The Descent Map from Automorphic Representations of Gl(n) to Classical Groups

by Ginzburg, David; Rallis, Stephen; Soudry, David

• ISBN13: 9789814304986
• ISBN10: 9814304980
• Format: Hardcover
• Copyright: 2011-06-30
• Publisher: World Scientific Pub Co Inc

Summary

This book introduces the method of automorphic descent, providing an explicit inverse map to the (weak) Langlands functorial lift from generic, cuspidal representations on classical groups to general linear groups. The essence of this method is the study of certain Fourier coefficients of the GelfandGraev type, or of the FourierJacobi type to certain residual Eisenstein series. An account of this automorphic descent, with complete, detailed proofs, leads to a thorough understanding of important ideas and techniques. The book will be of interest to graduate students and mathematicians, who specialize in automorphic forms and in representation theory of reductive groups over local fields. Relatively self-contained, the content of some of the chapters can serve as topics for graduate students seminars.

Table of Contents

Preface	p. v
Introduction	p. 1
Overview	p. 1
Formulas for the Weil representation	p. 7
The case, where H is unitary and the place v splits in E	p. 10
On Certain Residual Representations	p. 17
The groups	p. 17
The Eisenstein series to be considered	p. 19
L-groups and representations related to P¿	p. 19
The residue representation	p. 21
The case of a maximal parabolic subgroup (r = 1)	p. 22
A preliminary lemma on Eisenstein series on GLn	p. 24
Constant terms of E (h, f¿,s)	p. 26
Description of W{M¿, D¿)	p. 27
Continuation of the proof of Theorem 2.1	p. 33
Coefficients of Gelfand-Graev Type, of Fourier-Jacobi Type, and Descent	p. 41
Gelfand-Graev coefficients	p. 41
Fourier-Jacobi coefficients	p. 43
Nilpotent orbits	p. 45
Global integrals representing L-functions I	p. 48
Global integrals representing L-functions II	p. 51
Definition of the descent	p. 52
Definition of Jacquet modules corresponding to Gelfand-Graev characters	p. 58
Definition of Jacquet modules corresponding to Fourier-Jacobi characters	p. 62
Some double coset decompositions	p. 65
The space Qj\h(V)k/Qz	p. 65
A set of representatives for Qj\h(V)k/Qe	p. 70
Stabilizers	p. 72
The set Q\h(Wm, l)k/Ll, wo	p. 75
Jacquet modules of parabolic inductions: Gelfand-Graev characters	p. 81
The case where K is a field	p. 81
The case K = k ⊕ k	p. 108
Jacquet modules of parabolic inductions: Fourier-Jacobi characters	p. 121
The case where K is a field	p. 121
The case K = k ⊕ k	p. 137
The tower property	p. 151
A general lemma on "exchanging roots"	p. 152
A formula for constant terms of Gelfand-Graev coefficients	p. 157
Global Gelfand-Graev models for cuspidal representations	p. 169
The general case: H is neither split nor quasi-split	p. 170
Global Gelfand-Graev models for the residual representations Er	p. 170
A formula for constant terms of Fourier-Jacobi coefficients	p. 172
Global Fourier-Jacobi models for cuspidal representations	p. 179
Global Fourier-Jacobi models for the residual representations Er	p. 183
Non-vanishing of the descent I	p. 187
The Fourier coefficient corresponding to the partition(m, m,m' _ 2m)	p. 188
Conjugation of Sm by the element ±_m	p. 194
Exchanging the roots y1,2 and x1,1 (dimEV = 2m , m > 2)	p. 198
First induction step: exchanging the roots yi, j and Xj-iti, for 1 < i	p. 201
First induction step: odd orthogonal groups	p. 210
Second induction step: exchanging the roots yi, j and xj-i, i, for i + j [3*P\ (dmiEV = 2m)	p. 215
Completion of the proof of Theorems 8.1, 8.2; dimEV = 2m	p. 226
Completion of the proof of Theorem 8.3	p. 228
Second induction step: odd orthogonal groups	p. 229
Completion of the proof of Theorems 8.1, 8.2; h(V) odd orthogonal	p. 233
Non-vanishing of the descent II	p. 235
The case HA = Sp4n+2(A)	p. 235
The case H = SO4n+1	p. 238
Whittaker coefficients of the descent corresponding to Gelfand Graev coefficients: the unipotent group and its character; h{V) $$$ S04n+1	p. 243
Conjugation by the element fjm	p. 245
Exchanging roots: h(V) = S04N, U4n	p. 247
Nonvanishing of the Whittaker coefficient of the descent corresponding to Gelfand-Graev coefficients: h(V) = S04n, U4n	p. 254
Nonvanishing of the Whittaker coefficient of the descent corresponding to Gelfand-Graev coefficients: h(V) = U4n+2, So4n+3	p. 262
The Whittaker coefficient of the descent corresponding to Fourier-Jacobi coefficients: Ha ^ Sp4n+2(A)	p. 268
The nonvanishing of the Whittaker coefficient of the descent corresponding to Fourier-Jacobi coefficients: HA=Sp4n(A)Sp4n(A), U4n(A)	p. 270
Nonvanishing of the Whittaker coefficient of the descent corresponding to Fourier-Jacobi coefficients: h(V) = U4n+2	p. 274
Global genericity of the descent and global integrals	p. 281
Statement of the theorems	p. 282
Proof of Theorem 10.3	p. 285
Proof of Theorem 10.4	p. 293
A family of dual global integrals I	p. 299
A family of dual global integrals II	p. 304
L-functions	p. 306
Langlands (weak) functorial lift and descent	p. 313
The cuspidal part of the weak lift	p. 314
The image of the weak lift	p. 316
On generalized endoscopy	p. 319
Base change	p. 330
Automorphic induction	p. 333
Bibliography	p. 335
Index	p. 339
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