T H ~ in , fact any sufficiently large power of Fr,. This determines uniquely the Galois representation H , * ( T H ~of) , Gal@/Q), and in particular its dimension. The displayed formula of “MatsushimaMurakami” type will be used to estimate the absolute values of the eigenvalues of the action of the Frobenius on H , * ( T H ~ ) .
6. Galois Representations The decomposition (1) for I H then defines a map ~f H I H i ( x f ) from the set of irreducible representations Tj of G ( A Q f )for which there exists of G(AQ)with central and infinitesimal an irreducible representation rTT, characters equal to those of f 5 such ~ that 7rm 8 7rf is discrete spectrum, to the set of finite dimensional representations of Gal(Q/IE). We wish to determine the representation I H i ( n f ) associated with r f , namely its restriction to the decomposition groups at almost all primes. However, the cohomology with which we work in this paper is H: and not I H i ( S ’ ) . Let p be a rational prime. Assume that G is unramified at p, thus it is quasisplit over Qp and splits over an unramified extension of Qp. Assume that K j is unramified at p , thus it is of the form K;Kp where Kf” is a compact open subgroup of G(A&) and K p = G(Zp). Then IE is unramified at p . Let p be a place of E lying over p and X a place of L such that p is a unit in LA. Let f = fPfK, be a function in the Hecke algebra W K ~ , L where , f P is a function on G ( A b f )and f ~ is, the quotient of the
I. Preliminaries
226
characteristic function of K p in G ( Q p )by the volume of K p . Denote by Fr, a geometric Frobenius element of the decomposition group Gal(Q,/E,). Choose models of SK, and of Skf over the ring of integers of IE. For almost all primes p of Q,for each prime p of E over p , the representation H ~ ( S K ~ V,) of Gal(a/E) is unramified at p, thus its restriction to Gal(Qp/E,) factorizes through the quotient Gal(Qgr/E,) !x Gal(F/F) which is (topologically) generated by Fr,; here Q;r is the maximal unramified extension of Qp in the algebraic closure Q p , F is the residue field of E, and IF an algebraic closure of F. Denote the cardinality of F by q p ; it is a power of p . As a Gal(F/F)module H ; ( S K , @ E ~ , V Ais )isomorphic to H:(SKf @EF,VA). Deligne’s conjecture proven by Zink [Zi] for surfaces, by Pink [PI and Shpiz [Sh] for varieties X (such as S K ~which ) have smooth compactification 7 which differs from X by a divisor with normal crossings, and unconditionally by Fujiwara [Fu], and recently Varshavsky [Val) implies that for each correspondence f P there exists an integer j o 2 0 such that for any j 2 j o the trace of f p . Fr; on
@~a)
2 dim SK
CB
i=O
(1y H:(sK, @EF,V,)
has contributions only from the variety S K , and not from any boundary component of SL,. The trace is the same in this case as if the scheme
S K @~E
IF were proper over F, and it is given by the usual formula of the
Lefschetz fixed point formula. This is the reason why we work with HZ in this paper, and not with I H i ( S ’ ) .
11. AUTOMORPHIC REPRESENTATIONS 1. Stabilization and the Test Function Kottwitz computed the trace of f P . F r i on this alternating sum (see [ K o ~ ] , and [ K o ~ ]chapter , 111, for p = 1) at least in the case considered here. The as a conjecture, is a certain sum result, stated in [ K o ~ ](3.1) ,
cc
4 7 0 ;7 7 6 ) . O(Y, f P )
70
. TO(6,4j). trP(Yo),
(r,@
rewritten in [ K o ~ ](4.2) , in the form
where 0 and TO are orbital and twisted orbital integrals and & is a spherical ( K p = G(Zp)biinvariant) function on G(Q,). Theorem 7.2 of [Ko4] expresses this as a sum
over a set of representatives for the isomorphism classes of the elliptic endoscopic triples ( H , s , q :~fi 3 G) for G. The STF‘,eg(f$S’P)indicates the (G, H)regular Qelliptic part of the stable trace formula for a function on H(AQ). The function denoted simply by h in [Ko4], is 7 assuming the “fundamental lemma” and constructed in [ K o ~ ]Section , “matching orbital integrals”, both known in our case by [F5] and [W]. Thus f$’” is the product of the functions :f on H ( A & ) which are
fg”
fisip,
fgp
: by matching of orbital integrals, on H ( Q p )which is obtained from f a spherical function obtained by the fundamental lemma from the spherical function $j, and fsfb, on H(R) which is constructed from pseudocoefficients of discrete series representations of H(R) which lift to discrete 227
II. Automorphic Representations
228
series representations of G ( R ) whose central and infinitesimal characters = fgfgpf2’ Kottwitz’s coincide with those of Pc. We denote by function h = hPhph,, so that functions on the adhle groups are denoted by f,and the notation does not conflict with that of h : R@/wG, + G. ; s ) is missing on the right side of [ K o ~ ](7.1). , Here The factor ( a P ( y oy),
fiS”
a p=
a,,
where
a,(yo;7,) E X * ( Z ( ~ O ) ~ ( ~ ) / Z ( ~ O )
V # P F
as defined in [ K o ~ ]p., 166, bottom paragraph We need to compare the elliptic regular part STFrg(fkS”) of the stable trace formula with the spectral side. To simplify matters we shall work only with a special class of test functions f P = @v+p,oofv for which the complicated parts of the trace formulae vanish. Thus we choose a place vo where G is quasisplit, and a maximal split torus A of G over Q,,, and require that the component f,, of fP be in the span of the functions on G(Qwo)which are biinvariant under an Iwahori subgroup I,, and supported on a double coset IwouIwo, where a E A(Q,,) has Ia(u)I # 1 for all roots a of A . The orbital integrals of such a function f,, vanish on the singular set, and the matching functions fHvo on H(Q,,) have the same property. This would permit us to deal only with regular conjugacy classes in the elliptic part of the stable trace formulae STFrg(fg’p),and would restrict no applicability. To avoid dealing with weighted orbital integrals and the continuous spectrum, we note that these vanish if two components of the test funcare discrete, by which we mean that they have orbital integrals tion which are zero on the regular nonelliptic set. The component f z H has this property. If G = RF/QG1 is obtained by restriction of scalars from a group G1 defined over a totally real field F , then G(Q) = G1(F) and G(R) = G l ( F @ R ) = G I @ ) ;the last product has [F : Q]factors. Correspondingly the function fkPwis a product of [ F : Q]discrete factors. This gives the equality of the elliptic regular part of the stable trace formula with the discrete spectral side when F # Q. If F = Q,and in general, we may take some of the components f w of f P to be discrete, for example pseudocoefficients of discrete series representations, to achieve this vanishing of the weighted terms in the trace formula. Such a choice of course will limit our results to only those automorphic representations with the specified (by the fw) elliptic components.
f2lp
n
2. Automorphic Representations of PGSp(2)
229
2. Automorphic Representations of PGSp(2) We then need to describe the stable trace formulae. This we can do only in the special case, studied in [F6].We then use the notations of [F6]from now on, and in particular the group denoted so far by G will be denoted from now on by H‘ = RF/QH, where F is a totally real number field and H is the projective group PGSp(2) of symplectic similitudes over F . When describing the automorphic representations of H’, note that H’(Q) = H ( F ) and H’(AQ) = H ( A ) ,where A denotes now the ring of adkles of F (and AQ of Q). It is more convenient to describe the automorphic representations of H / F . Working with PGSp(2) is the same as working with GSp(2) and functions transforming trivially under the center. A detailed description of the automorphic representations of H J F is given in [F6]. We recall here only the most essential facts. The group
H = PGSp(2) is the quotient of (we put J = GSp(2) = ((9, A) E GL(4) x G,;
( w O
“1 ,
w=
(; i))
tgJg = XJ}
by its center {(X,X2); A E S,}. It has a single proper elliptic endoscopic
group Co = PGL(2) x PGL(2) over F . The group H itself is one of the two elliptic endoscopic groups of G = PGL(4) with respect to the involution 8, B(g) = Jl tg’ J . The other &twisted elliptic endoscopic group of G is
The automorphic representations of H are described in [F6] in terms of liftings, defined by means of the natural embeddings of Lgroups. The groups GI HI C , Co are split. Hence their Lgroups ( L G l . . . ) are the direct product of the connected component of the identity (GI. . . ) with the Weil group. Let 9 be the involution on G defined by the formula which defines 8. Writing Ze(s8) for the group of g in G with ie(g)d’ = g, the Lgroup homomorphisms are A : I? = Sp(2, C) = ZG(8) L) G = SL(4, C),
230
II. Automorphic Representations
Here BO = diag(1, 1, 1, l),B = diag(l,1, 1, l),and
A@B=
d consists of the
(zi iz),where (A = (z :) , B ) ranges over GL(2, C)x GL(2, C)
with det A.det B = 1, modulo ( z , ~  ~z )E, C x . These homomorphisms of complex groups define liftings of unramified representations via the Satake transform. They are extended in [F6] to ramified representation by character relations involving packets and quasipackets (which are introduced in [F6]). The packets and quasipackets define a partition of the discrete spectrum of H ( A ) . To define a global (quasi) packet P = { T } , fix a local (quasi) packet P, = {T,} at every place 21 of F , such that P, = {n,} contains an unramified representation at almost all places. Then P = {.} consists of all products @IT, over all v, where T, E P, = {T,} for every v and T , = T,” for almost all v. Before we recall the definition of local packets, we state that the discrete spectrum of H(A) is the disjoint union of what we call the stable and unstable spectra. The lifting X defines a bijection from the set of packets and quasipackets of discrete spectrum representations in the stable spectrum to the set of self contragredient discrete spectrum (cuspidal or residual) representations of G(A) which are not in the image of X I . In particular, X maps one dimensional representations of H ( A ) to one dimensional representations of G(A), stable non onedimensional packets of H(A) to cuspidal self contragredient representations of G(A), and the quasipackets in the stable discrete spectrum of H ( A ) , each of which has the form {L(
.:
<
3. Local Packets
231
In particular, the image of X in the discrete spectrum selfcontragredient representations of PGL(4, A) is precisely the complement of the lifting XI from C(A). Similarly, the lifting XO defines a bijection to the set of packets and quasipackets in the unstable spectrum of H ( A ) from the set of unordered pairs {7r1 x 7 r 2 , 7rz x 7 r ' ; 7r' # 7 r 2 } of discrete spectrum automorphic representations of PGL(2,A). This last set is bijected by X with the set of automorphic (irreducible) representations I(7r1, 7 r 2 ) normalizedly induced from the representation T' @7r2 on the Levi subgroup of G(A) of type (2,2), where T ~ 7r2, are discrete spectrum on PGL(2,A) with r1 # 7 r 2 . In fact if 7r1 x 7r2 is cuspidal it is mapped by A0 to a packet, while if not, that is when 7r1 or 7r2 are onedimensional, Ao(7r1 x 7 r 2 ) is a quasipacket. To repeat, the global liftings are defined by the Lgroup homomorphisms for almost all components, which are unramified, and it is a theorem that the liftings extend to all places in terms of packets and quasipackets, and have the properties listed above. The stable part of the discrete spectrum, defined above by means of the bijection A, has the property that the multiplicity in the discrete spectrum of H ( A ) is stable, namely constant over each packet. Thus each member gU7rvof a packet { T } which Alifts to a discrete spectrum representation 7r 2: f r of PGL(4,A) occurs in the discrete spectrum of H ( A ) with multiplicity one. The same is true for the stable quasipackets, each of which is of the form { L ( ( v ,v'I27r2)}.
3. Local Packets The multiplicity is not constant on the unstable packets, but it is bounded by one. It is possible that a member in an unstable packet will not occur in the discrete spectrum of H(A). Then its multiplicity is zero. To specify the multiplicity, we need to describe the local packets. For this purpose we recall the main local theorem of [F6]. It has 4 parts. Let F be a local field. (1) For any unordered pair 7r1, 7 r z of irreducible square integrable representations of PGL(2, F ) there exists a unique pair T;, 7r; of tempered
II. Automorphic Representations
232
(square integrable if 7r1 # 7r2, cuspidal if 7r1 # tations of H ( F ) , 7r& is generic, 7r; is not, with
7r2
are cuspidal) represen
tr(n1 x .r2)(fc,) = trr&(fH) trrh(fH) trIG(7r1,7r2;f x 0) = t'7r&(fH)
+ tr7rJfH)
, , for a suitable choice of an for every triple of matching functions f,f ~fc, operator ~ ( d )7r, = I ~ ( 7 r ' , 7 r ~intertwining ), 7r with ' 7 r 1 and having order 2 . We define the packet of 7r; and of 7rg to be { 7 r $ , 7 r , } . The packet of any other irreducible representations of H ( F ) is defined to be a singleton. More details are known. If 7r1 = 7r2 is cuspidal, 7r& and 7r; are the two inequivalent constituents of the induced representation 1 >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 XIlift is a Alift 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 u1/27r2 of HI where 7r2 is a cuspidal (irreducible) representation of GL(2, F ) with central character # 1 = t2and En2 = 7r2, Alifts to the square integrable (resp. nontempered) constituent
<
s ( u ' /u~ ~~ /~~ ,T ~(resp. )
J ( u ' / ~ Tu~ ', / ~7.r
1)
of the induced representation I G ( v ~ /u ~ ~~ /~' T~ ~of , ) G = PGL(4, F ) .
We define a quasipacket only for the nontempered irreducible representations 7 r i l and L = L ( u J , J >a c r ~  ' / ~ )to , consist of { 7 r G 1 7 r ; } , and of { L ,X}, x = X( 0 and k l , k2 are odd, are discrete series representations of PGL(2,R), and 7rA is the generic 7rTrkz,7rk is the holomorphic 7 r ~ ~ ~ which are discrete series when kl > 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(u<,<x c r ~  ~ / are ~ ) , not cohomological, and will not concern us in this work.
<
<
234
II. Automorphic Representations
4. Multiplicities We are now ready to describe the multiplicities of the representations in the packets and quasipackets in the unstable spectrum of H ( A ) . Each member of a stable packet occurs in the discrete spectrum of PGSp(2,A) with multiplicity one. The multiplicity m ( 7 r ~of) a member T H = @ 7 r ~in ~an unstable [quasi] packet Xo(7r’ x 7r2) (7r1 # 7r2) is not (“stable”, namely) constant over the [quasi] packet. If 7r1 x 7r2 is cuspidal then
) the number of components 7rGw of T H ( n ( 7 r ~is) bounded Here ~ ( T H is by the number of places where both 7rt and 7r,” are square integrable). If m ( 7 r ~= ) 1 then T H is cuspidal. If 7r2 is a cuspidal representation of PGL(2,A) and a is a character of A x / F X A X 2the , multiplicity m ( 7 r ~of) T H = @ 7 r ~ , in a quasipacket Xo(7r2 x a 1 2 ) is
where ~ ( T H is ) the number of components 7 r i w of T H , and E = &(a7r2, f) is 1 or 1, being the value at f of the &factor occurring in the functional equation of the Lfunction L(an2,s)of an2. This E is 1 if and only if 7rff = @7rGw ( n ( 7 r ~ =)0) is discrete series. Finally we have ~ ( T H = ) f (l + (  l ) n ( K H ) ) for T H = @ 7 r in ~ ~X o ( a C l 2 x 0 1 2 ) with ~ ( T H components ) T H ~ = X,. Here T H = @L, ( n ( 7 r ~=) 0) is residual.
5. Spectral Side of the Stable Trace Formula We are now in a position to describe the spectral side of the stable trace formula for a test function f H = @ f H v with at least two discrete components, on H ( A ) . Thus STFH(fH) is the sum of five parts: I ( H , l ) ,. . . ,I(H,5
5. Spectral Side of the Stable Dace Formula
235
The first, I ( H , l),is the sum of three subterms: I ( H , l ) i , i = 1,2,3, each of which is a sum of products
21
where tr{7rHV} indicates the sum of tr 7 r over ~ all ~ T H in ~ a packet or quasipacket {7rHv}, over all packets and quasipackets in the stable spectrum. I ( H , 1)1 ranges over the packets { T H } which Alift to cuspidal self contragredient representations 7r of PGL(4, A) not in the image of A'. I ( H , 1 ) 2 ranges over the discrete series quasipackets {L(cv, (which Alift to the residual J(v1I27r2,v'l27r2),cuspidal 7r2 with quadratic central character E # 1 with E7r2 = 7r2). I ( H , 1)s is a sum over the one dimensional representations T H of H(A). The second part, I ( H ,2), of S T F H ( ~ His) ,the sum of
over all unordered pairs (7r' , 7r2) of distinct cuspidal representations of PGL(2,A). Here { 7 r ~ }is the Xolift of 7r1 x 7 r 2 , that is Ao(7r; x T,") = {7rL,,~,,} for all u,and 7rGw is zero if T: and T," are not both discrete series. The third part, I ( H , 3 ) , is the sum of
over all pairs (o,7r2),where 7r2 is a cuspidal representation of PGL(2,A) and 5 is a character of A X / F X A X 2For . each u the pair {7r;fvl 7rjjw} is the quasipacket A0(7r," x 0 ~ 1 2 ) when 7r," is discrete series, while it consists only of 7 r i v (and 7rGv is zero) when 7r," is not discrete series. The fourth part, I ( H , 4 ) , is the sum of
over all unordered pairs (atla)of characters of A x / F X A Xwith 2 For each u the pair { L H ~ , X His ~the } Alift of a w E w 1 2x 0 ~ 1 2 .
E # 1.
II. Automorphic Representations
236
The fifth part, I ( H , 5 ) , is the sum over all discrete spectrum representations 7rz of PGL(2,A) of the terms
At each place u where T," is properly induced (hence irreducible), the normalized intertwining operator R, is the scalar 1, and tr(1 >a 7r;)(fH,) = tr(.lr; x n;)(fco,) for a matching function fc,,on Co(F,). If T," is square integrable (or one dimensional), our local results assert that the two constituents of the composition series of 1XT," can be labeled 7r2, (or ~2,)and 7 r H , , such that for matching functions tr(7r," x ~,")(fc,,) is tr7rf;,(fHv) trn;,(fH,) (or tr7riw(fH,) trn;,(fHw)). Moreover, R, acts on 7r$, as 1 and on T;, as 1 when T,"is square integrable, and as 1 on both .lr$, and 7rEwwhen T," is one dimensional.
+
6. Proper Endoscopic Group The spectral side of the other trace formula which we need is for a function fc, = @fcowon Co(A) = PGL(2,A) x PGL(2,A). It comes multiplied by the coefficient Since PGL(2) has no proper elliptic endoscopic groups, this trace formula is already stable. Thus STFc,(fo) = TFco(fo). It is a sum of three sums, I(C0,i ) ,i = 1,2,3. The first, I(C0, l),is a sum of
b.
over all ordered pairs (d, r 2 )of cuspidal representations of PGL(2,A). The second part, I(C0,2), is a sum of
over all pairs ( a ,n 2 ) ,where 7r2 is a cuspidal representation of PGL(2, A) and a is a character of A x / F X A x 2 The . third, I(Co,S),is the sum over all ordered pairs (a,
IJ tr(ow<w
12
2)
x
0 ,
12)(fc0w).
6. Proper Endoscopic Group
237
At all places w # p , oc) the component fcOwis matching f H v , so the local factor indexed by u in each of the 3 cases can be replaced by
111. LOCAL TERMS 1. Representations of the Dual Group Part of the data which is used to define the Shimura variety is the G(@)conjugacy class Int(G(@))(ph)of the homomorphism PI, : G, + G over CC. Let C k denote the set of conjugacy classes of homomorphisms p : G, + G over a field k. The embedding Q + C induces an Aut(C/Q)equivariant map CQ + Cc. This map is bijective. Indeed, choose a maximal torus T of G defined over Then Hom G m , T ) / W + Ca is a bijection, where Q L W is the Weyl group of T in G(Q). Similarly Homc(G,,T)/W + Cc is a bijection. Since Homg(G,,T) + Homc(G,,T) is an Aut(@/Q)equivariant bijection, so is Ca + Cc. The conjugacy class of ph over C is then a point in HorndG,, T)/W. The subgroup of Gal(a/Q) which fixes it has the form Gal(Q/IE), where IE is a number field, named the reflex field. It is contained in any field El over which G splits, since can be chosen to split over El. In our case (G is) H’ = RF/QH,where H is PGSp(2) over a totally real field F . Thus H’ is split over Q,and E = Q. Note that HI(()) = H ( F ) and H’(R) = H(R) x ... x H(R) ( [ F : Q]times). The dimension of where 3 is half the real the corresponding Shimura variety is 3 [ F : Q], dimension of the symmetric space H ( R ) / K H p ) . Let (rE,V,) be the representation of LHk = H >a WE determined by Int(H‘(CC))ph (see [L5] and section 1). It is determined by two properties. (1) The restriction of rE to ?I is irreducible with extreme weight  p . Here p = ph E X * ( T ) = X , ( T ) is a character of a maximal torus T of I?’, uniquely determined up to the action of the Weyl group. (2) Let y be a splitting ( [ K o ~ ]Section , 1) of I?’. Assume that y is fixed by the Weil group WE of IE. Then WEc LHk acts trivially on the highest weight space of V, corresponding to y. Put r = r, for the representation induced from r: on H‘ x WE to H I >a w,. We proceed to specify this representation explicitly in our case, as the twisted tensor 4[F:Q1dimensionalrepresentation of Sp(2, C)[F:Ql M WE, lE =
a.
238
1. Representations of the Dual Group
Q. In particular, r = To. Consider first H = PGSp(2)/Q. Take h : FtqwG,
+ bi) = (
nr>
+ Hnp
239
to be defined
a l bI bI
, I = 1 2 . Over C,the homomorphism h can be diagonalized to ( z ,w) H diag(z1, wl).We claim that the representation T of H = Sp(2,C) is its natural embedding in GL(4,C). Let 5"; be the diagonal torus in H , and TH the diagonal torus in H . Then X*(?H) = { ( a ,b, b, a); a , b E Z}and
by h(a
X*(?H) = ((2, y, z , t )mod(n, m, m, n ) ;5 , y, z , t E Z}. Here (z,y, z , t ) takes diag(a, b, bl, a') in phism u : X*(?H) s X * ( T & )
+
TH to
+
+
aZ'bY'.
The isomor
+
is given by u : ( x , y , z , t ) H (x y , x z,y t , z t ) , with inverse ul : ( a , , / 3 , y 1 6H ) ( a  y , a  P , O , O ) . Now X,(T&) is spanned by the cocharacters a. = (O,O, 1 , l ): x ++ diag(1, 1,x,z), a1 = ( L O , 0,1) : x a2 =
H diag(x, 1,1,x'),
(0,1, 1,O) : z H diag(l,x,zl,1).
An extremal weight of r is (YO, viewed as a character of T H , thus u1(ao)= (1, o,o, 0). The orbit under the Weyl group W = ((23),(12)(34)) of 00 is
+ +
(23)(14) a0 = cyo a1 a2 = (1,1,0,0). Their images under ul are (1, O,O, 0), (0, 1,O, 0) (equivalently (O,O,O,l), (0,0,1,0)), (O,l,O,O), and (~,0,0,0). The representation r with these weights is the natural embedding r : H = Sp(2,C ) + GL(4, C). The unramified representation x ~ ( p 1p2) , of H = PGSp(2, Q,)contained in the composition series of the representation normalizedly induced of the upper trifrom the character n . diag(cy,P, y,6) H pl(a/y)p2(a/P) angular subgroup is parametrized by the conjugacy class of t x Fr, in
III. Local Terms
240
L H = H x W(Qir/Q,). Here F'r, is the F'robenius element, which generates the unramified Weil group W(Q;./Q,). Further t = t(rH(p1,pZ)) = diag(p1, p2, pT1 , p;') in I? = Sp(2, C)(where we write here pi for pi(n)). The matrix T(t(rH(p11 p2))) has the eigenvalues p1, p2, pT1 , pT1 I the values of the weights (1,0,0,0), (O,l,O,O), . . . at t = diag(pllp2,p;1,p;1). Let now F be a totally real number field, H = PGSp(2) over F , n R. Then the and H' = RF/QH. Fix an embedding L : F set S of archimedean places of F can be identified with the coset space G a l ( a / Q ) / G a l ( o / F ) by T t+ L,, where L, : F 0 is z t+ T L ( ~ ) . Then HI(()) = H ( F ) and H ' ( R ) = H ( R ) . The connected dual group H' is Z? = Sp(2,C), and the Lgroup is the semidirect product H' = H' x WQwhere the Weil group WQacts by translation of the factors via its projection to Gal(F/Q). The homomorphism h : Rc/RG, 4 HL is TAKEN TO BE

0
nSH,
ns
( [ F : Q]copies on the right). Up to conjugacy by the Weyl group, the weight p : + ex, where ?Ht is the diagonal torus in H' (product of the IS(= [F : Q]diagonal tori in H ) , has the form
The group Gal(o/Q) stabilizes p, thus the reflex field E is Q. Let r1 be the natural embedding of H = Sp(2,C) in GL(4,C). Then the representation r = r p of LHL = H f >a 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 Qgroup 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+$, 0 5 j o < nu, j = jo(modn,). A vector is fixed by iff
F’ri
it is equal t o BieiTz3) = @,eij0 e ( i ) , thus l ( i )depends only on i m o d j (and imodn,), namely only on imodj,, where j , = (j,n,). Then (tUx
Fr,p
=
(. . . ,
n O
This is
t,,i+k,..
.) x
w$ .
2. Local Terms at p
243
It acts on vectors of the form
The product of the first j , vectors is repeated n,/j, times. On the vectors with superscript 1 the class (t, x Frp)ju acts as
t u , i = tu = diag(plu,
=
~21211,CLI;~),
1
+
+
and so (t, x F r p ) j acts as g/ju.The trace is then p { p u p i p u pi:/irr + p;:'ju. The same holds for each superscript, so we get the product of j , such factors. P ut j , = (j,n,). We then have
The spherical function f& is defined by means of Lgroup homomorphisms LCA + LH' + LH;l where Hi = ~ J Q ~ and H ' Q, denotes the of degree j . Since the groups C A and H' unramified extension of Q p in are products of groups C;, = RF,/Q~CO and HL = RF,/Q,H, it suffices to work with these groups. Thus H; = HL,, where HL, = RQJQ,HL.
op
The function Now
fgopwill be @f&,
nulp
for analogously defined f&.
and Fr, acts on
BY
It suffices to work with LHLj = (fiL)j M (Frp).
III. Local Terms
244
Let s l , .. . , s j be Fr,fixed elements in Z(Ch,), thus si = (si,.. . , si), si E Z(C0) = ( f I 2 ) x ( f I 2 2 ) repeated n, times, with s 1 . .. sj = s = (s,. . . ,s), s = diag(1, 1,  l , l ) . Define
LCA, = C .:
:
fjj
x (R,) + LHLj = (l?L)j x (F'r,)
by
t
t+
",
( t , .. . , t ) ,
. . ,Sj) x Fr,,
H (s1,sz,.
thus Frf
H (SIS2.
. . s 2 , s z . . . S i + l , . . . ,sjs1...sa1)
The diagonal map HL + HLj defines LHLj tl . . . tj x Fkf. The composition qj : LCh,
+
$
tx
The homomorphism
fjj
"f
H
tjsi x
x
Fr;.
' H L , ( t l , .. . , tj) x Fr$ H LHL gives
"6.
defines a dual homomorphism
fz,
of Hecke algebras. The function is defined to be the image by the relation tr T H u ( f j j ( 41(4jU) = tr 7r'Cou ( t )(f&) of the function q5j of [ K o ~ ]p. , 173, or rather the ucomponent q5,j of 4j, which is the characteristic function of K,j . p~~( p  l ) . K,j. Theorem 2.1.3 of [Ko3] (see also [ K o ~ ]p. , 193) asserts that the product over ulp in F of these traces is the product of p 4
f
with the product over ulp of
Similarly for s = I we have that the analogous factor (with Co replaced by H ) is the product with factors
3. The Eigenvalues at p
245
3. The Eigenvalues at p We proceed to describe the eigenvalues pl,, p2, for the various terms in the formula, beginning with STFH( f H ) , according to the five parts which make it. Note that X(nH(plu, p2u)) = z G ( p l u , p2zl, p;:, p,,'), where G = PGL(4,Fu). We choose the complex numbers p, to have IpmuI 2 1 (otherwise we replace as we may pmu by p;:; m = 1 , 2 ) . Write t, for diag(p1,, p2u, p;:, p;;). The first part of S T F H ( ~ Hdescribes ) the stable spectrum. It has 3 types of terms. (1) For the packets { T H } which Alift to cuspidal T G j i ($ ~ ImXl), t , is diag(p1,,
p2u, pi:,
p;:)
with
112 Qu
< IPmuI < 4:12,
since TG is unitary and so its component TG, is unitarizable. Note that the unramified component X G , is generic (since T G is), hence fully induced. (2) For the quasipackets { L ( ( v ,vp1l27r2)}, they Alift to the residual J(v1127r2, v1/27r2), n2 cuspidal with central character ( # 1 = t2 satisfying [ r 2 = n2. The component 7r: of n2 at u is unramified of the form ~ ' ( z l , , ~ 2 , ) . This is an unramified generic representation of GL(2, F,), hence fully induced, normalizedly from the character
(:
z)
H
val(a) val(c)
zlu z 2 , . We have that z1,z2, = (,(7rU) has square 1. If (, # 1 then {z~,,z2,} = {1,1}. If E, = 1, since n2 is unitary its component 7r: is unitarizable, and so qL1l2< Iz,l < q:I2. In both cases we have
t, = diag(q:/2zl,,
4:/2z2u, q,112 22, 1
, q;1/2z;,).
Better estimates are known for the (zmul (the exponent 1/2 can be reduced to 1 / 4 by the theory of the symmetric square lifting), but for our T' we shall show below that lzmul = 1. (3) For one dimensional representations n ~X (,7 r ~ )= T G is one dimenX(det g), where x is a character of order 2, and sional representation g t ( w u ) = di&lu, P Z U , P;;, P;;) is

diadxuq:'2,
xud12
7
xu4,1/2, xu4, 3 1 2 ) ,
III. Local Terms
246
where xu.= x(T,) has square 1. Since ~ F His a quadratic character we have that plu = &&I2,p2, = *q;I2. The second part of S T F H ( ~ His) a sum of terms indexed by { T H } = Ao(7r' x r 2 ) .Here d, 7r2 are cuspidal representations of PGL(2,A), and t r .ir;,(fHu) = 0 as fH, is spherical. Then the component of 7rm ( m = 1,2) at u is the unramified generic thus fully induced n," = 7r2(zm,, z;:), and t, = diag(zl,, z2,, z;:, z,,'), where lz,,Ifl 5 q,112 . The terms in the third part of S T F H ( ~ Hcorrespond ) to Ao(,/r2 x a l z ) , where 7r2 is a cuspidal representation of PGL(2, A) and u is a character of A x / F x A x 2 .The factors at ulp of 7r2 are 7r:(zu,z;'), q i 1 / 2 < lzil < q:l2. So t, = diag(a,qA/2, z,, z l l , auqZL1l2),where a, = D,(T,) has square 1. The terms in the fourth part correspond to A o ( a t l 2 x alz), where o,6 are characters of A x / F X A X 2with # 1. Pu t o, = o,(n,). Then
<
autuqzL1/2), tu=
t, = diag(o,t,q$I2, auq$12,
tU.(~,).
The fifth part consists of terms indexed by T H = 1 >a n2 where r 2 is a cuspidal representation of PGL(2,A). At u the factor 7r: = T ~ ( , Z , , , Z ; ~ ) is fully induced with Jz,Jfl < &I2 and A ( 1 >a 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 4tuples 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 Lgroup 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
cf
(
( ~ 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 $kl,kz : WR + LG = G x WR, G = sL(4,C), given
<
by
z
H
diag((z/lzl)kl,( z / l ~ l ) k z(I4/4", ,
(14/4k1) x z
and (T
H
( ") x
0
w
(: a)
w
0
(odd kl > k2 > 0)
(T
or x
0
(even I C ~> 1c2 > o),
factorizes via (LCo +) L H = Sp(2,C) x WR precisely when ki are odd. When the ki are even it factorizes via LC = SO(4,C) x WR. When the ki are odd it parametrizes a packet {7rz:k2, of discrete series representations of H(R). Here 7rwh is generic and 7rho1 is holomorphic and antiholomorphic. Their restrictions to H o are reducible, consisting of 7rpt and n$' 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 , sgnk1 vk/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 , klkz ) = ( 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 Quasipackets 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,wlll,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,rl)) 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
IV. Real Representations
252
of PGL(4, R) is parametrized by
$ : WRx SL(2, C)4 SL(4, C),
thus
& ( z ) = d i a g ( r 3 , r , r  l , r w 3 )x z ,
&,(a) = [ (  1 ) I 4
x
CT.
This parameter factorizes via $ : W, x SL(2, C)+ Sp(2, C), which parametrizes the one dimensional representation [ H of H ( R ) , h ++ [ ( X ( h ) ) where X(h) denotes the factor of similitude of h, whose infinitesimal character is ( 2 , l ) = Ca,oa. We have
i(1,
for ( i , j ) = (O,O), (1,1),(2,2), (3,3). Of course 1~ # sgn,, and + sgn,) is the characteristic function of H o in H(R). Moreover, the charsgnH) 7rT: 7r;: vanishes on the regular elliptic set of acter of H ( R ) ,as (EH 7rTF 7r;;’) IHo is a linear combination of properly induced (“standard”) representations ([Vo], [Ln]) in the Grothendieck group.
i(1, +
+
+
+
+
4. The Cohomological L(V sgn, v
 ~ /
The nontempered nonendoscopic representation L ( v sgn, v1/27r2k) of the group H(R) (k 2 1) is the Langlands quotient of the representation v sgn >avl/’7rZk induced from the Heisenberg parabolic subgroup of H . It Alifts to J(V’/27r2k, V1/27r2k), 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
($1
SL(2,C))
The Cohomological L(u sgn, V1/27T2k)
253
(E 1) = (E; ii), defines
which factorizes via
fi = Sp(2, C)
L$
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~r1)
c,
thus $+ defines a representation of C(R) (which X1lifts to the representation J(v1'27T2k+1, v1/27T2k+1)
of PGL(4,W)), but not a representation of H ( R ) . As in [Ty] write T:k,o for L(SgnV,v1/27T2k+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 >aY1/27r2k, note that it is a constituent of the induced u sgn >av1/21(uk,sgnvk) 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 ) .
+
IV. Real Representations
254
5. The Cohomological L
( ~ u ~ /, <~~  T 1 / 2~) ~ + ~
The nontempered endoscopic representation L ( < Y 1 / 2 7 r 2 k + l , J v  ’ / ’ ) o f the group H ( R ) is the Langlands quotient of the representation < V 1 / 2 7 r 2 k + l >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 Alifts 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 <7r2>’ = 7rZiE and 7r2,EIHois irreducible. In the Grothendieck group the induced decomposes as
Here
7rz:l,l
is generic, discrete series if k 2 1, tempered if k = 0. Our constituent of the induced
>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 ) k1). 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>
(3) Stable nontempered case: (a,b) = ( 2 k , O ) ( k 2 1) and x is hol x2:3,2k+l
7
x2k+3,2k+l,
7rh,0
= L(’
Sgn, v  1 / 2 x 2 k + 2 ) .
+
(4) Tempered case: any other ( a ,b ) , thus a > 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 Qrational 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
IV. Real Representations
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 7rz:,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$, Va,b), even a,  b,, for all v E S . Suppose 7 r occurs ~ in the stable spectrum, namely in I ( H , 1)l. The choice of the function f m H guarantees that the components T H , lie in the packet 7 r ~ ~ ~ , k z , k} lr , = a, b, 3 , ka, = a,  b, 1, at each archimedean place v E S. We start by fixing a cuspidal representation T H with TH, in the set {7rz:,kz, } for all w in S and with 7KrHi# 0. In particular the component a t p of such T H is unramified, of the form @ u l p ~ ~ ( p pl ,~, , ) . We use a correspondence which is a KfPbiinvariant function on H(APf).Since there are only finitely many discrete series representations of H ( A ) with a given infinitesimal character (determined by p ) and a nonzero K Kffixed vector, we can choose to be a projection onto {T,;}. Writing
{~z;,~~~,
7
+ +
+
4p:>,,,
fg,
fg
K
t,,
for p m U ( r U )m , = 1, 2, the trace of the action of Fri on the { r H f f isotypic component of H , ' ( S K ~@IFF, V p )(which vanishes outside the middle dimension 3[F : is multiplication by (we put j, = (j,n u ) )
a]),
Note that H,$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$F:Q1(namely the 7rZisotypic part for each member of the packet) is of the form { r z }@ p ( { r ~ } ) .Here 258
1. Tempered Case
259
p( ( 7 r ~ ) ) is a 4[F:Qldimensionalrepresentation of Gal(a/Q). The 4#{”IP) nonzero eigenvalues t of the action of Fr, are p$[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 L2cohomology,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 L2cohomology 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 IHversion due to Gabber) asserts that the eigenvalues t of the Frobenius Fr, acting on the tadic 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 “mixedpurity” (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
V. Galois Representations
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 ,v1/27r2)} which Alifts to the residual representation J(v1l27r2,v1/27r2) 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$isotypic part of the cohomology, Fr, acts with the 4#{uIP) eigenvalues p i where
dim sK
nu,, a,,
Note that T H , = L(sgnv,, ~ ~ ~ / ~ 7 r ~ has k , +H 2i 3) ( ~ ~ , € 3 V a , , b ,# ) 0 only when a, = 2k,, b, = 0, and ( i , j ) = (2,0), (0,2), (3, l ) ,(1,3).
2. Nontempered Case
263
Next we deal with the case of T H which occurs in I ( H , 3 ) , namely in a quasipacket Xo(7r2 x 5 1 ~ where ) ~ 7r2 is a cuspidal representation of PGL(2,A) whose real components are r 2 k v + 3 , and E is a character of A x / F X A x 2 . There are corresponding contributions in I(C0,2) from 7r2 x 5 1 2 and from E l 2 x 7r2. The infinitesimal character of T H , is ( k , , k , ) , k , 2 0, for all 21 E 5'. We fix a correspondence f$ 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 quasipacket {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
V. Galois Representations
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, I3iacts 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 7rHfisotypic 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
Note that the functions f ~ != ,C3JWEshHw ~ satisfy
We conclude that the representation P ( T H ~ )of Gal(a/Q) on H,* attached to ?rHfis 4[F'Q]dimensional. Its restriction to Gal(a,/Q,) is unramified. Its trace is equal to the trace of @,~pV,1/2r,(R,).Here r,(R,) acts on the twisted tensor representation (T,, (C4)[Fu:Qp1) as t(E,) x Fr,, t(E,) = ( t l ,. . . , t,,), t , diagonal with
I. O N AUTOMORPHIC FORMS 1. Class Field Theory Underlying the discipline of Automorphic Representations is a hypothetical reciprocity law that would generalize to the context of connected reductive groups G over local or global fields F the deep Class Field Theory, which simply asserts that W,$b 2i C F , and is to be viewed as the special case of GL(1) = G,. We review some of the key notions here, starting with basics. Key topics are in bold letters, and new terms are in italics. Number Theory concerns number fields F , finite extensions of the field Q of rational numbers. The completion of F at each of its valuations, Y,is denoted by F,. It is the field C of complex numbers or the field R of real numbers if v is archimedean (Iz g/J,I )z), Iylv), or a finite extension of Qp for a prime p if Y is nonarchimedean (Iz yl, 5 max(lzl,, Iyl,)). There is a positive characteristic analogue, where F is the function field of a curve C over a finite field IF,, the places v are the closed points of C and F, is IFqT( ( t ) )the , field of power series over a finite extension of IF,. In the nonarchimedean case denote by R, the ring of integers of F, (defined by 1x1, I 1). The ring of Fadhles, denoted AF or simply A, is the union over all finite sets S of valuations of F containing the archimedean ones, of the products F, x R,. Thus an adkle is a tuple (z,), z, E F, for all Y and x, E R, for almost all Y (finite number of exceptions). The field F embeds diagonally (xu = z for all Y ) in A as a discrete subgroup, and Amod F is compact. By AF,f , or Af,we denote the ring of adkles without archimedean components. Thus A = A, F, where 03 is the set of archimedean places of F . The multiplicative group of A is the group of idkles, A X , consisting of (z,) with x, E F," for all Y, z, E R,X for almost all w, where the multiplicative group R,X of R, is the group of units, defined by IzI, = 1. Thus A X = US F," x R,X . The multiplicative group F Xembeds diagonally as a discrete subgroup in A X , and A ' / F X is compact, where
+
+
+
nwESnWes
nvEm
nvES n,,,
269
2 70
I. On Automorphic Forms
A' consists of the (z,) in A X with flu (z,(, = 1. The product formula flulzlu = 1for z in FXimplies that F X c A'. Denote by ?T, a generator of the maximal ideal R, R,X of the local ring R,, when v is nonarchimedean. The field R,/(?T,)is finite, of cardinality qv and residual characteristic p,. The quotient space A X / F Xis called the idkle class group and is denoted by C F . When F is a local field put CF = F X . For more on valuations, adble, idkles, see, e.g., PlatonovRapinchuk [PR]. Class Field Theory can be stated as providing a bijection between the set of characters x of finite order of the profinite Galois group Gal(F/F) of F (7 denotes a separable algebraic closure of F ) , and the set of characters T of finite order of C F . When F is global, the bijection is defined as follows. The decomposition group D, of w,which consists of the g in Gal(F/F) which fix an extension V of w to F, is isomorphic to Gal(F,/F,). (In fact D, depends on v. Replacing 3 by ;il' leads to a subgroup D;! conjugate to D;. Thus D, is determined by v only up to conjugacy). Its inertia subgroup I , consists of the g E D, which induce the identity on Rv modulo its maximal ideal. The quotient group D v / I , is Gal(~,,,/F,,). Any element of D, which maps to the generator z H zq' of the Galois group of F,, is called a Robenius at w, denoted Fr,. Now x is unramified at almost all v, which means that its restriction to D, is trivial on I,. It is then determined by its value x(Fr,) at Fr,. Chebotarev's density theorem asserts that x is uniquely determined by x(Fr,) at almost all v. On the other hand, the character 7r of A X is the product @,7r,, where 7rv is the restriction of 7r to F," (F," is embedded in A X as (zw), z, = 1 if w # v). Since 7r is continuous, almost all components 7rv are unramified, namely trivial on R,". Thus they are determined by their value T , ( A , ) at the generator A, of the maximal ideal R,  R," in the local ring R,. By the Chinese Remainder Theorem F . F," is dense in Ax.Hence the character 7r of A x / F X is uniquely determined by 7 r , ( ~ , ) for almost all v. The bijection of global Class Field Theory is x H 7r if x(F'r,) = 7r(n,) for almost all v. The bijection of local class field theory can be derived from this on embedding a local situation in a global one, thus starting from xu or 7 r , one can construct global x and 7r with components xu and 7rv at v, when xu or 7rv are ramified.
rives
1. Class Field Theory
271
In fact, CFT provides a homomorphism CF + Gal(F/F)ab, named the reciprocity law, where the maximal abelian quotient Gal(F/F)ab of Gal(F/F) is the inverse limit of Gal(E/F) over all abelian extensions E of F in F (if G is a topological group, Gab is its quotient by the closure G" of its commutator subgroup). However, in this form the statement is unsatisfactory, as it applies only to characters of finite order, and indeed these are all the continuous characters of the compact, profinite group Gal(F/F). However A X / F Xis not lzvlw.To compact, and has characters of infinite order, e.g. z H IIzlI = extend CFT to characters of CF of any order, Weil introduced the group WF that we describe next, following Deligne [D2] and Tate [Tt].
n,
To introduce Weil groups, note that a Weil datum for F/F, F local or global and F a separable algebraic closure, is a triple (WF, p, {YE}). Here WF is a topological group and p : WF + Gal(F/F) is a continuous homomorphism with dense image; E ranges over all finite extensions of F in 7. Put WE = cpl(Gal(F/E)). It is open in WF for each E since p is continuous and {Gal(F/E)}E makes a basis of the topology of Gal(F/F). As Im cp is dense in Gal(F/F), cp induces a bijection of homogeneous spaces
for each El and a group isomorphism WFIWE 1 Gal(E/F) when EIF is Galois. The r E : CE WEb are isomorphisms. A Weil datum is called a Weal group if 'E
( W l ) For each E, the composition CE 2 W E b ~ G a l ( F / E ) a b is the reciprocity law homomorphism of CFT. (W2)For
each w E WF and any E , commutative is the square
I
Int(w)
I. On Automorphic Forms
272
(W3) If E'
c E , commutative is the square
The transfer map on the right is defined as follows. Suppose H is a closed subgroup of finite index in a topological group G, s : H \ G t G a section. For any g E G, x E H \ G , define h,,z E H by s(x)g = h,,,s(xg), and tr(gG") = h,,,(modHC). Then tr : Gab + Hab is a homomorphism. (W4) Put W E / , for W F / W ~The . natural map W F + lim WE,, is an
nzEH,G
t
isomorphism of topological groups. It follows that if ( W F ,cp, { r E } ) is a Weil group for F/F and E is a finite extension of F in F , then WE^ PIWE,{ r E l } E 1 3 E ) is a Weil group for F I E . We usually abbreviate the triple to W F . Note that via T E , the norm N E / E 1 : CE + C E (for ~ F c El C E ) becomes the map Wzb + W,$ induced by the inclusion W E c WE^. Note also the exactness of
whenever E / F is a Galois extension. When F is local archimedean, if F = C we take W F = C",p : C x + {l},r F = id. If F = R we take WR to be the subgroup C" u j C x of W x where W is the Hamiltonian quaternions. It is ( z , j ;z E C x , j 2= 1,jz = Z j ) where z is the complex conjugate of z . Then cp : WR + Gal(C/R) takes C x t o 1 and j C x to the nontrivial element in Gal(C/R). Further rc = 1 and r R : IRx + W i b is x H &Wi if x > 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 localtoglobal 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 ndimensional 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 $~='=,X, the representation Ip(7r1,. . . , 7 r T ) normalizedly induced from the . .@T, . of the parabolic P (trivial on its unipocuspidal representation ~ 1 8 tent radical) of type (dim XI,. . . , dim A,.), where A, H T,. We postpone the explanation of the new terms, but note that this new correspondence is defined similarly to the case of CFT, which is that of n = 1. The local analogue has recently been proven (by HarrisTaylor [HT], Henniart [He]) in the nonarchimedean case, and by Lafforgue [Lf] in the function field case. Once the connection between ndimensional representations of W F and admissible (locally) or automorphic (globally) representations is accepted, one would like to include all admissible and automorphic representations. For that the group WF has to be replaced by a bigger group, which is the WeilDeligne group WF x SU(2,R), an extension of W F by a compact group (see [D2], [Tt], KazhdanLusztig [KL], and Kottwitz [Ko2], 312) when F is nonarchimedean (when F is archimedean the group remains W F ) . This is necessary for inclusion of the square integrable but noncuspidal representations of GL(n, F) in the reciprocity law. The representations X of Wp x SU(2,R) of interest are analytic in the second variable, thus extend to SL(2,C). We embed Wp in WF x SL(2,C) by w H w x diag()w)1/2, )wJ~/~ where ), 1.1 : W p 4 F X + Cx is the composition of the usual absolute value with W$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 ndimensional 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 Fgroup 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
I. On Automorphic Forms
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 Lgroup 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 1parameter 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 Ftorus (up to isomorphism), since the Fisomorphism 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 Fsplit torus. Let X , X v be free Zmodules of finite rank, dual in a Zvalued 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 4tuple ( 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 4tuple 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 Fgroup 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 gxgl) 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 Fisomorphism 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 Fisomorphism 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.
I. On Automorphic Forms
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 = elez, 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] = zysly') 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 Lgroup, 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 Lgroup LG = >a WF depends on the choice of C, but a different choice gives rise to an isomorphic Lgroup. If v is a place of the number field F there is a natural WFconjugacy 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 Lparameters. 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 1cocycle 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) 1cocycle 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 Fpoints of the connected reductive Fgroup 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 (BernsteinZelevinsky [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 Ufixed 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 Mmodule. 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 Gmodule by ( 7 r ( g ) f ) ( h )= f(hg). Another useful construction is that of the module ITN of Ncoinvariants of an admissible Gmodule 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 Mmodule, with action ‘ T N . Put I T N = b i 1 / 2 t j 7 N . The functor IT ++ T N of Ncoinvariants is exact and leftadjoint 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 Fparabolic 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. HarishChandra 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 “Lparameters”. 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 HarrisTaylor [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 LabesseLanglands [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 3variables ([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]).
I. On Automorphic Forms
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 Fgroup G1 generated by G and a central torus, whose center is a torus. The normal homomorphism G cf 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 zextension 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 +
<
<
e,
Gsc
5 G
II
1 1+ D
+
H
II
1
D
H/Gsc
 G 4 1
has as dual the commutative diagram A
1 4 (2
/I
e

(H/GSC)" 5
1
h
H
1

6 II
5
4
1
A
5
(GSc)" A
As (Gsc)" = Gad, Z((2) = kerG. A diagram chase implies that Z ( G ) = kerG. Hence there is a map
H 1 ( L ~Z ,( e ) )+ ker[H1(LF,( H / G S C ) " ) H 1 ( L ~5)] , f
Thus given C E H 1 ( L F Z , ( e ) )there is a character a~ of ( H / G S C ) ( Fwhich ) is trivial on D ( F ) , hence a character
4. shown that
& is independent
wcx = J p x ,
Unrumijied Case
285
of the choice of D. We have
(' E H 1 ( L ~Z ,( e ) ) ,
X E R(G/F).
Further conditions on X H IIx, R ( G / F ) + I I ( G / F ) ,are: (1) The central character w, of 7r E II, is wx. (2) If A' = (A [A', A' E R ( G / F ) , (' E H ' ( L F , Z ( ~ ) ) ] then , 7r
IIx!
=
{Cp;
E IIA}.
Note that (Ec . T ) ( 9 ) = Er(9)7r(g).
4. Unramified Case Local functoriality for tori leads to functoriality for unramified representations. This is necessary for the global theory, as each irreducible admissible representation 7r of G(A) decomposes as the restricted product @xu of representations 7rw of G(F,,) over all places w of F , where 7ru is unramified for almost all 21. Thus assume that F is local padic with residual field F,. Suppose G is (connected reductive) unrumijied over F , namely G is quasisplit over F and split over an unramified extension of F . Then the inertia subgroup IF of WF acts trivially on 6, hence x (Fr) is defined. An Lparameter X is called unrumzjied if it reduces to (Fr) 4 6 x (Fr). It is determined by X(Fr) = t x Fr where t is semisimple in 6. The set F ' ( G / F ) of equivalence classes of unramified Lparameters is the set of Gconjugacy classes in LG of elements t N Fr, where t is semisimple. This set is naturally bijected with the set IIur(G/F) of equivalence classes of unrumijied representations 7r of G(F ) (namely the irreducible admissible representations ( T , V )of G ( F )which have a nonzero Kfixed vector, where K is a fixed hyperspecial ([Ti]) maximal compact subgroup K of G ( F ) . Note that all such K are conjugate under G a d ( F ) ) . Let us explain the isomorphism h U r ( G / F )= IIur(G/F) when G is an Ftorus T. There is an isomorphism
e
where T(R) is the maximal compact subgroup of T(F). The isomorphism is defined by (u(t))(X)= OrdF(X(t)).
286
I. On Automorphic Forms
Here ordF is the map F X + Z, valF(mn) = n if 2 is in the group R X of units (1x1= 1). The surjectivity of u follows on using an unramified splitting field E of T and descending using Hilbert's theorem 90, which implies thus
Let S denote the maximal Fsplit torus in T. Then
X,(S)
= X,(T)Ga'(F/F), so
s^ = Hom(X,(T)Ga'(F/F),C " ) = Hom(T(F)/T(R),C " ) = IIUr(T/F). h
The inclusion X,(S) X,(T) defines the exact sequence 1 + T1& + T^ s^ + 1. But s^ = T^/T^lFr is T^ x Fr / Int(T^) = A u ' ( T / F ) . When G is an unramified reductive group, let S be a maximal F split torus in G, and T a maximal Ftorus containing S . There is a unique econjugacy class of embeddings of in compatible with L : 3 T^ such S ( c )s S ( G ) " . Choose such an embedding and a Borel that ( g , T^) is fixed by the Galois action. Then we get LT LG and a map s^ = R U ' ( T / F ) Aur(G/F). The Weyl group WF(T) (= normalizer of T ( F ) in ,. G ( F ) , quotient by T ( F ) ) of T ( F ) in G ( F ) preserves S and acts on S by duality. The map factorizes to an isomorphism Aur( T / F )/WF(T) = A"'(G/F). On the representation theoretic side there is a bijection +
+
T^
c
~f
+
nu'(T/F)/WF(T) 5IIur( G / F ) , H ~ ( x constructed ), by means of the unramified principal series I ( x ) as follows. Let B be a Borel subgroup containing T, and N its unipotent radical. Then B ( F ) = T(F)N(F) and G ( F ) = B ( F ) K . Extend x E nUr(T/F) to a character of B ( F ) trivial on N ( F ) . The induced representation I ( x ) of G ( F ) acts by right translation on the space of locally constant functions f : G ( F ) + C with f ( n a g ) = S 1 / 2 ( u ) x ( u ) f ( gfor ) all a E T(F), n E N ( F ) , g E G ( F ) ,where S(a) = I det(Ad(a)ILieN)I. Since G ( F ) = B ( F ) K ,I ( x ) is admissible and contains a unique (up to a scalar multiple) nonzero Kinvariant vector. Hence I ( x ) has a unique unramified irreducible constituent, denoted ~ ( x )Every . unramified irreducible representation of G ( F ) is of the form ~ ( xfor) some unramified x : T(F) + C " ,
x
4. and
IT(X) 'v
Unramijied Case
287
~ ( x 'iff) x' = x o Int(w), w being a representative in G ( F ) for
WF ( T ) . The Hecke algebra W(G) of G ( F )with respect to K is the convolution algebra of compactly supported Zvalued Kbiinvariant functions f on G ( F ) . One has Hc(G) = W(G) @ C The Satake transform f t+ f V 1 IT) = t r r ( f d g ) on IIUr(G/F),is a map from Wc(G) to the space of functions on the affine variety S^/WF(T), whose coordinate ring is C [ X , ( S ) ] W F ( T )It . is an algebra isomorphism. Let F be a global field, and G a connected reductive group over F . A (smooth) representation IT of G(A) is a vector space V which is both a g,(, K,)module ( K , = K,, G, = G,, goo denotes the Lie algebra of G,, co signifies the set of archimedean places of F ) and a (smooth) G(Af)module (each vector of V is fixed by some open subgroup of G(Af)), such that the action of G(Af) commutes with that of K , and goo. Let K , be a maximal compact subgroup of G(F,) at each place u of F , which is hyperspecial ([Ti]) at almost all places, and put K f = K,, K = K,Kf. A representation IT is called admissible if it is smooth and for each isomorphism class y of continuous irreducible representations of K , the yisotypic component of V has finite dimension. Every irreducible admissible representation (IT, V )of G(A) is factorizable as the restricted tensor product of admissible irreducible representations (IT,, V,) of G(F,), over all u , where IT, is unramified for almost all u. Thus we fix a nonzero K,fixed vector at each place v where IT, is unramified, and the space V of IT is spanned by the products @<,, where E IT, for all u and = <," for almost all u. We write IT = @,IT,; the local components IT, are uniquely determined by IT up to isomorphism. Suppose F, is nonarchimedean, and G(F,) acts on a Hilbert space H , by a unitary representation IT,. The space H," of K,finite vectors is stable under the action of G(F,). If H , is irreducible, H," is admissible. Unitary IT^,, IT^, are unitarily equivalent iff the admissible IT:, IT,: are equivalent. If {H,} is a family of Hilbert spaces, fix a unit vector x, in H , for almost all u. The Hilbert restricted product H = G Z v H , is a Hilbert space with basis & h V , h, E P, for all u , h, = x, for almost all u,where P, is an orthonormal basis of H,, including 5, for almost all w. If IT is a continuous irreducible unitary Hilbert space representation of G(A) then
nvEm
nvE,
n,,,
ct
<,
<,
288
I. On Automorphic Forms
there exist such representations 7r, of G(F,), unramified for almost all v, unique up to isomorphism, with 7r Y gnu. For each isomorphism class 7 of continuous irreducible representations of K , the yisotypic component of 7r has finite dimension. The space 7ro of Kfinite vectors in 7r is an admissible irreducible G(A)module. Then no = @7r,", and 7r," is isomorphic as an admissible G(F,)module to the space of K,finite vectors of xu. For references and further comments see Flath [Fl]. By Schur's lemma ([BZl]),an admissible irreducible representation 7r, has a central character, w,. Thus if Z(F,) is the center of G(F,), 7rv(zg)= w,(z)n,(g) for all z E Z(F,), g E G(F,). Similarly, an admissible irreducible n of G(A) has central character, w.
5. Automorphic Representations Very few of the admissible representations 7r of G(A) are of number theoretic significance. Those which are of interest are the automorphic representations. Let Z denote the center of G, and let w be a unitary character of Z ( A ) / Z ( F ) . Let L = LL(G(F)Z(A)\G(A))be the space of smooth functions q5 on G(F)\G(A) with q5(zg) = w(z)q5(g) ( z E Z(A)) and Jlq5(g)12dg < 00, where dg is the unique up to scalar invariant measure on G(F)Z(A)\G(A). The completion of this space in the L2norm is a Hilbert space of the q5 which are measurable (not smooth: right invariant under an open subgroup of G(Aj)). The space L is a G(A)module under right translation: (r(g)4)(h)= $(hg). Any irreducible constituent, or subquotient, of ( T , L), is called an automorphic representation. The space L decomposes as a direct sum of irreducible representations only when the homogeneous space G(F)Z(A)\G(A) is compact. In this case G is called anisotropic, and all elements of G ( F ) are semisimple. In general Langlands theory of Eisenstein series [L3] decomposes L as a direct sum of two invariant subspaces, the discrete spectrum Ld, and the continuous spectrum L,. The discrete spectrum is the sum of all irreducible subspaces of L. Each irreducible summand, 7 r , in Ld, occurs with finite multiplicity, rn(7r). The continuous spectrum L, is the direct integral of families of representations induced from parabolic subgroups of G(A). The discrete spectrum splits as the direct sum of the cuspidal spectrum Lo, and the residual spectrum L,. The cuspidal spectrum consists of the
5. Automorphic Representations
289
4 in L with J N ( F ) , N ( W ) $ ( n g ) d n = 0 for the unipotent radical N of any proper Fparabolic subgroup P of G, and any g E G(A). The residual spectrum is generated by residues of Eisenstein series associated with proper parabolic subgroups. The irreducible constituents in L,, named residual representations, are quotients of properly induced representations. They are determined in MoeglinWaldspurger [MWl] for G = GL(n), in terms of the divisors d of n and cuspidal representations of GL(d,A) (and the parabolic subgroup of the type ( d , . . . , d ) ) . Cuspidal representations are the constituents of Lo. Langlands [L4] has shown that the constituents of an induced representation I(cr) from a cuspidal representation (T = @cr, of a parabolic subgroup P ( A ) (cr trivial on the unipotent radical N(A)) are the @ , 7 r , , where T , is a constituent of I(.,) for all w,and 7r, is the unique unramified constituent of I(a,) for almost all w. Moreover, an admissible irreducible representation x of G(A) is automorphic iff T is a constituent of I ( a ) for some P and some cr. The global principle of functoriality relates parameters X : LF t 'G with irreducible automorphic representations 7r of G(A). The relation is such that for almost all places, where the restriction A, of X to LF, ~f L F is unramified and the component 7r, of 7r is unramified, the Gconjugacy class A,(Fr,) = t(X,) x Fr, in G x (Fr,) corresponds to T , = 7r(xv),xu in h
h
rIur(T/F,)/Wpu(T)= II"'(G/F,) = @"'(G/F,) = @ u ' ( T / F , ) / W ~ V ( T ) . In other words, the unramified components of X and 7r correspond under the correspondence for unramified representations. For split groups, LG is a direct product, and the unramified A, and 7r, are parametrized by semisimple conjugacy classes in G. For the group G = GL(n) the principle can be stated as asserting that there is a bijection between the set of ndimensional irreducible representations X : LF t GL(n, C ) , and the set of cuspidal (irreducible) representations 7r of GL(n,A). Here X is uniquely determined by A, for almost all w by the Chebotarev density theorem: the set of Frobenii at almost all is dense in Gal(F/F). The cuspidal 7r is uniquely determined by almost all of its unramified components, by the rigidity theorem for GL(n) ([JS]). When the global field F is a function field, this principle was proven by Lafforgue [Lf]. This case has as an application the (Emil) Artin conjecture, which predicts that the Lfunction of an irreducible nontrivial representation X of A
I. On Automorphic Forms
290
Gal(F/F) is entire. Indeed, if X c) T then L(s,X) = L(s,T), and the Lfunction of a cuspidal T is entire. Note that if X i ~ i (5l i 5 Ic) then @iXi c) Wini, where W i r i indicates the representation I ( T I , .. . , ~ k normalizedly ) induced from r1@ . . . C3 T k on the parabolic subgroup P(A)of G(A) of type (dim A, . . . ,dim Xk) which is trivial on the unipotent radical N of P(A).The normalizing factor is cT1/’, where S(m)= I det(Ad(m)l LieN)I.

For general reductive connected group G over a global field F , a weak form of the principle would assert the existence of an automorphic representation 7r of G(A) for each parameter X : LF + LG, such that A, ++ T, for almost all v, and conversely, given such 7~ there is a A. The last claim, that T defines A, is false even for GL(n), and the group LF has to be increased to L F x SL(2,R). Before we explain this, let us present a strong form of the conjectural principle of functoriality, in terms of all places. Let P, be a packet of admissible irreducible representations of G(F,) for each place v of the global field F , such that P, contains an unramified representation T,” for almost all v. The global packet P = P({P,},) consists of all G(A)modules @,T, with T, E P, for all v and T, = T,” for almost all v. It is the restricted product of the P, with respect t o { T : } ~ . The global packet is called autornorphic (discrete spectrum, cuspidal, . . . ) if it contains such a representation. The example of SL(2) shows that not all irreducibles in an automorphic packet need be automorphic. A strong form of the principle would assert that there is a bijection between h ( G / F ) ,the set of equivalence classes of parameters X : LF 4 LG, and the set of automorphic packets P = {T} = @{T,}, such that A, c) {T,} for all v. Moreover it would specify which members of P = PA are automorphic.
6. Residual Case As noted above, the group LF does not carry sufficiently many parameters X : LF + LG to account for all discrete spectrum, or even cuspidal, automorphic representations. These X correspond, by the Ramanujan conjecture, to those discrete spectrum representations whose local components are all tempered. A bigger group than LF has to be introduced to
6. Residual Case
291
account for the discrete spectrum, including cuspidal, representations of G(A) which are not tempered, in fact at almost all places. To present it, we consider first the case of G L ( n ) . The discrete spectrum representations of G L ( n ,A) have been determined by Moeglin and Waldspurger [MWl] in terms of the divisors d of n, and the cuspidal representations r of GL(d,A). Denote by P the standard parabolic subgroup of G L ( n ) of type d = ( d , d , . . . , d ) , and by N its unipotent radical. P u t S p u ( p ) = 1 det(Ad(p)I LieN)I, for p E P(F,). Thus
where m d
= n , for
gi E GL(d,F,). Thus
where v,(g) = 1 det gl,. The normalizedly induced representation
is realized in the space of smooth functions f : G(F,) is the space of 7,) with
4
V, @ . . . @ V, (Vv
where diag(g1, . . . ,gm) is the Levi component of p . It has a unique quotient 771
7 7  1
7,) when r, is generic (or tempered), by J ( b1/2 , r,d ) = J(uvTr,,. . . ,v, [Z]. The discrete spectrum representations of G L ( n ,A) are precisely the
J(6;l2rd) = J ( u F r , v y r , . . . v
771 
2
r)
as d ranges over the divisors of n , m = n / d , and r range over the cuspidal representations of GL(d,A). If the cuspidal representations 7r of G L ( n , A ) are parametrized by the X : L F 4 LG = G L ( n , @ )x W F , namely ndimensional representations X : L F + GL(n,C), the discrete spectrum representations can be
I. On Automorphic Forms
292
parametrized by the equivalence classes of the irreducible complex representations LY : LF x SL(2, C ) 4 GL(n, C), where a is the tensor product a,, 18 a u n i p . Here ass: LF 4 GL(d, C ) and aunip : SL(2,C ) 4GL(m, C ) are irreducible representations with n = d m . In particular
au,ip
((A)) is a regular unipotent element in G L ( ~C, )
(single Jordan block). The cuspidal representations can then be viewed as the semisimple ones, while the unipotent representations are those with ass= 1. The associated discrete spectrum representation J is the trivial representation of GL(n, A). Further, the map
is the ndimensional representation of LF which parametrizes J ( C ~ ; ’ ~ T = 7(cxSs), by the principle of functoriality. Here 1. 1 is the composition of LF 4 WF + Wgb N CF and the absolute value on C F . The group GL(n) has the special property that the decomposition of its discrete spectrum into the cuspidal and residual parts is conjecturally the same as its decomposition into tempered and nontempered representations. Indeed, the Ramanujan conjecture predicts that all local components of any cuspidal representation of GL(n,A) are tempered. From the explicit description given above of the residual spectrum it is clear that each component of a residual representation of GL(n, A) is nontempered. Such partition, cuspidal equals temperedness and residual equals nontemperedness, does not hold for groups which are not closely related to GL(n), such as inner forms or SL(n).
To describe a conjectural picture of the automorphic representations of G(A) for a reductive connected group G over a global field F , Arthur ([A2], [A3], [A4]) introduced the notion of what we call Aparameter. It is a homomorphism LY : LF x
SL(2,C) 4LG
whose restriction to LF is an essentially tempered Lparameter (the projection to 8 of Q ( L F )is bounded modulo Z(8),the composition of ~ I L F with the projection LG 4 W F is the natural map LF 4W F ,prE OCY(W) is
6. Residual Case
293
semisimple for every 'u1 E L F )and whose restriction to the factor SL(2, C) is a homomorphism SL(2, C)+ G of complex algebraic groups. Moreover, a is globally relevant: if prE of its image lies in a parabolic subgroup of A
the corresponding parabolic subgroup of G has to be defined over F . Thus a tempered Lparameter X is an Aparameter; an Aparameter a whose restriction to the second factor SL(2,C) is trivial (thus a is also an Lparameter) is tempered; and the restriction of cy to LF, x SL(2,C) defines a local parameter (Y, up to equivalence, for each v . Two Aparameters a1 and a2 are called equivalent if there exist g in 21 and a 1cocycle z of LF in Z(@ with Int(g)al = z ( ~ 2such that the , is locally trivial (lies in ker[H1(LF, + class of z in H ~ ( L Fz(G)) H 1 ( L ~ ,Z, ( E ) ) ] )If. Gal(F/F) (hence L F , W F )acts trivially on Z(e) then H 1 ( L ~Z(@) , = HOm(LF, Z(c)) and Chebotarev density theorem for Lgb = W$b implies that t is trivial. Denote by N ( G / F ) the set of equivalence classes of Aparameters for G over F . For any cy the parameter
z(@)
n,
X,(W)
=Q
( IWy
(w,
,W,:l,2))
lies in h ( G / F ) . Here w E LF, and L F + W F 4 Wgb N CF together with the absolute value on CF defines w ++ Iw(. The map Q ++ A, injects N ( G / F ) in A ( G / F ) ,h ( G / F ) is the subset of N ( G / F ) of Q with CI: = 1. Locally, to each CI: E N(G/F,) there should be associated a finite set of irreducibles, containing HA,. The set named Apacket or quasipacket, does not partition the set of representations. Examples of U(3, E / F ) ([F4])and PGSp(2, F ) ([F6])show that a quasipacket has nontrivial intersection with a packet of cuspidal representations. Quasipackets come up in character relations which define liftings, by means of the trace formula. They do however define a global partition of the discrete spectrum. We define a global quasipacket as the restricted product over all v of a family of local quasipackets for all w which contain a fixed unramified for v where a, = irreducible 7rt for almost all w. In fact nt is (YI(LF,x SL(2,C)) is unramified. However, not every irreducible in a quasipacket is discrete spectrum, or automorphic.
n,
n,,
n,,,
I. On Automorphic Forms
294
e
Let S, = S,(G) be the set of s E such that s~:(w')s~= t(w')a:(w') for all w' in L F x SL(2,C), where z(w')E Z ( e ) depends only on the LFfactor w of w',and the class of the cocycle t in H 1 ( L ~ , Z ( e is) )locally trivial, namely in the kernel ker[H1(LF,Z ( e ) ) + f f ' ( L ~ , ~ Z ( e of ))] all localization maps. Put 3, = S,/S:. Z(6)= 7r0(Sa/Z(G)).Then S, is surjective, where = 7ro(Sx/Z(e))and Sx is the group of sE with sX(w)s' = z(w)X(w) (wE L F ) ,where Z(W) E Z(e)defines a locally trivial element in H ' ( L F , z(G)).
n,
f
sx,
sx
The composition of the map LF, x SL(2, C)+ LF x SL(2, C)with a: defines a parameter a , E N(G/F,). There are natural maps S, 4 Sawand S, + S,,. Arthur ([A2], 1.3.3) then expects to have a finite set, of irreducible representations of G(F,), containing , and a function E,, : + {fl}which is 1 on and which is 1 if a, is tempered,
n,,,
n,,,
n,, n,,,, and a pairing (., .), S,, x n,, C',with various properties, including: (i) E n,,,(cn,,)iff (., is a character of T,, pulled viaSav * 3x,, from a character of sx, . (ii) The invariant distribution cTEnU, tr7r is stable (de
:
+
7r),
7r
~,,(7r)(1,7r)
pends only on the stable orbital integrals of the test measure fdg).
nx,,
(iii) contains an unramified irreducible 7r," whenever a:, is unramified (trivial on the inertia subgroup of W F , )and G is unramified over F,. There should also be a function c, : Sav/Z(e) + {fl}which is conjugacy invariant, such that the map 7r H c,(s)(S, T ) on ~ is independent of the pairing (.,),. Here s is the projection of s to Tau.It is used in endoscopy. We name the quasipackets. When a:, is trivial on the factor SL(2,C) the quasipacket is simply a packet. The quasipackets do not partition the set of (equivalence classes of) irreducible admissible representations. The examples of U(3, E / F ) ([F4])and PGSp(2) ([F6])show that often a quasipacket consists of a nontempered irreducible together with a cuspidal representation, and the cuspidal lies in a packet of cuspidals. These examples show that quasipackets naturally occur in character relations describing liftings, and are necessary to describe the discrete spectrum automorphic representations.
n,,
n,,
n,,
n,
Given a: E N ( G / F ) we define the quasipacket as the restricted tensor product of the local quasipackets with respect to the unramified
n,,
7. Endoscopy 7rt
E
295
nAev for almost all u. There should be a global pairing (., .)
:
3, x II,
+ c1,
(s,7 r ) = r(z,, 7 r , ) , 21
where 5, is the image of 5 in
s,,. Further there should be a function
Almost all E, (T,) should be 1, and (Zv, T , ) , = 1 for almost all w. Further one expects that for s E S , / Z ( e ) the product c,(s,) is 1, where s, is the image of s in Sav/2(e). It is expected of the quasipackets, parametrized by a E N ( G / F ) , to partition the automorphic representations of G(A). The automorphic 7r in occur in the discrete spectrum iff S, is finite. If S, is finite there should exist an integer d , > 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 ~ ] .
I. On Automorphic Forms
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 WFmap. 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 Fgroup, 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 econjugacy 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 WFactions 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 Fisomorphism Q : H I + H z satisfying: (1) 171 0 6 and 7 2 are Gconjugate. (6 is defined up to HIconjugacy; 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
I. On Automorphic Forms
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 Fisomorphism of Fgroups. 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 Lgroup LG depends only on the class of inner forms of G. In each such class there exists a unique quasisplit form. The Lgroup determines the Fisomorphism class of the quasisplit form. The Galois action on is trivial iff G is an inner form of a split group. The Lparameters of G are only those which factorize through L P for an Fparabolic subgroup P of the quasisplit form G* of G , provided P is relevant, namely is an Fparabolic 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 pequivarient isomorphism p’ : I ~ , ( G ) ~ I ~ , W ( eGhave ) . r(CL(a))(ll0f)(.) = 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 Egroup, its restriction of scalars G‘
299
=
RE/FG is the Fgroup
~ 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 Eparabolic subgroups of G to the set of Fparabolic 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 Aparameter 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).

~
I. On Automorphic Forms
300
8. Basechange As an example, let us consider the case of basechange lifting. It concerns an Fgroup 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 Fgroup in order to compare Lparameters of the Fgroup G with those of what should describe G ( E ) . Such a group is given by G’ = R E / F G ,which is an Fgroup with G’(F) = G ( E ) . As for Lparameters, 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 Lparameter 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 Lgroup 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 ArthurClozel [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 regularIwahori 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 regularIwahori 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 ,e1 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 Lparameter X E : W E + GL(m, @).
302
I. On Automorphic Forms
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 wtwisted 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 LabesseLanglands [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, PiatetskiShapiro, Shalika ([JPS])showed Let E/F be a nonGalois 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[lq,".(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 Lfunction 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 gEGH. 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 twodimensional, 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 twodimensional. 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 3dimensional representations whose restrictions X3 to A4 remain irreducible and equal to one another. The selfdual partition (2,2) parametrizes the 2dimensional 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 3Sylow subgroup).
partition
Young T
chains
dim
(4)
2x22
(xxx,2 2 , x)
1
2
1
The 3dimensional representation X3 of A4 is induced. Indeed, consider the 2Sylow 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 3dimensional 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 =
[c2,
It remains to show that 7r as in Proposition 6 exists. Since u X = ~ X E for all 0 in G a l ( F / F ) , we have u 7 r ( X ~ )= ~ ( X E ) . Hence there exists a cuspidal 7r of GL(2, AF) with bCE/F(7r) = (.A),. This 7r is unique only up to a twist by a character of A:/FXNEIFAg = 2/3. From bCEIF(7r) = ~ ( X E we ) get w, o NEIF = det X o NEIF, hence w,w = det X for some character w of A:/FXNE,FAg. As wnBwz = w,w4 = wnw, we may and do choose 7r with
II. O n Artin’s Conjecture
307
7r1 = Sym2(7r) and 7r2 = 7r(Sym2OX) are equal, namely that the classes t(7r1,) and t(7r2,) are equal for almost all u. For this we use the following theorem of Jacquet and Shalika [JS].
w, = det A. It remains to show that
7. LEMMA.Let 7 r 1 1 7r2 be automorphic representations of GL(n,AF) with 7r2 cuspidal, such that t(n1,) 8 t(ir2,) = t(7r2,) 8 t(irzv) for almost all u, where 5 2 , denotes the representation contragredient to ~ 2 , . Then 7r1
= 7r2.
We take n = 3 , and note that t(7rlv)= t(7r2,) when u is split in E . It remains t o verify the requirement of the Lemma when v stays prime in E . In this case the image of F’r, E Gal(F/F) in A4 has order 3, namely t(r2,) = Syrn2(A(Fr,)) = [1,<,c2] for some # 1 = Hence X(Fr,) = [a,
<
c3.
The next case, completed by Tunnel1 [Tu] after some work of Langlands, is that of 8. THEOREM. Let X sentation with
:
Gal(F/F)
Im(Sym2OX) = S 4
+
GL(2,C) be a n irreducible repre
(N
PGL(2,IFs)).
Then T ( X ) exists as a cuspidal representation of G L ( ~ , A F ) . Suppose ker(Sym20A) = Gal(F/N), thus N / F is an S4Galois extension. The subgroup So of 5’4, generated by (12)(34), (13)(24), (14)(23), is normal in S 4 , isomorphic to 2 / 2 @ 2/2, and there is an exact sequence
As So is normal in 5’4, its fixed field, M , is a Galois extension of F of type S3. The sgn character on S3 defines a character of 5’4; let E be the
308
II. On Artin’s Conjecture
quadratic extension of F defined by its kernel. Let K be the nonGalois cubic extension of F fixed by a fixed 2Sylow subgroup containing So. Since Im(Sym2 O A K ) is a dihedral group (A, = XI Gal(F/K)), ;.(AK) exists as a cuspidal representation of G L ( ~ , A K )Since . Im(Sym2 OX), is Aq, A(.), exists as a cuspidal representation of G L ( 2 , k i ~ ) .As usual, by bcAIF(7r) we mean the basechange of 7r from GL(2, Ap) to G L ( ~ , A A ) . We have the following diagram of fields
K N
M
F
\ / E
9. LEMMA.Let 7r be a cuspidal representation of GL(2, A,) bCEIF(7r) = ~ ( X E )and bcKIF(7r) = ~ ( X K ) . Then 7r = .(A).
such that
PROOF.At a place v of F where both T and X are unramified, put t(7rv)= [a,b] and X(Fr,) = [ a ,p ] . If ‘u splits in E or if K , = K 8 3 8 F, has F, as a direct summand, we have t(7r,) = X(Fr,) (equality of conjugacy ) X(FI,)~. ~ classes in GL(2, C ) ) . If not, we get t(7rv)’ = X(Fr,)’ and t ( ~ , = If t(7rv)and A(Fr,) share an eigenvalue, say u = a , then b2 = p’ and b3 = p3 imply b = /3 and t(7rv)= X(Fr,). If t(7rw)# X(Fr,) then they do not share an eigenvalue, and we may assume that a = a. As t ( ~ , = )~ X(Frv)3, we have /3 =
II. O n Artin’s Conjecture
309
representations 7r‘ of GL(2, AK), and these two differ by a twist with xM/K. Hence 7r’ are 7 r ( X ~ )and 7 r ( X ~ )‘8 XM/K. At this stage we require a theorem of Jacquet, PiatetskiShapiro and Shalika [ JPS].
10. PROPOSITION. Let K I F be a field extension of degree 3 which is nonGalois. Then the basechange bcKIF(7r) of a cuspidal representation 7r of GL(2, A F ) exists and is a cuspidal representation of GL(2, AK ). 0 This is proven by means of the converse theorem. In particular bcKlF(7rl) and b C ~ , ~ ( 7 r 2exist. ) They lift to 7 r ( X ~ ) . Indeed, basechange is transitive, and is compatible with the Langlands correspondence X H 7r( A). Hence
But and bassechange with twisting,
By the compatibility fo
Hence bcKIF(7ri) = 7 r ( X ~ ) for either i = 1 or i = 2. This properties required by Lemma 9, hence theorem 8 follows.
7ri
has the
0
REFERENCES J. Arthur, On a family of distributions obtained from Eisenstein series 11: Explicit formulas, Amer. J. Math. 104 (1982), 12891336. J . Arthur, On some problems suggested by the trace formula, Lie group representations 11, Springer Lecture Notes 1041 (1984), 149. J. Arthur, Unipotent automorphic representations: conjectures, in: Orbites unipotentes et representations, 11, Asterisque 171172 (1989), 1371. J. Arthur, Unipotent automorphic representations: global motivation, in Automorphic forms, Shimura varieties, and Lfunctions, Vol. I (Ann Arbor, MI, 1988), 175, Perspect. Math., 10, Academic Press, Boston MA 1990. J. Arthur, A note on the automorphic Langlands group, Canad. Math. Bull. 45 (2002), 466482. J. Arthur, L. Clozel, Simple algebras, base change, and the advanced theorg of the trace formula, Annals Math. Studies 120, Princeton, NJ, 1989. xiv+230 pp. M. Artin, Algebra, Prentice Hall, Englewood Cliffs, N.J., 1991. J. Bernstein, A. Zelevinskii, Representations of the group GL(n, F ) where F is a nonarchimedean local field, Russian Math. Surveys 31 (1976), 168. J . Bernstein, A. Zelevinsky, Induced representations of reductive padic groups, Ann. Sci. Ecole Norm. Sup. 10 (1977), 441472. D. Blasius, On multiplicities for SL(n), Israel J . Math. 88 (1994), 237251. A. Borel, Automorphic Lfunctions, in Automorphic forms, representations and Lfunctions, Proc. Sympos. Pure Math. 33 11, Amer. Math. SOC.,Providence, R.I., (1979), 2763. 311
312
References A. Borel, Linear algebraic groups, 2nd enl. ed., Graduate texts in math. 126, SpringerVerlag, 1991. A. Borel, W. Casselman, Lzcohomology of locally symmetric manifolds of finite volume, Duke Math. J. 50 (1983), 625647. A. Borel, H. Jacquet, Automorphic forms and automorphic representations, in Automorphic forms, representations and Lfunctions, Proc. Sympos. Pure Math. 33 I, Amer. Math. SOC., Providence, R.I., (1979), 189202. A. Borel, N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, 2nd Ed., AMS, Providence, R.1: , (2000). W. Casselman, Characters and Jacquet modules, Math. Ann. 230 (1977), 101105. P.S. Chan, Invariant representations of GSp(2), thesis, the Ohio State University, 2005. L. Clozel, Characters of nonconnected, reductive p a d ic groups, Canad. J. Math. 39 (1987), 149167. L. Clozel, J. P. Labesse, R. Langlands, Seminar on the trace formula, Lecture Notes, Institute for Advanced Study, Princeton. P. Deligne, Travaux de Shimura, in Se'minaire Bourbaki, 238me annke (1970/71), Exp. No. 389, 123165, Lecture Notes in Math. 244, Springer, Berlin, 1971. P. Deligne, Formes modulaires et reprksentations de GL(2), in Modular functions of one variable, Lecture Notes in Math. 349 (1973), 55105. P. Deligne, Les constantes des equations fonctionnelles des fonctions L, Modular Functions of One Variable 11, Springer Lecture Notes 349 (1973), 501597. P. Deligne, Le support du caractkre d'une represkntation supercuspidale, C. R. Acad. Sci. Paris 283 (1976), 155157. P. Deligne, Vaxiktks de Shimura: interprtation modulaire, et techniques de construction de modkles canoniques, in Automorphic forms, representations and Lfunctions, Proc. Sympos. Pure Math. 33 11, 247289, Amer. Math. SOC.,Providence, R.I., 1979.
References
313
P. Deligne, La conjecture de Weil 11, Publ. Math. IHES 52 (1980), 137252. D. Flath, Decomposition of representations into tensor products, in Automorphic forms, representations and Lfunctions, Proc. Sympos. Pure Math. 33(1) (1977), 179183, Amer. Math. SOC.,Providence, R.I., 1979. Y. Flicker, I. On twisted liftings, Trans. Amer. Math. SOC. 290 (1985), 161178; 11. Rigidity for Automorphic forms, J . Analyse Math. 49 (1987), 135202. Y . Flicker, I. Regular trace formula and basechange lifting, Amer. J. Math. 110 (1988), 739764; 11. Regular trace formula and basechange for GL(n), Ann. Inst. Fourier 40 (1990), 130. Y . Flicker, O n the symmetric square lifting, research monograph, 2003; On the symmetric square. IV. Applications of a trace formula, Trans. Amer. Math. SOC.330 (1992), 125152; V. Unit elements, Pacific J. Math. 175 (1996), 507526; VI. Total global comparison; J. Funct. Analyse 122 (1994), 255278. Y. Flicker, Automorphic representations of the unitary group U(3, E / F ) , research monograph, 2003; I. Elementary proof of a fundamental lemma for a unitary group, Canad. J. Math. 50 (1998), 7498; 11. Packets and liftings for U(3), J. Analyse Math. 50 (1988), 1963; 111. Base change trace identity for U(3), J. Analyse Math. 52 (1989), 3952; IV. Characters, genericity, and multiplicity one for U(3), J. Analyse Math. 94 (2004), 115. Y. Flicker, Matching of orbital integrals on GL(4) and GSp(2), Mem. Amer. Math. SOC.no. 655, vol. 137 (1999), 1114. Y. Flicker, Lifting automorphic forms of PGSp(2) and SO(4) to PGL(4), part 1 of this book; Automorphic forms on PGSp(2), Arbeitstagung 2003, Max
314
References PlanckInstitut fur Mathematik, Bonn, June 2003, 114; Automorphic forms on PGSp(2), Elect. Res. Announc. Amer. 10 (2004), 3950, http://www.ams.org/era; Math. SOC. Automorphic forms on S0(4), Proc. Japan Academy 80 (2004), 100104. Y. Flicker, Zeta functions of Shimura varieties of PGSp(2), part 2 of this book. Y. Flicker, D. Kazhdan, Metaplectic correspondence, Publ. Math. IHES 64 (1987), 53110 Y. Flicker, D. Kazhdan, A simple trace formula, J. Analyse Math. 50 (1988), 189200. Y. Flicker, D. Kazhdan, Geometric Ramanujan conjecture and Drinfeld reciprocity law, in Number theory, trace formulas and discrete groups (Oslo, 1987), 201218, Academic Press, Boston MA, 1989. Y. Flicker, D. Zinoviev, Twisted character of a small representation of PGL(4), Moscow Math. J. 4 (2004), 333368. K. Fujiwara, Rigid geometry, LefschetzVerdier trace formula and Deligne’s conjecture, Invent. Math. 127 (1997), 489533. W. Fulton, J. Harris, Representation Theory, A First Course, Graduate Text in Mathematics 129, SpringerVerlag (1991). S. Gelbart, An elementary introduction to the Langlands program, Bull. Amer. Math. SOC. 10 (1984), 177219. D. Ginzburg, S. Rallis, D. Soudry, On generic forms on S 0 ( 2 n + 1); functorial lift to GL(2n), endoscopy and base change, Internat. Math. Res. Notices 14 (2001), 729763. G. Harder, Eisensteinkohomologie und die Konstruktion gemischter Motive, Springer Lecture Notes 1562 (1993). HarishChandra, Admissible invariant distributions on reductive pa dic groups, notes by S. DeBacker and P. Sally, AMS Univ. Lecture Series 16 (1999); see also: Queen’s Papers in Pure and Appl. Math. 48 (1978), 281346. M. Harris, R. Taylor, The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, 151, Princeton University Press, Princeton, NJ, 2001. G. Henniart, Progrks rbcents en fonctorialitb de Langlands,
References
315
Skminaire Bourbaki, Vol. 2000/2001. AstBrisque 282 (2002), Exp. No. 890, ix, 301322. J. Humphreys, Linear algebraic groups, Graduate texts in math. 21, SpringerVerlag, 1975. H. Jacquet, R. Langlands, Automorphic forms o n GL(2), Springer Lecture Notes 114 (1970). H. Jacquet, J. Shalika, On Euler products and the classification of automorphic forms (I and) 11, Amer. J. Math. 103 (1981), (499558 and) 777815. H. Jacquet, I. PiatetskiShapiro, J. Shalika, Relkvement cubique non normal, C. R. Acad. Sci. Paris Se'r. I Math. 292 (1981), no. 12, 567571. D. Kazhdan, On lifting, in Lie groups representations II, Springer Lecture Notes 1041 (1984), 209249. D. Kazhdan, Cuspidal geometry of padic groups, J. Analyse Math. 47 (1986), 136. D. Kazhdan, G. Lusztig, Proof of the DeligneLanglands conjecture for Hecke algebras, Invent. Math. 87 (1987), 153215. H. Kim, Residual spectrum of odd orthogonal groups, Internat. Math. Res. Notices 17 (2001), 873906. A. Knapp, Representation Theory of Semisimple Groups, Princeton University Press (1986). R. Kottwitz, Rational conjugacy classes in reductive groups, Duke Math. J . 49 (1982), 785806. R. Kottwitz, Stable trace formula: cuspidal tempered terms, Duke Math. J. 51 (1984), 611650. R. Kottwitz, Shimura varieties and twisted orbital integrals, Math. Ann. 269 (1984), 287300. R. Kottwitz, Shimura varieties and Xadic representations, Automorphic forms, Shimura varieties, and Lfunctions, I (Ann Arbor, MI, 1988), 161209, Perspect. Math. 10, Academic Press, Boston MA, 1990. R. Kottwitz, On the Xadic representations associated to some simple Shimura varieties, Invent. Math. 108 (1992), 653665. R. Kottwitz, Points on some Shimura varieties over finite fields, J . Amer. Math. SOC.5 (1992), 373444.
316
References
R. Kottwitz, D. Shelstad, Foundations of twisted endoscopy, Asterisque 255 (1999), vitl90 pp. S. Kudla, S. Rallis, D. Soudry, On the degree 5 Lfunction for Sp(2), Invent. Math. 107 (1992), 483541. L. Lafforgue, Chtoucas de Drinfeld, formule des traces d’Arthur Selberg et correspondance de Langlands, Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), 383400, Higher Ed. Press, Beijing, 2002. J.P. Labesse, R. Langlands, Lindistinguishability for SL(2) , Canad. J . Math. 31 (1979), 726785. R. Langlands, Problems in the theory of automorphic forms, in Lectures in modern analysis and applications, 111, 1861, Lecture Notes in Math. 170, Springer, Berlin, 1970. R. Langlands, Euler products, Yale Mathematical Monographs, 1. Yale University Press, New Haven, Conn.London, 1971. v t 5 3 pp. R. Langlands, O n the functional equations satisfied by Eisenstein series, Lecture Notes in Math. 544. SpringerVerlag, 1976. R. Langlands, On the notion of an automorphic representation, in Automorphic forms, representations and Lfunctions, Proc. Sympos. Pure Math. 33 I, Amer. Math. SOC.,Providence, R.I., (1979), 203207. R. Langlands, I. Qn the zeta functions of some simple Shimura varieties, Canad. J. Math. 31 (1979), 11211216; 11. Shimura varieties and the Selberg trace formula, Canad. J. Math. 29 (1977), 12921299; 111. Automorphic representations, Shimura varieties, and motives, in Automorphic forms, representations and Lfunctions, Proc. Sympos. Pure Math. 33 11, (1979), 205246. R. Langlands, On the classification of irreducible representations of real algebraic groups, in Representation theory and harmonic analysis o n semisimple Lie groups, 101170, Math. Surveys Monogr., 31, Amer. Math. SOC.,Providence, RI, 1989. R. Langlands, Base change f o r GL(2), Annals of Math. Studies, 96. Princeton University Press, Princeton, N.J., 1980.
References
317
R. Langlands, Representations of abelian algebraic groups, Pacific J. Math. (1997), special issue, 231250. M. Larsen, On the conjugacy of elementconjugate homomorphisms, Israel J . Math. 88 (1994), 253277; 11, Quart. J. Math. Oxford Ser. (2) 47 (1996), 7385. G. Laumon, Sur la cohomologie B supports compacts des varihths de Shimura pour GSp(4,Q), Compos. Math. 105 (1997), 267359. C. Moeglin, J.L. Waldspurger, Le spectre rhsiduel de GL(n), Ann. Sci. Ecole Norm. Sup. 22 (1989), 605674. C. Moeglin, J.L. Waldspurger, Spectral decomposition and Eisenstein series. Une paraphrase de l’e‘criture, Cambridge Tracts in Math. 113, Cambridge University Press, Cambridge, 1995. I. PiatetskiShapiro, On the SaitoKurokawa lifting, Invent. Math. 71 (1983), 309338; A remark on my paper: “On the SaitoKurokawa lifting” , Inwent. Math. 76 (1984), 7576. I. PiatetskiShapiro, Work of Waldspurger, Lie group representations 11, Springer Lecture Notes 1041 (1984), 280302. V. Platonov, A. Rapinchuk, Algebraic groups and number theory, Pure and Applied Math. 139, Academic Press, Boston, MA, 1994. [PI R. Pink, On the calculation of local terms in the LefschetzVerdier trace formula and its application to a conjecture of Deligne, Ann. of Math. 135 (1992), 483525. B. Roberts, Global Lpackets for GSp(2) and theta lifts, Doc. Math. 6 (2001), 247314. F. Rodier, Modble de Whittaker et caractkres de reprhsentations, Noncommutative harmonic analysis, Springer Lecture Notes 466 (1975), 151171. F. Rodier, Sur les rhpresentations nonramifikes des groupes rhductifs padiques; l’example de GSp(4), Bull. Soc. Math. France 116 (1988), 1542; Errata: http://xxx.lanl.gov/pdf/math.RT/O4lO273. P. Sally, M. Tadic, Induced representations and classifications for GSp(2, F ) and Sp(2, F ) , Mem. SOC. Math. France 52
318
References (1993), 75133. R. Schmidt, Generalized SaitoKurokawa liftings, Habilitation 2002. E. Shpiz, Deligne’s conjecture in the constant coefficients case, Ph.D. thesis, Harvard 1990. J.P. Serre, Cohornologie Galoisienne, Springer Lecture Notes 5, 5“ 6d. (1994). F. Shahidi, On certain Lfunctions, Arner. J. Math. 103 (1981), 297355. F. Shahidi, A proof of Langlands’ conjecture on Plancherel measures; complementary series for p a d ic groups, Ann. of Math. 132 (1990), 273330. F. Shahidi, Langlands’ conjecture on Plancherel measures for pa di c groups, in Harmonic analysis on reductive groups, Progr. Math., 101, pp. 277295, Birkhauser, Boston, MA, 1991. F. Shdika, A theorem on semisimple p a d ic groups, Ann. of Math. 95 (1972), 226242. A. Silberger, The Langlands quotient theorem for p a d i c groups, Math. Ann 236 (1978), 95104. D. Soudry, A uniqueness theorem for representations of GSO(6) and the strong multiplicity one theorem for generic representations of GSp(4), Israel J. Math. 58 (1987), 257287. T . A. Springer, Linear algebraic groups, 2nd ed., Progress in math. 9, 1998. J. Tate, Number theoretic background, in Automorphic forms, representations and Lfunctions, Proc. Sympos. Pure Math. 33 11, Amer. Math. SOC.,Providence, R.I., (1979). R. Taylor, On the ladic cohomology of Siege1 threefolds, Invent. Math. 114 (1993), 289310. J. Tits, Reductive groups over local fields, in Autornorphic forms, representations and Lfunctions, Proc. Sympos. Pure Math. 33(1), 2969, Amer. Math. SOC.,Providence, R.I., (1979). J. Tunnell, Artin’s conjecture for representations of octahedral type, Bull. Arner. Math. SOC.(N.S.) 5 (1981), 173175. Y. Varshavsky, LefschetzVerdier trace formula and a generali
References
319
zation of a theorem of F’ujiwara, http://arXiv.org/abs/math/0505564, 2005. D. Vogan, The KazhdanLusztig conjecture for real reductive groups, in Representation Theory of Reductive Groups, P. Trombi, Ed. , Birhauser (1983), 223264. D. Vogan, G. Zuckerman, Unitary representations with nonzero cohomology, Compos. Math. 53 (1984), 5190. J.L. Waldspurger, Un exercise sur GSp(4, F ) et les reprhsentations de Weil, Bull. SOC.Math. France 115 (1987), 3569. J.L. Waldspurger, Correspondances de Shimura et quaternions, Forum Math. 3 (1991), 219307. J.L. Waldspurger, Le lemme fondamental implique le transfert, Compositio Math. 105 (1997), 153236. R. Weissauer, http://www.mathi.uniheidelberg.de/Nweissaue A. Zelevinsky, Induced representations of reductive padic groups 11, Ann. Sci. Ecole N o m . Sup. 13 (1980), 165210. Th. Zink, The Lefschetz trace formula for an open algebraic surface, in Automorphic forms, Shimura varieties, and Lfunctions, Vol. I1 (Ann Arbor, MI, 1988), 337376, Perspect. Math. 11, Academic Press, Boston MA, 1990. S. Zucker, Lacohomology of Shimura varieties, in Automorphic forms, Shimum varieties, and L functions, Vol. I1 (Ann Arbor, MI, 1988), 377391, Perspect. Math., 11, Academic Press, Boston, MA, 1990.
This page intentionally left blank
INDEX Aparameter, 292 Apacket (= quasipacket), 293 adhles, AF, 3, 6, 269 adjoint group, Gad, 279 admissible representations, 6, 281, 287 anisotropic group, 288 Arthur’s conjectures, 28, 294 Artin conjecture, 289 associate parabolic subgroups, 49 automorphic representations, 4, 288 N ( G / F ) , 293
Clifford theory, 304 cohomological representations, 176 conjugacy class, 50 connected dual group, 277 control of fusion, 31 cuspidal local representation, 16 cuspidal global representation, 21, 101, 289 cuspidal spectrum, 288
e,
BaileyBorelSatake compactification, 222 basechange, bcEIF, 300 Borel pair, ( G , T ) , 277 Borel subgroup, 6
decomposition group, D,, 270 Deligne’s conjecture, 208 derived group, 50 dihedral group, 51, 73 discrete spectrum, 8, 288 discrete series representations, 18, 174, 176, 282 dual group, 4, 5, 6, 35
canonical models, 207 central exponents, decay, bounded, 282 character, 9, 39 character computation, 28, 40, 87 character relations, 10, 11, 16, 24 Chebotarev’s density theorem, 2 70 Chinese Remainder Theorem, 270
elliptic endoscopic group, 5, 13 elliptic function, 18 elliptic representations, 9, 16 endoscopic datum, (s,p ) , elliptic, 296 endoscopic group, (HIs, v), 296 endoscopic lifting, endEIF, 13, 302 essentially tempered, 49
32 1
322
Frobenius, Fr,, 270, 273 functoriality for tori, 281 functoriality for unramified reps, 285 fundamental lemma, 285 fusion control, 31 Galois cohomology, 150 generalized linear independence, 19, 83, 102, 136 generic representations, 18, 24, 25, 148 global class field theory, CFT, 270 global packet, II, 290 globally relevant, 293 (g, K)module, 282 Hecke algebra, 46 Hecke eigenvalues, 8 highest weight, 210 Heisenberg parabolic subgroup, 15 Hodge types, 216 holomorphic representations , 18, 177 hypercohomology, 53 idkles, A;, 269 induced group, I i , (G), 298 induced representations, 39, 47, 97, 281 inertia group, I,, 270 inner form, 298 inner twist, 298 intersection cohomology, 223
Index irreducible representation, isogeny, 278
7
Jacobian, 40, 113 Lfunctions, 149 Lparameter, 280 Langlands classification, 49 Langlands dual group, 'GI 277, 279 Langlands group, LF, 276 Langlands parameter, 8 Lefschetz fixed point formula, 208 lifting, 3, 6, 7, 8, 13, 20, 44, 275 local Langlands conjecture, 282 locally conjugate, 31 h ( G / F ) , 282 A ( H ,s, r l ) , 296 N.,P>, 296 matching orbital integrals, 14 module of coinvariants, 133, 281 monomial representation, 9, 11, 12, 70 multiplicity, rn(r), 295 multiplicity one for GL(n), 8 multiplicity one for SL(2) , 11 multiplicity one for U(3), 25, 147 multiplicity in a packet, 21 norm, 15, 35, 53, 113 normal morphism, 277 normalized induction, 6 orientation, 27 orthogonal group, 5, 86, 149
Index orthogonality relations, 129 packets, 17, 20, 169 parabolic induction, 5 parameters, equivalent, 280 principal series representations, 6 principle of functoriality, 4, 5, 282, 289 pseudocoefficients, 103, 134 282 IIUr( G / F ), 285
ww)),
quasipackets (= Apacket), 18, 20, 169, 294 quaternions, 174 radical, 279 Ramanujan conjecture, 23, 209, 275 rank, 19 reciprocity law, 271 reductive group, 277 reflex field, 212 regular conjugacy classes, 50 regular Iwahori functions, 19 relevant parabolic, 298 representations, 7r: admissible, 6, 281, 287 algebraic, smooth, 281 automorphic, 4, 288 cohomological, 176 cuspidal, 16, 21, 101, 282, 289 discrete series, 18, 174, 176, 282
discrete spectrum, 8, 288 elliptic, 9, 16
323
equivalent, 282 factorizable, 287 generic, 18, 24, 25, 148 holomorphic, 18, 177 induced, 39, 47, 97, 281 irreducible, 7 monomial, 9, 11, 12, 70 principal series, 6 residual, 22, 289 square integrable, 16, 17, 282 tempered, 16, 282 unramified, 6, 285 residual representation, 22, 289 residual spectrum, 288 rigidity property, 31 rigidity theorem, 8, 9, 34, 289 root datum, based, dual, reduced, 276 roots, coroots, 73, 176, 276 Satake isomorphism, 46 selfcontragredient, 4 semisimple element, 4 semisimple group, 279 Shimura varieties, 28, 207, 222, 275 Siege1 parabolic subgroup, 15 simple group, 279 simply connected group, 50, 279 simple trace formula, 217 smooth functions, 7 smooth sheaf, 211 split , quasisplit group, 277 splitting, 280 square integrable representations, 16, 17, 282
324
Index
stable conjugacy, 50, 122 stable distribution, 294 stable 0conjugacy, 52 stable spectrum, 20, 169 “supercuspidal” , 16 Sylow subgroup, 31 symmetric square, 11, 94 symplectic group, 5, 149 0conjugacy classes, 52 0invariant, 4 0semisimple, 62 TateNakayama isomorphism, 50, 122 tempered representation, 16, 282 thetacorrespondence, 23, 167 trace formula, 60, 66 transfer, 122 transpose, 4 twisted endoscopic group, 4
unipotent radical, 6, 279 unramified group, 285 unramified Lparameter, 285 unramified representations, 6, 285 unstable spectrum, 20 weak lifting, 148 weak matching, 44 Weil datum, 271 Weil group, W F , 5, 12, 70, 105, 174, 271 WeilDeligne group, 272 Weyl integration formula, 40, 42, 123 Young Tableau, 304 zextension, 284
AUTOMORPHIC FORMS AND SHIMURA VARIETIES OF PGSp(2) by Yuval Z. Flicker (The Ohio State University, USA) The area of automorphic representations is a natural continuation of the 19th and 20th centuries studies in number theory and modular forms. A guiding principle is a reciprocity law relating the infinite dimensional automorphic representations, with finite dimensional Galois representations. Simple relations on the Galois side reflect deep relations on the automorphic side, called “liftings”. This monograph concentrates on an initial example of the lifting, from a rank 2 symplectic group PGSp(2) to PGL(4), reflecting the natural embedding of $ 4 2 , C ) in SL(4,C). It develops the technique of comparison of twisted and stabilized trace formulae. Main results include: 0 0 0
0
A detailed classification of the representations of PGSp(2). A definition of the notions of “packets” and “quasipackets” . A statement and proof of the “lifting” by means of character relations. Proof of multiplicity one and rigidity theorems for the discrete spectrum.
These results are then used to study the decomposition of the cohomology of an associated Shimura variety, thereby linking Galois representations to geometric automorphic representations.
To put these results in a general context, the book ends with a technical introduction to Langlands’ program in the area of automorphic representations. It includes a proof of known cases of Artin’s conjecture. This research monograph will benefit an audience of graduate students and researchers in number theory, algebra and representation theory.