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Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula SIMPLE ALGEBRAS, BASE CHANGE, AND THE ADVANCED THEORY OF THE TRACE FORMULA BY JAMES ARTHUR AND LAURENT CLOZEL ANNALS OF MATHEMATICS STUDIES PRINCETON UNIVERSITY PRESS Annals of Mathematics Studies Number 120 Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula 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 Mathematics Studies are edited by Luis A. Caffarelli, John N. Mather, John Milnor, and Elias M. Stein Clothbound editions of Princeton University Press books are printed on acid-free paper, and binding materials are chosen for strength and durability. Paperbacks, while satisfactory for personal collec- tions, are not usually suitable for library rebinding Printed in the United States of America by Princeton University Press, 41 William Street Princeton, New Jersey Library of Congress Cataloging-in-Publication Data Arthur, James, 1944- Simple algebras, base change, and the advanced theory of the trace formula / by James Arthur and Laurent Clozel. p. cm. - (Annals of mathematics 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. Title. III. Series. QA171.A78 1988 512'.2-dcl9 88-22560 CIP ISBN 0-691-08517-X (cl.) ISBN 0-691-08518-8 (pbk.) Contents Introduction vii Chapter 1. Local Results 3 1. The norm map and the geometry of u-conjugacy 3 2. Harmonic analysis on the non-connected group 10 3. Transfer of orbital integrals of smooth functions 20 4. Orbital integrals of Hecke functions 32 5. Orbital integrals: non-inert primes 48 6. Base change lifting of local representations 51 7. Archimedean case 71 Chapter 2. The Global Comparison 75 1. Preliminary remarks 75 2. Normalization factors and the trace formula 86 3. The distributions IM(-) and IM(7) 92 4. Convolution and the differential equation 100 5. Statement of Theorem A 107 6. Comparison of I (7, ) and IM(7, f) 111 7. Comparison of germs 116 8. The distributions IM('r,X) and If(ar,X) 124 9. Statement of Theorem B 131 10. Comparison of I (~r,X,f) and IM(r, X, f) 136 11. More on normalizing factors 146 12. A formula for If(f) 153 13. The map eM 160 14. Cancellation of singularities 167 15. Separation by infinitesimal character 179 16. Elimination of restrictions on f 187 17. Completion of the proofs of Theorems A and B 193 vi Contents Chapter 3. Base Change 199 1. Weak and strong base change: definitions 199 2. Some results of Jacquet and Shalika 200 3. Fibers of global base change 201 4. Weak lifting 202 5. Strong lifting 212 6. Base change lift of automorphic forms in cyclic extensions 214 7. The strong Artin conjecture for nilpotent groups 220 Bibliography 225 Introduction The general theory of automorphic forms is inl some ways still young. It is expected eventually to play a fundamental unifying role in a wide array of arithmetic questions. Much of this can be summarized as Langlands' functoriality principle. For two reductive groups G and G' over a number field F, and a map LG' __ LG between their L-groups, there should be an associated correspondence between their automorphic representations. The functoriality principle is very deep, and will not be resolved for a long time. There is an important special case offunctoriality which seems to be more accessible. It is, roughly speaking, the case that LG' is the group of fixed points of an automorphism of LG. In order that it be uniquely determined by its L-group, assume that G' is quasi-split. Then G' is called a (twisted) endoscopic group for G. Endoscopic groups were introduced by Langlands and Shelstad to deal with problems that arose originally in connection with Shimura varieties. Besides being a substantial case of the general question, a proper understanding of functoriality for endoscopic groups would be significant in its own right. It would impose an internal structure on the automorphic representations of G, namely a partition into "L-packets", which would be a prerequisite to understanding the nature of the general functoriality correspondence. However, the problem of functoriality for endoscopic groups appears accessible only in comparison with the general case. There are still a number of serious difficulties to be overcome. When the endoscopic group G' equals GL(2), Jacquet and Langlands [25], and Langlands [30(e)], solved the problem by using the trace formula for GL(2). In general, it will be necessary to deal simultaneously with a number of endoscopic groups G', namely the ones associated to those automorphisms of LGO which differ by an inner automorphism. One would hope to compare a (twisted) trace formula for G with some combination of trace formulas for the relevant groups G'. There now exists a (twisted) trace formula for general groups. The last few years have also seen progress on other questions, motivated by a comparison of trace formulas. The purpose viii Introduction of this book is to test these methods on the simplest case of general rank. We shall assume that G' equals the general linear group GL(n). A special feature of this case is that there is essentially only one endoscopic group to be considered. There are two basic examples. In the first case, G is the multiplicative group of a central simple algebra. Then G' is the endoscopic group associ- ated to the trivial automorphism of LGO GL(n, C). This is the problem of inner twistings of GL(n). In the second case, G is attached to the gen- eral linear group of a cyclic extension E of degree f over F. In order to have uniform notation, it will be convenient to write Go = RE/F(GL(n)) for the underlying group in this case, while reserving the symbol G for the component Go > 0 in a semidirect product. The trace formula attached to G is then just the twisted trace formula of G°, relative to the automorphism 0 associated to a generator of Gal(E/F). In this second case, the identity component of the L-group of Go is isomorphic to e copies of GL(n,C), and G' comes from the diagonal image of GL(n, C), the fixed point set of the permutation automorphism. This is the problem of cyclic base change for GL(n). In both cases we shall compare the trace formula of G with that 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 Chapter 2) is that these two sets of terms are equal. This means, more or less, that 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 f E C°(G(A)) on the subspace of L2(G0(F)\G0(A)1) which decomposes discretely. However, this is only one of several such col- lections of terms, which are parametrized by Levi components M in G. Together, they form the "discrete part" of the trace formula (1) Idisc,t(f) = disc,(f) II '"I 1IIW det(s - )M tr(M(s,O)pp,t(O,f)) M 11o sEW(aM)reg in which pp,, is a representation induced from the discrete spectrum of M, and M(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 Chapter 2 implies an identity between the discrete parts of the trace formulas of G and G'. We shall describe this more precisely. Introduction ix Let S be a finite set of valuations of F, which contains all the Archi- medean and ramified places. For each v E S, let f, be a fixed function in Cc(G(F,)). We then define a variable function f =IIfv v in C,(G(A)) by choosing functions {f, :v S} which are spherical (i.e. bi-invariant under the maximal compact subgroup of G°(F,)). For each valuation v not in S, the Satake transform provides a canonical map fA -- f' from the spherical functions on G(F.) to the spherical functions on G'(F,). Our results imply that there are fixed functions fv E C'(G'(Fv)) for the valuations v in S, with the property that if Vf= then (2) Idisc,t(f) = t( ) Given the explicit nature (1) of the distribution Id8Ct and the fact that the spherical functions {fv : v 0 S} may be chosen at will, we can see that the identity (2) will impose a strong relation between the automorphic representations of G and G'. In particular, we shall use it to establish global base change for GL(n). Chapter 1 is devoted to the correspondence fv -- fv. We shall also establish a dual correspondence between the tempered 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 Af, we shall show that there exists a function f: E C (G'(Fv)) whose orbital integrals match those of f, under the image of the norm map from G(Fv) to G'(Fv). 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.
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