Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula. (AM-120), Volume 120 9781400882403

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Annals of M athem atics Studies Number 120

Simple Algebras, Base Change, and the Advanced Theory of the Trace Form ula by James Arthur and Laurent Clozel

PRINCETON UNIVERSITY PRESS

PRINCETON, NEW JERSEY 1989

Copyright © 1989 by Princeton University Press ALL RIGHTS RESERVED

The Annals of M athem atics Studies are edited by Luis A. Caffarelli, John N . M ather, John Milnor, and Elias M. Stein

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L ib r a r y o f C o n g r e s s C a t a lo g in g -in - P u b lic a t io n D a t a

Arthur, James, 1944Simple algebras, base change, and the advanced theory of the trace formula / by James Arthur and Laurent Clozel. p.

cm. - (Annals of m athem atics studies ; no. 120)

Bibliography: p. ISBN 0-691-08517-X :

ISBN 0-691-08518-8 (pbk.)

1. Representations of groups. 2. Trace formulas. 3. Automorphic forms.

I. Clozel, Laurent, 1953-

. II. T itle. III. Series.

Q A171.A78 1988 512’.2 -d c l9

88-22560 CIP

ISBN 0-691-08517-X (cl.) ISBN 0-691-08518-8 (pbk.)

Contents

I n tr o d u c tio n C hapter 1. L o ca l R e s u lts 1. 2. 3. 4. 5. 6. 7.

The norm map and the geometry of 1 0 in a semidirect product. The trace formula attached to G is then ju st the twisted trace formula of G°, relative to the autom orphism 9 associated to a generator of G a l(F /F ). In this second case, the identity component of the L-group of G° is isomorphic to i copies of G L (n ,C ), and G' comes from the diagonal image of G L (n ,C ), the fixed point set of the perm utation automorphism. This is the problem of cyclic base change for GL(n). In both cases we shall compare the trace formula of G with th at of G'. For each term in the trace formula of G, we shall construct a companion term from the trace formula of G'. One of our main results (Theorems A and B of C hapter 2) is th at these two sets of term s are equal. This means, more or less, th a t there is a term by term identification of the trace formulas of G and G'. A key constituent in the trace formula of G comes from the right convo­ lution of a function / £ C£°(G (A )) on the subspace of L 2(G °(F )\G °(A )1) which decomposes discretely. However, this is only one of several such col­ lections of terms, which are param etrized by Levi components M in G. Together, they form the “discrete p a rt” of the trace formula

(i) w

/ )

= /d L ,« (/) =

E ll^ llll^ ir 1 J2 l|det(S - l ) (la | r 1 tr(M (5,0)ppit(0)/ ) ) ) M «€W(aAf)reg in which pp t is a representation induced from the discrete spectrum of M , and Af(s, 0) is an intertwining operator. (See §2.9 for a fuller description of the notation, and, in particular, the role of the real number t.) Theorem B of C hapter 2 implies an identity between the discrete parts of the trace formulas of G and Gf. We shall describe this more precisely.

Introduction

ix

Let 5 be a finite set of valuations of F , which contains all the Archi­ medean and ramified places. For each v E S', let f v be a fixed function in C£°(G(FV)). We then define a variable function

/= n* v in C £°(G (A )) by choosing functions { f v : v £ S } which are spherical (i.e. bi-invariant under the maximal compact subgroup of G 0(FV)). For each valuation v not in S , the Satake transform provides a canonical map f v —► / ' from the spherical functions on G(FV) to the spherical functions on G'(FV). Our results imply th at there are fixed functions / ' E C^°(G,(FV)) for the valuations v in 5 , with the property th at if r = n /:. V

then ( 2)

/ & , , ( / ) = ^ , «(/')•

Given the explicit nature (1) of the distribution and the fact th at the spherical functions {/„ : v £ 5 } may be chosen at will, we can see th a t the identity ( 2) will impose a strong relation between the automorphic representations of G and Gf. In particular, we shall use it to establish global base change for GL(n). C hapter 1 is devoted to the correspondence f v —►/ ' . We shall also establish a dual correspondence between the tem pered representations of G(FV) and G'(FV). For central simple algebras, the local correspondences have been established by Deligne, Kazhdan and Vigneras [15]. We can therefore confine ourselves to the case of base change. The correspondence is defined by comparing orbital integrals. For a given /„ , we shall show th a t there exists a function / ' E C£°(G'(FV)) whose orbital integrals m atch those of f v under the image of the norm map from G(FV) to G \ F V). At the p-adic places we shall do this in §1.3 by an argument of descent, which reduces the problem to the known case of a central simple algebra. The main new aspect of C hapter 1 is the proof in §1.4 th at the matching of orbital integrals is compatible with the canonical map of spherical functions. The proof is in two steps. We first define a space of “regular spherical functions” ; if one represents a spherical function as a finite Laurent series, they are defined by the condition th a t certain singular exponents do not occur. For these regular functions, the required identities of orbital integrals can be proved inductively by simple representation-theoretic arguments.

Introduction

X

An argument of density using the version of the trace formula due to Deligne and Kazhdan then shows th at the identities hold for all spherical functions. This argument relies in an essential way on a result of Kottwitz, which proves the identities of orbital integrals for units in the Hecke algebra. Once the comparison theory of spherical functions has been established, it will be easy to obtain the local correspondence of tempered representations (§1.5). It takes the familiar form of a lifting from the representations of G'(FV) to the representations of G 0(irv) th at are fixed by 9. We shall also prove identities between local L-functions and ^-factors related by lifting (§1.6). For the Archimedean places, the local lifting of representations is already known ([32], [11(a)]). We shall establish the matching of orbital integrals, as well as a Paley-W iener theorem, in §1.7. In C hapter 2 we shall compare two trace formulas. The trace formula £ |W o M| | W f r 1 .

M

,

£ 7 € ( M ( F ) ) m>s

t

w) =

£

£

| W

o

M | K

r

1

/

for G will be matched with a formula

£ | w 0Ml K r 1 M (3)£

£

7 € (M (F ))

£t £M iWtf'MWtfr1 /

aM'£ ( S , 7 ) I £M ( 7 , f ) = m

,s

aM>£ ( * ) l £M ( w , f ) d i r

obtained by pulling back the trace formula from Gf to G. Theorem A establishes an identification of the geometric terms on the left-hand sides of the two formulas, while Theorem B gives parallel identities for the spectral term s on the right. (It is the identity of global spectral term s aG,e(ir) and aG( 7r) which gives the equation ( 2), and leads to the global correspondence of autom orphic represntations.) The two theorems will be proved together by means of an induction argument. We shall assume th at all the identities hold for groups of strictly lower dimension. This hypothesis will actually be needed in §2.12 to construct the right-hand side of (3)^. It will also give us considerable scope for various descent arguments. These arguments lead to the identity of aM>€ ( j ) and aM ( j ) in most cases (§2.5), of aM'€ (7c) and aM ( 7r) in most cases (§2.9), and of I M,e(TTjf) and J M( 7r , / ) in all cases (§2.10). They also provide partial information relating 7M,£:( 7 ,/ ) and I M ( j yf ) (§2.5, §2.6, §2.7). However, some intractible terms remain in the end, and these must be handled by different m ethods. In §2.13 and

xi

Introduction §2.14, we shall show th a t for suitable / , 7 € M(Fs),

is the orbital integral in 7 of a function on M(Fs)- This allows us to apply the trace formula for M . We obtain a relation between the spectral sides of (3), of (3)5 , and of the trace formula for M . By comparing the resulting distributions at both the Archimedean and discrete places, we are then able to deduce vanishing properties for the individual term s (§2.15, §2.16). We shall finally complete the induction argument, and the proofs of the two theorems, in §2.17. As an application of the identity (2), we shall establish base change for GL(n) in C hapter 3. For GL(2), the complete spectral decomposition of the space of autom orphic forms is known, and this makes it possible to compare very explicitly the discrete spectra of GL(2, A f ) and GL(2, A # ). Such explicit information is not available for n > 3. If it were, and in particular, if there was a strong enough version of multiplicity one, we would have no trouble deducing all the results on base change directly from the formula (2). We must instead restrict the category of automorphic representations considered to those th at are “induced from cuspidal” , a natural notion coming from the theory of Eisenstein series. To prove th a t the lifting exists, and preserves this special kind of automorphic forms, we use (2) in combination with the very precise results obtained by Jacquet and Shalika about the analytic behavior of L-functions associated to pairs of autom orphic representations. Assume th a t E / F is a cyclic extension of number fields, of prime degree t, with Galois group { l , a ,