Cohomology of Drinfeld modular varieties. Part II. Automorphic forms, trace formulas and Langlands correspondence 9780521470612

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CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS : 56 EDITORIAL BOARD D.J.H. GARLING, W. FULTON, K. RIBET, T. TOM DIECK, P. WALTERS

COHOMOLOGY OF DRINFELD MODULAR VARIETIES, PART II

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Cohomology of Drinfeld Modular Varieties, Part II Automorphic forms, trace formulas and Langlands correspondence Gerard Laumon Universite Paris-Sud, CNRS

with an appendix by Jean-Loup Waldspurger

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PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE

The Pitt Building, Trumpington Street, Cambridge CB2 1RP, United Kingdom CAMBRIDGE UNIVERSITY PRESS

The Edinburgh Building, Cambridge CB2 2RU, United Kingdom 40 West 20th Street, New York, NY 10011-4211, USA 10 Stamford Road, Oakleigh, Melbourne 3166, Australia www. c ambri dge.org Information on this title: www.cambridge.org/9780521470612 © Cambridge University Press 1997 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1997 A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication data available ISBN-13 978-0-521-47061-2 hardback ISBN-10 0-521-47061-7 hardback Transferred to digital printing 2005

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Contents

Preface

ix

9. Trace of /A on the discrete spectrum

1

(9.0) Introduction

1

(9.1) Automorphic representations

2

(9.2) Cuspidal automorphic representations 2

14

(9.3) L -automorphic representations

24

(9.4) One dimensional automorphic representations

30

(9.5) Trace of /A into the discrete spectrum

33

(9.6) Main theorem

45

(9.7) Comments and references

46

10. Non-invariant Arthur trace formula: the geometric side

47

(10.0) Introduction

47

(10.1) The kernel K(h, g)

47

(10.2) Integrability of K(h,g) along the diagonal

50

(10.3) Harder's reduction theory revisited

52

(10.4) Proof of the integrability of k(g)

62

(10.5) The distributions Jgeom and Jo

71

(10.6) Kazhdan's trick

73

(10.7) The distributions J6

80

(10.8) Reduction of (10.7.6)

86

(10.9) Proof of lemma (10.8.5)

97

(10.10) Flicker-Kazhdan simple trace formula

114

(10.11) Comments and references

119

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G. LAUMON

11. Non-invariant Arthur trace formula: the spectral side

120

(11.0) Introduction

120

(11.1) Arthur's truncation operators

120

(11.2) Jgeom(f) as the integral over the diagonal of the truncated kernel

132

(11.3) Expansion of the kernel following the cuspidal data

146

(11.4) Evaluation of Jj(f) (11.5) Evaluation of Jj(f)

in a special case: the operators Mf7r(X) in a special case: (G, M)~-families

150 168

(11.6) Evaluation of Jj{f) intertwining operators

in a special case: normalization of 185

(11.7) Evaluation of J j ( / ) in a special case: Waldspurger's theorem 195 (11.8) A "simple" spectral side for the trace formula

199

(11.9) Comments and references

200

12. Cohomology with compact supports of Drinfeld modular varieties

201

(12.0) Introduction

201

(12.1) Cohomological correspondences and the Deligne conjecture

201

(12.2) Application of the Deligne conjecture to Drinfeld modular varieties

204

(12.3) Application of the non-invariant Arthur trace formula to Drinfeld modular varieties

209

(12.4) Langlands correspondence

213

(12.5) The virtual module W |

221

(12.6) Comments and references

222

13. Intersection cohomology of Drinfeld modular varieties: conjectures

224

(13.0) Introduction

224

(13.1) A conjectural trace formula

224

(13.2) Some particular cases of conjecture (13.1.6)

228

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DRINFELD MODULAR VARIETIES

vii

(13.3) A cohomological interpretation of the constants Vd'iT)

242

(13.4) Intersection cohomology

244

(13.5) Comments and references

248

Appendices D. Representations of unimodular, locally compact, totally discontinuous, separated, topological groups: addendum 249 (D.9) Restricted tensor products

249

(D.10) Rationality properties

256

(D.ll) The Grothendieck group of admissible representations

259

(D.12) Comments and references

261

E. Reduction theory and strong approximation

262

(E.O) Introduction

262

(E.I) Reduction theory

262

(E.2) Strong approximation

272

(E.3) Comments and references

274

F. Proof of lemma (10.6.4)

275

(F.O) Notations

275

(F.I) Reductions

275

(F.2) A geometric construction

276

(F.3) The Harish-Chandra lemma

277

(F.4) An application of the Harish-Chandra lemma

279

(F.5) End of the proof of (10.6.4)

281

(F.6) Comments and references

282

G. The decomposition of I?G following the cuspidal data

283

(G.0) Introduction

283

(G.I) Cuspidal data

283

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ii

G. LAUMON

(G.2) Paley-Wiener functions

288

(G.3) Fourier transformation

290

(G.4) Eisenstein series

291

(G.5) Intertwining operators

299

(G.6) The constant terms of the Eisenstein series

310

(G.7) Pseudo-Eisenstein series

315

(G.8) The scalar product of two pseudo-Eisenstein series

320

(G.9) Analytic continuation of Eisenstein series

325

(G.10) The spectral decomposition of t?G x^ cuspidal data c

329

c

for regular

(G.ll) Comments and references

334

References

335

Some residue computations by J.-L. Waldspurger Index

341 365

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Preface

The second volume of Drinfeld modular varieties is devoted to the ArthurSelberg trace formula and to the proof in some cases of the RamanujanPetersson conjecture and the global Langlands correspondence for function fields. As in the first volume we fix a function field F together with a place oo of F and a positive integer d. The group F£\GLd(A) acts by right translation on the Hilbert space L2GLd>lao=L2(F^GLd(F)\GLd(A)) and any locally constant function / with compact support on F£)\GLd(A) induces by convolution an operator RGLd,ioo(f) on L%Ld i^- This operator admits a kernel K(h,g) = 2_] f(h~lrY9) 7eGLd(F)

and, at least formally, its trace is the integral

J(f)= [

K(s,g)

dg

Unfortunately, for an arbitrary function / the operator RGL^IOOU) is not of trace class and the integral J(f) is not absolutely convergent. To tide over this difficulty Arthur has introduced a truncated version JT(f) of the above integral which is absolutely convergent. It depends on some truncation parameter T in the positive Weyl chamber a j . Let us fix some level / and some place o which is prime to /. If / = /oo/°°'°/o5 where / ^ is the very cuspidal Euler-Poincare function introduced in chapter 5 of the first volume, /°°>o is an arbitrary element of the Hecke algebra of level / and fo is the Drinfeld function of level r (for some positive integer r) introduced in chapter 4 of the first volume, we will see that J(f) is convergent and that we have

JT(f) = Af) for any value of the truncation parameter T which is far enough from the walls of ajj". Moreover, if we take r large enough with respect to /°°'° (Kazhdan's trick) we will see that J(f) is equal to the number Lef r (/°°' 0 ) of fixed points Downloaded from https://www.cambridge.org/core. University College London (UCL), on 29 Dec 2017 at 03:38:43, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511661969.001

x

G. LAUMON

of Frob£ x/°°'° acting on the reduction in "characteristic" o of the Drinfeld modular variety Mf. But, on the one hand, if r is large enough with respect to /°°'° we will see that the Lefschetz number Lefr(/°°'°) is equal to the trace of Frob£ x/°°'° acting on the £-adic cohomology with compact supports of F®F Mf. This is due to Grothendieck's theorem if f°°'° is the trivial Hecke operator and it is due to Deligne's conjecture, proved by Fujiwara and Pink, for a general /°°'°. On the other hand, following Arthur the truncated integral JT(f) admits a spectral expression and, using some residue computations, which have been done by Waldspurger and which are included as an appendix at the end of this volume, we will make explicit this spectral expression. Putting together all these results we will obtain an explicit expression for the alternating sum of traces, M

f,

in terms of cuspidal automorphic representations of F£\GLd'(A) {d! = i,...,d). Finally, by a standard procedure we will deduce the Petersson conjecture and the Langlands correspondence for cuspidal automorphic representations of F£)\GLd(A) the local component at oo of which is isomorphic to the Steinberg representation. The numbering of this volume is the continuation of the numbering of the first one. Here is a brief description of its contents. In chapter 9 we review some basic definitions and results about the cuspidal spectrum of L2GLdl^. In chapter 10 we study the geometric side of Arthur's non-invariant trace formula for our function / = foo f °°'° fo a n d we prove that, if r is large enough with respect to /°°'° (and /oo), it has a simple form. In fact / ^ may be any very cuspidal function and, as a special case, we obtain the Flicker-Kazhdan simple trace formula. The arguments are adapted from those used by Arthur in the number field case. In chapter 11 we study the spectral side. Again we adapt Arthur's arguments. But here we have not been courageous enough to transpose all of his arguments to the function field case. Actually JT(f) is a sum over the cuspidal data of expressions Jj(f)> Using Waldspurger's residue computations we obtain an explicit formula for Jj(f) when c is a regular cuspidal datum. For the other cuspidal data we only state a conjectural formula. This formula has been recently proved by Lafforgue. In chapter 12 we deduce the Ramanujan-Petersson conjecture and the global Langlands correspondence (for cuspidal automorphic representations of F£\GLd(A) the local component at oo of which is isomorphic to the Downloaded from https://www.cambridge.org/core. University College London (UCL), on 29 Dec 2017 at 03:38:43, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511661969.001

DRINFELD MODULAR VARIETIES

Steinberg representation) from the results of the previous chapters. We also give a complete description of the virtual (Gal(F/F) x GLd(A°°))-module

Up to this point we have only considered the cohomology with compact supports of Drinfeld modular varieties. We may also consider the intersection cohomology of the Satake compactification of Mf. In chapter 13 we give a conjectural description of the intersection complex of this Satake compactification. We have discovered this conjectural description by transposing to the function field case a formula for the L2-Lefschetz number of a Hecke operator which has been proved by Arthur in the number field case. There are four appendices. An addendum to appendix D contains some rationality results and a definition of the Grothendieck group of admissible representations. The main results of reduction theory are reviewed in appendix E. Our proofs differ from Harder's original ones in the way that we systematically use the Harder-Narasimhan filtration. In appendix F we give the proof of Harish-Chandra's results on orbital integrals which are needed in chapter 10. In appendix G we present some of the basic results concerning the spectral decomposition of Langlands and Morris. In particular we explain the first step in Langlands' computation of the scalar product of two pseudo-Eisenstein series associated with cuspidal automorphic forms of Levi subgroups. I would like to thank J.-L. Waldspurger once more for his help during the elaboration of this project. His residue computations are fundamental for the results of the second volume. I would also like to thank R. Pink for his comments on chapter 13. During the preparation of the manuscript I visited the University of Toronto (Winter 1993). My thanks go to J. Arthur for his kind hospitality and for the numerous discussions that I had with him. Special thanks go to the editors who again did a beautiful job for this second volume.

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9 Trace of fA on the discrete spectrum

(9.0) Introduction In this chapter we will use again the notations of chapters 1, 2, 3 and 6 of the first volume. So F is a function field of positive characteristic p, oo and o are two distinct places of F, I is a proper, non-zero ideal of the ring A = {a e F | x(a) > 0, Vx e \X\ - {oo}} such that o ^ V(I) and A is the ring of adeles of F. In fact, in this volume we will use the notation J for the ideal / in order to avoid any confusion with the subsets / of A. The purpose of this chapter is to compute the trace of the compactly supported, locally constant function /A — / o o /

Jo

acting on L2-automorphic irreducible representations of F£\GLd(A). Here

is our very cuspidal Euler-Poincare function (see (5.2.1)), /°°'° G C™(GLd{A°°>0)//K^0) is an arbitrary Hecke operator and fo E

C?(GLd(Fo)//Ko)

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2

DRINFELD MODULAR VARIETIES

is the Drinfeld function with Satake transform

for some fixed positive integer r.

(9.1) Automorphic representations Let M be a standard Levi subgroup of GLd- We have M — Mj for some / C A and, if dj = ( d i , . . . ,d s ) is the corresponding partition of d, M is canonically isomorphic to GL&X x • • • x GLds. We denote by (9.1.1)

C%=C°°(M(F)\M(A),C)

the C-vector space of the complex functions tp on M(A) which are invariant under left translation by M(F) and which are invariant under right translation by some compact open subgroup of M(A) (depending on RM)-

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9. TRACE OF / A ON THE DISCRETE SPECTRUM

3

Proof : Up to isomorphism the automorphic irreducible representation (V, TT) of M(A) is equal to for some subrepresentations Let us arbitrarly fix^G W2 — W\. As (V, TT) is irreducible the subrepresentation of (W2, P2) generated by ip maps onto (V, TT). Therefore we may assume that (W2, P2) is generated by %>o such that

for every family {z^x)xexM

^ (ZM(&))XM

• Let us prove that

by induction on the number of elements in the support of \i. If Supp(//) = {x} we have v G VX)gen- If x' ¥^ x" a r e m Supp(/i) let us fix ^A G Z M ( A ) such that X'(ZA) ¥" X* (ZA)- Then by the Bezout theorem there exist polynomials P\T), P"(T) G C[T] such that P'(T)(T - X'(zA)yM

+ P"(T)(T -

Let us set v' = (P'(n(zA))(7r(zA) - X' Downloaded from https://www.cambridge.org/core. University College London (UCL), on 29 Dec 2017 at 03:40:18, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511661969.002

9. TRACE OF / A ON THE DISCRETE SPECTRUM

and v" = {P"(n(zA))(n We have v

= v' + v"

and

and

for every family (ZA,X)X£XM ^ ^ M ( A ) ' Y M . By the induction hypothesis we have

and the corollary follows.

•

Let /7OO

c

zM

/7OO / rr = C

( T?\\

y^M \b)

>7

\ZM

/ A\

/f^N

\A)' ^)

be the C-vector space of the complex functions ip on ZM(A) which are invariant under ZM(F) and under some compact open subgroup of ZM(A) (depending on (CFM)x,m, ¥ > ^ V % is an isomorphism. Therefore we may assume that x — 1Let C(ZS, C) be the C-vector space of the complex functions on Zs and let r be the action by translation of If on this space. It is well known and easy to prove by induction on s and m that {

P ( n ) (use the fact that is a basis of C[X]fc for each positive integer k). Therefore we have and the equality holds if we can prove that any

aj + 1 , • • •, fft — 1} and where \(3 — a\ = 3= 1

(Pi — OL\) H

\-{Ps — as)- Therefore we obtain

and dimc(V K / ) < +oo. Therefore v is ZM(A)-finite. If (V, TT) is irreducible we have V = V x?m for any x G ^ M and any positive integer m such that V x , m ^ (0) (Vx>m is stable under T T ( M ( A ) ) ) . Therefore there exists a unique x G ^ M such that V x , m ^ (0) for some m. Let m be the smallest positive integer such that V x , m i=- (0). For any zA G ^ M ( A ) we have (TT(ZA) — x(^A))(V x?m ) C V x , m _i = (0) and therefore ra = 1. See also (D.I.12). • Downloaded from https://www.cambridge.org/core. University College London (UCL), on 29 Dec 2017 at 03:40:18, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511661969.002

12

DRINFELD MODULAR VARIETIES

For each \ £ %M let us set (9.1.14) PROPOSITION

AM,X = AX(M(F)\M(A),C)

=

(9.1.15). — The C-linear map (9.1.10) induces an isomor-

phism from

0

C[Xu...,Xa]®cAMtX

onto AM • More precisely, for each x £ %M and each positive integer m, ix maps C[X\,... ,Xg] m 0 c AM,X isomorphically onto (AM)x,m and we have AM = ®

(^Af)x,gen.

Proof: First of all let us show that the image of i is contained in AM- Let P{X) E C[Xi,... ,Xs]m and let (p G AM,X ^or s o m e positive integer m and some x £ %M- It is sufficient to check that the subrepresentation of (C[Xi,..., Xs) c ^4M,X» ( r ° de gM) ® ^ M ) which is generated by P(X) 0 w e n a y e dimc(C[Xi,... ,Xj] m ) < -hoc and (V,.RM,X|V) is admissible. As any compact open subgroup of M(A) is contained in M(A) 1 and acts trivially on C[Xi,..., Xs] our assertion follows. The same argument proves that P(degM)V> e AM for every P(X) G C [ I i , . . . , I s ] and every i\) e AMNow if if G (AM)x,m we may decompose ip into

with ipa G C ^ x for every a G {0,1,..., m - 1}S (see the proof of (9.1.11)). From remark (9.1.12) it follows that (pa G AM,X f° r e v e r y OL and we have proved that (p G Lx(C[Xi,...,X3\m®cAM,x)' Finally lemma (9.1.13) implies that AM —

(AM)/-

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9. TRACE OF / A ON THE DISCRETE SPECTRUM

13

COROLLARY (9.1.16). — An irreducible representation of M(A) is automorphic if and only if it is isomorphic to a subquotient of (AM,XI RMIX) for some x £ %M {this x, if it exists, is the central character of the representation and is thus uniquely determined).

Here we again denote by RM,X ^ ne restriction of RM,X ^° ^M,X ^ ^M x' Proof: The "if part is obvious. Conversely, let (V, TT) be an automorphic irreducible representation of M(A). It is admissible (see (9.1.13)) and trivial on ZM(F). Therefore we have V = Vx,i for some x € %M (see (9.1.13)). Let us choose admissible subrepresentations

such that (V, TT) is isomorphic to

(see (9.1.3)). Replacing Wi and W2 by (Wi) x , m and (W2)x,m respectively for some sufficie large enough positive integer m we may assume that

for some m (see (9.1.15); choose m such that the image of (W 2 ) x , m in V is non-zero). Let Kf be a compact open subgroup of M(A) such that VK ^ (0). Then we have Wf ^ (0) (the functor (-)K' is exact, see (D.I.5)). Let (Wg,/^) b e a subrepresentation of (W2,p2) such that W 2 K maps onto VK and has the smallest possible dimension for this property. Let (W^, p2) be the intersection of all the subrepresentations of (W2, p2) containing W 2 K . Then if (W^", p2") is a proper subrepresentation of (W^, p2) the map (W^7, p 2 0 —> (V, TT) is zero (otherwise it would be surjective and (W 2 ;/ )^ —• VfK would be surjective too, so that the inclusion

(Wt')K> c {Wt)K> = (W2)K' would be an equality and we would have a contradiction). Therefore, replacing (W2,/92) by {W^p'i) and (Wi,pi) by its intersection with (W2',p2) w e may assume that any proper subrepresentation of (W 2 ,p 2 ) is contained in (Wi,pi). Now to prove the corollary it is sufficient to construct a non-zero homomorphism (W2^P2) —> (AM,XI ^MIX)' -^ ut aPP^ymg proposition (9.1.15) we obtain a non-zero homomorphism (W2, p2) —> (C[Xi,..., Xs]m ® c AM,X, (r o det M ) ® RMIX) Downloaded from https://www.cambridge.org/core. University College London (UCL), on 29 Dec 2017 at 03:40:18, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511661969.002

14

DRINFELD MODULAR VARIETIES

and we leave it to the reader to check that there is at least one filtration (0) = (U0,a0) C (Wi,(7i) C • • • C (UL,aL) = (C[*i, • • • ,Xs)m,r

odetM)

such that the successive subquotients (Ue,ae)/(Ue-i,at-i)

(£ =

1,...,L)

are isomorphic to the trivial representation (C, 1) of M(A). By the proof of the corollary is completed. •

(9.2) Cuspidal automorphic representations Let M = Mi be a standard Levi subgroup of GLd as in (9.1). Let P' C M be a standard parabolic subgroup of M with its standard Levi decomposition P ' = M'N', P' = P j , NT = Mj and N' = N$ = Nj n Mx for some J C /. LEMMA

(9.2.1). — The topological space Nf(F)\N'(A)

is compact

Proof: Let R^ and K^, be the sets of positive roots for (M, T,B D M) and (M', T , B n Mf) and for each (3 = e{ - €j e R^ - R^, let

be the corresponding 1-parameter subgroup (JS^- is the elementary matrix with all entries 0 except the entry on the z-th row and the j-th column which is equal to 1). Then we have

There exists a total ordering /?i < fa < • • * < PL on K^ — K^, such that, for each £ = 0 , 1 , . . . ,L,

is a normal closed algebraic subgroup of N'. As V^/Vt-i is isomorphic to G a for £ = 1,..., L and as F\A is compact (the group F\A/O = H1(X,OX) is finite), Vg(F)\Vg(A) is compact for £ = 1,..., L (induction on £) and the lemma is proved (VL = iV7). • Downloaded from https://www.cambridge.org/core. University College London (UCL), on 29 Dec 2017 at 03:40:18, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511661969.002

9. TRACE OF / A ON THE DISCRETE SPECTRUM

15

For any (p G C^ we set (9.2.2)

(mA)

where dnfA is the Haar measure on Nf(A) which is normalized by vol(iV(A) fi Kz, dnfA) = 1 and where dv1 is the counting measure on Nf(F). The function C

is called the constant term of (p along P'. It is invariant under right translation by some compact open subgroup of M(A). The function ip G C^ is said to be cuspidal if cusp (resp. AM,CUSP) where we have set ' ' u M,cusp

(resp. **

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DRINFELD MODULAR VARIETIES

Proof : The lemma follows from remark (9.1.12): for any P(X) G C[Xi,... ,X S ], any ip G C^ and any standard parabolic subgroup P' of M we have D THEOREM (9.2.6) (Harder). — Let K' be a compact open subgroup of M(A). Then there exists an open subset CK> in M(A) such that ZM(&)M(F)CK'Kr = CW/, tfie quotient ZM(A)M(F)\CK'/Kf is finite and

Before proving the theorem we need to recall some basic results from reduction theory. Recall that M = Mi. For any integers c\ < c2 let

be the open subset of T(A) defined by the conditions deg(a(U)) < c2

(Va G J)

(resp. d < deg(a(tA)) < c2 (Va G /)). Let gx be the geometric genus of X, i.e. the genus of an arbitrary connected component of X wp k where k is an algebraic closure of ¥p. We have dhn¥p(H1(X,Ox)) = fgx. LEMMA

(9.2.7) (Harder). — For any integer c2 > 2gx we have M(A) = M(F)C//(A)T(A)f_oo?C2]M((9)

(recall that U1 = U n Afj).

D

A proof of (9.2.7) is given in (E.I.I). (9.2.8) (Harder). — For any compact open subgroup Kf of M(A) and any integer c2 there exists an integer c\ < c2 having the following property: if a E I and N' — Nj_ray we have LEMMA

TV'(A) = N'(F)(N'(A) fl

biK'ibiy1)

for any b{ = u{tA G BT(A) = f/7(A)T(A) with tA G T(A)]7_OO Ca] and deg(a(tA)) < ci.

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9. TRACE OF / A ON THE DISCRETE SPECTRUM

17

Proof : We use the same notations as in the proof of (9.2.1). For each (3 E R^i x^1{^') C A is an open subgroup and for each 6^ = ujj^A £ B J (A) - C/7(A)T(A) we have

x-^ibiK^bi)-1) = (3(tA)x^(Kf) C A. Let a e I and let us set M' = Mi_{ay If (3 G i ? ^ — ^ M ' > i-e- ^ ^ occurs in Lie (AT'), /3 — a is a sum of simple roots P — a = ai -\

\- ar

with a?i, • • •, ar G / and r < d — 1. Therefore we have deg(/J(*A)) < (d " 1) sup(c 2 ,0) + deg(a(t A ))

(VtA G T(A)f_OO)C2,).

If ZY C A is an open subgroup there exists 6A € A x such that b&O C.U and for any a A G A X we have

where the divisor i) on X is defined by D=

By the Serre duality we have

H\X,OX{-D)) - H°(X,^x/¥p(D)r

= (0)

as long as deg(D) - deg(a A 6 A ) < 2gx - 2. Therefore we have F + a A 6 A O = F + aAZV - A as long as deg(a A ) support, the representation of M(A) which is induced by RM on the C-vector space (

f^\

(poo\

\ r^noo

y Sjj \LM)x^(x)j ' IUM,cusp xe*M is admissible. In particular a function

0 for any (V,TT). THEOREM (9.3.7) (Harish-Chandra; Borel and Jacquet). — For any admissible irreducible representation (V, n) of M(A) the dimension of the C-vector space Hom RePs ( M ( A ))((V,7r), (CM

is finite.

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26

DRINFELD MODULAR VARIETIES

COROLLARY (9.3.8). — (i) For any smooth irreducible representation (V, TT) of M(A) which admits \ as central character m2 (TT) is finite. (ii) Let Tljrf be a system of representatives of the isomorphism classes 2 of L -automorphic irreducible representations of M(A) with central character X- Then the smooth representation (-4^ ^ d i s c , i ? ^ x d i s c ) of M(A) is (noncanonically) isomorphic to

Moreover, if we choose IIM,X,CUSP and ^2Mx in such a way that ITM,X,CUSP C ^ x , we can find an isomorphism between (*4^ x?disc ,i?^ x?disc ) and

such that the decomposition (9.3.5) corresponds to the decompositions m2(n)

= m c u s p (7r) + mr2es(7r)

(V(V, TT) G

U2Ma).

Proof of the corollary assuming the theorem : Part (i) is a direct consequence of (9.1.3) and (9.3.7). If we apply (D.6.9) to (A/w\x,res,^M,x,res) we get a (non-canonical) isomorphism (^M,x,res, i?M,X,res) =

0

(V, 7r

(thanks to part (i), m2es(7r) < m2(n) is finite) and together with (9.2.14) this implies part (ii) of the corollary. • We will deduce theorem (9.3.7) from the following stronger result. Let us fix a place x oi F. We consider the convolution algebra C (8) C^°(M(FX)) with respect to the Haar measure dmx which is normalized by vol(M(Ox), dmx) — 1. It acts on any smooth representation of M(FX) (see (D.I)). The restrictions of RM to the closed subgroups M(FX) and M(AX) of M(A) are smooth representations on C^. Therefore, if Jx is a left ideal of C 0 C^°(M(FX)) and if K'x is a compact open subgroup of M(AX) we may consider the C-vector subspace M[JX,&

J C LM

of the functions

C^, u ^

U(L(VX

® vx))

is injective (V is the C-linear span of TT(M(A))(L(VX®VX)) by irreducibility of (V,TT)) and its image is contained in C^[Jx,Kfx]. Therefore (9.3.7) follows from (9.3.9). • Before proving (9.3.9) let us give some properties of admissible left ideals (9.3.10). — Let Jx be an admissible left ideal o/C® C£ (i) There exist a compact open subgroup K'x of M(FX) and an ideal Xx of the group algebra C[ZM{FX)] of finite codimension and having the following property: for any smooth representation (Vx,nx) of M(FX) and any vector v x £ Vx which is annihilated by Jx we have LEMMA

vx e v 5 and *xPx)(vx) = (0) (C[ZM(FX)} C C[M(FX)] acts on Vx by nx). (ii) If P' is a standard parabolic subgroup of M with standard Levi decomposition P' = M'N' there exists an admissible left ideal J'x of C ® C^°(M/(FX)) having the following property: for any smooth representation (VxjTTa) of M(FX) and any vector vx e Vx which is annihilated by Jx the canonical image vx of vx in the Jacquet module V'X = VX/VX(N'(FX)) is annihilated by

KX(JX)-

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28

DRINFELD MODULAR VARIETIES

Proof: Let us set WX =

C®C?(M(FX))/JX

and let px be the natural representation of M(FX) on Wx (M(FX) acts by right translation on itself). Then (Wx,px) is admissible and there exists a compact open subgroup Kx of M(FX) such that, if we set e ^ = lK>x/vol(K'x,dgx), we have fx * eK>x ~fxejx

(V/x G C ® C~(M(FX)))

(pick an admissible representation (Vx,nx) of M(FX) and a vector vx G V^ such that Jx is the annihilator of vx, choose for Kx any compact open subgroup of M(FX) fixing vx and consider the monomorphism

Let w* = eK>x +JX

eWx.

Then K;^ is fixed by K'x and there exists an ideal Xx of C[ZM(FX)] of finite codimension such that Px(Xx)(wx) = (0) (we have C W 5 )• Px{C[ZM{Fx)}){wx) If (Vx, TTX) is a smooth representation of M(FX) and if u^ G Vx is annihilated by Jx then vx is fixed by K^,. Indeed vx is fixed by some open subgroup Kx of K'x and e^i - e/f" = ZKX' * e ^ - eK» G J^Therefore the morphism ^x : (WXJPX) -»• (Vx,7rx), fx+ Jx^>

ftx(fx)(vx),

maps i/;x onto ^x and we have *Tx(Z*)(v*) = «x(Px(Ia:)K)) = (0). This completes the proof of part (i). Let (Wx = WX/WX(N'(FX)), \f/x = px\M'{Fx) modulo WX(N'{FX)) Downloaded from https://www.cambridge.org/core. University College London (UCL), on 29 Dec 2017 at 03:40:18, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511661969.002

9. TRACE OF / A ON THE DISCRETE SPECTRUM

29

be the Jacquet module of (Wx,px) and let wx G Wx be the canonical image of wx. Then \vvx'>Px)

— \rM(Fx)

v9 \y">op>(Fx))

\yvxiPx))

is admissible (see (7.1.4)(i)) and the annihilator J'x oiw'x in C®C™(M'(FX)) is therefore admissible. By functoriality we get a morphism

which maps wx onto the canonical image vx of vx in V'x — VX/VX(N'(FX)). Therefore J'x annihilates v'x and part (ii) is also proved. • Proof o/(9.3.9) : First of all let us consider C OO

I" /T

T^fXl

M,cusp Wx )A

/^OOr

rr

T^fXl

J — U M [Jx ,J\

o

/7OO

J II ^M,cusp •

Let us fix a compact open subgroup K'x of M(FX) and an ideal Xx C of finite codimension as in (9.3.10)(i). Then there exists a function fi : XM -^ %>o with finite support such that C[ZM(FX))

c%[jx,K'x] c ( ©

(c%)xMx))n(c%)K>

where we have set ' = K'XK'X.

Indeed the group ZM(FX)ZM(F)\ZM(A)/{ZM(AX)

n Kfx)

is finite (for any compact open subgroup IAX of (A^)x the group F X X F X \A X /^ X is finite); therefore there exists an ideal of finite codimension, JA C C[ZM(F)\ZM(A)/(ZM(A)

n

K')],

which annihilates any (p in CM\JX,KIX\ and we can apply (9.1.5) (or more precisely its proof). Hence it follows from (9.2.10) that C^cusp[Jx,K/x] is finite dimensional over C. Next let us prove the theorem by descending induction on the integer s such that di — (di,..., ds) if M — M/. For s = dwe have C^ — C^ cusp and the theorem is already proved. Let us assume the theorem for any s' > s and Downloaded from https://www.cambridge.org/core. University College London (UCL), on 29 Dec 2017 at 03:40:18, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511661969.002

30

DRINFELD MODULAR VARIETIES

let us prove it for s (s < d). For each proper standard parabolic subgroup P' of M let Cp> be a system of representatives of the double cosets in P1(A)\M(A)/K'. It is finite (we have the Iwasawa decomposition M(A) = P'(A)M((9), [Jx,Kfx] is the intersection of the kernels of the C-

see (4.1)). Then C^c linear maps

JN lN'{F)\N'(h)

aL

for all CA G Cp' and all proper standard parabolic subgroups P' of M with standard Levi decomposition P' = M1 N' (dn'A and dv' are as in (9.2.2)). It is therefore sufficient to show that, for any given P' and c& G Cp>, the image of the corresponding C-linear map is finite dimensional. Replacing Jx by {mx i—• fx{c~lmxcx) x

x

/x

x

and K' by c K (c )~ morphism

1

| fx G Jx}

we may assume that CA(— CXCX) = 1. But the

factors through the Jacquet module ^ F , ) modulo Therefore, if J^ is an admissible left ideal of C®C™{M'(FX)) as in (9.3.10)(ii), J'x annihilates (pP,\M'(Fx) for all tp G C%}[JX,K'X]. It follows that v? «-» (pp/lAf 7 ^) maps C^[J X ,^ / X ] into ^ [ J ^ M ^ A ^ ) n K /x ] and, since this last space is finite dimensional over C by our induction hypothesis, the proof of the theorem is completed. • (9.4) One dimensional automorphic representations We denote by EM the abelian group of smooth complex characters f : M(A) —> C x which are trivial on M(F) and by det M : M(A) —> (Ax)s the group homomorphism defined by = (det(#A,i)> • • • > for all mA - (flA|i, • • -,0A,*) G M(A) = GLdl(A) x • • • x GLds(A) if M - M7 and dj = (d 1 ? ... ,d a ). Downloaded from https://www.cambridge.org/core. University College London (UCL), on 29 Dec 2017 at 03:40:18, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511661969.002

9. TRACE OF / A ON THE DISCRETE SPECTRUM

31

LEMMA (9.4.1). — The map

C i—• C °

det

M

induces a group isomorphism from the abelian group of smooth complex characters (:(FX\AX)S ^ C x onto Sjvf. Proof: We may assume that s — 1. Then d e t ^ = det is open, continuous and surjective and maps GLd(F) onto Fx (it admits the continuous section

, aA

Therefore, if £ is a complex character of A x , £ o det is smooth (resp. trivial on GLd(F)) if and only if £ is smooth (resp. trivial on Fx). Now to finish the proof of the lemma it is sufficient to show that any is trivial on Ker(det) = SLd(A). But SLd(A) is generated by U N A _ { a }(A) where a = e± - e2 G A (see the proof of (8.5.5)). Moreover, if we fix aA G A x such that O A - 1 G A X , for any

(resp. ' 1

0

V 2A

we then have 1 «A\ 0 ld-i)

=

(ak V0

0 \ (I U-i) \0

v A \ faA ld-i/ V 0

0 U-iJ

\0

(resp.

1

0 \ _faA

0 \ f 1

0 \ faA

0 ^"Y 1

0

0 where vA = uA/(aA — 1) (resp. of GLd(A) is trivial on SLd(A).

^A

=

«A^A/(1

— ctA)). Therefore any character D

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32

DRINFELD MODULAR VARIETIES

Any £ G S ^ may be viewed as a complex function on M(F)\M(A) is then an automorphic form for M. More precisely, if we set \ — we have x

and

It follows that (C, £) is an automorphic irreducible representation of M(A) with central character \ (it is isomorphic to the subrepresentation (C£, For each \ £ 8) = X ( ( « A , I ) D S • • •, (dA,s)Ds) for every (&A,s) G ( A X ) S , where we have identified Z^f(A) with (A x ) s in the usual way). As vol(Z M (A)M(F)\M(A),- d m A ^ is finite it follows that SM, X C A?M have

X

i

for any unitary x ^ XM- Therefore we C

^

for any unitary \ Now if x is unitary and if £ G SM, X let £CusP be the orthogonal projection of £ into the orthogonal direct summand *4M,X,CUSP °f ^ M X disc ( see (9-3.5)). We have #M,x,cusp(raA)(£cusp) = £(^A)£cusp

(VmA G M(A))

(the orthogonal projection of w4|^ x?disc onto ^4M,X,CUSP is M(A)-equivariant). Therefore, if there exists m^ G M(A) such that £ C US P (^A) / Owe have £CUSP(^A) 7^ 0 for every m& G M(A). But this contradicts (9.2.6) unless ZM(A)M(F)\M{A) is compact. If M g T, Z M (A)M(F)\M(A) cannot be compact and £cusp = 0 for any £ G S M)X , so that ^M,x,triv C A2Mxres for any unitary x ^ XM- Then the other assertions of part (iv) follows from parts (i) and (ii) which are already proved and from (D.6.6). •

(9.5) Trace of /A into the discrete spectrum For any admissible irreducible representation (V, TT) of M(A) we have a "unique" decomposition into a restricted tensor product

xe\x\

where, for each x G \X\, (V^,^) is an admissible irreducible representation of M(FX) and where dimc(Vx ) = 1 for almost every x G \X\ (for each x G |X| the Q-algebra

C?(M(FX)//M(OX)) = C Stoo of the Steinberg representation stoo (resp. of the vector 1 G C of the trivial representation l ^ ) (see the first step of the proof of (8.1.2)). Then J^ and J^ are admissible left ideals and C GLd [JLiKx*] a n d CGLd [J£» Kf\ a r e finite dimensional C-vector subspaces oiC%Ld (see (9.3.9)). But, applying (9.5.1), we obtain that, for any unitarizable admissible irreducible subrepresentation (V, TT) of {C(QLd, RGLd) (with oo^F^ = 1) such that we have C

00

GLd

The lemma follows (use (9.3.8) (ii)).

n

We can now intoduce the formal trace (9.5.3)

ftri^Ld,x,disc(/A)

=

E

"i2(7r)tr7r(/A

(this sum is finite thanks to (9.5.3) and (9.3.8)(i)). REMARK (9.5.4). — In fact Moeglin and Waldspurger have given a complete description of (A%LdXdisc,RtQLd disc ) in terms of the cuspidal automorphic forms for certain standard Levi subgroups M of GLd (see [Mo-Wa 1] (Introduction, Theoreme)). It follows from their result and from (9.2.10) that (^GLd,x,disc^GLd,x,disc) *s a n admissible representation of GLd(A). In particular the operator ^?GLd,x,disc(/A) n a s a well-defined trace (see (D.2)). Obviously this trace coincides with the above formal trace. It also follows from their result and from [Sha 1] (see (9.2.15)) that m2(7r) = 1

f l l ( V ) n ^ .

•

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36

DRINFELD MODULAR VARIETIES

(9.5.5). — Let \ G %GLd be such that xWZo = 1 and let (V?71") be an (admissible) irreducible subrepresentation of (A%LdX,R%LdX). (i) // its component at oo is isomorphic to the Steinberg representation (StoojStoo) of F£D\GLd(Foo) then the subspace V of A?GLd x is automatically contained in AGLd,x,cusp> In particular (V,TT) is a cuspidal automorphic irreducible representation o/GL^(A). (ii) If its component at oo is isomorphic to the trivial representation (C, loo) of F£\GLd(Foo) then the subspace V of A2GLdX is automatically equal to the 1-dimensional C-vector subspace C£ of A?GLdX for some £ G — 1- In particular (V,TT) is isomorphic to (C,£) for some THEOREM

COROLLARY

(9.5.6). — Let x € XGL* be such that x\F£> = 1 and let

(i) If the component at oo of (V, TT) is isomorphic to the Steinberg representation (StoojStoo) o/FcJ\GLrf(F00) then (V, TT) Z5 automatically cuspidal and 2 m

(7r) =m c u s p (7r).

(ii) // ^/ie component at oo o/ (V, TT) Z5 isomorphic to the trivial representation (C, loo) of F^GLdiFoo) then (V, TT) Z5 automatically isomorphic to (C,£) /or some