Automorphic Representations and L-Functions for the General Linear Group

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ISBN: 9780521474238 | 052147423X
Cover: Hardcover
Copyright: 5/30/2011

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This graduate-level textbook provides an elementary exposition of the theory of automorphic representations and L-functions for the general linear group in an adelic setting. The authors keep definitions to a minimum and repeat them when reintroduced so that the book is accessible from any entry point, and with no prior knowledge of representation theory. They also include concrete examples of both global and local representations of GL(n), and present their associated L-functions. The theory is developed from first principles for GL(1), then carefully extended to GL(2) with complete detailed proofs of key theorems. Several of the proofs are here presented for the first time, including Jacquet's simple and elegant proof of the tensor product theorem. Finally, the higher rank situation of GL(n) is given a detailed treatment. Containing numerous exercises, this book will motivate students and researchers to begin working in this fertile field of research.

Contents for Volume II	p. xiii
Introduction	p. xv
Preface to the Exercises	p. xix
Adeles over Q	p. 1
Absolute values	p. 1
The field Qp of p-adic numbers	p. 2
Adeles and ideles over Q	p. 7
Action of Q on the adeles and ideles	p. 8
p-adic integration	p. 12
p-adic Fourier transform	p. 15
Adelic Fourier transform	p. 18
Fourier expansion of periodic adelic functions	p. 23
Adelic Poisson summation formula	p. 30
Exercises for Chapter 1	p. 31
Automorphic representations and L-functions for GL (1,AQ)	p. 39
Automorphic forms for GL(1,AQ)	p. 39
The L-function of an automorphic form	p. 45
The local L-functions and their functional equations	p. 55
Classical L-functions and root numbers	p. 60
Automorphic representations for GL (1,AQ)	p. 65
Hecke operators for GL (1,AQ)	p. 68
The Rankin-Selberg method	p. 69
The p-adic Mellin transform	p. 70
Exercises for Chapter 2	p. 72
The classical theory of automorphic forms for GL (2)	p. 76
Automorphic forms in general	p. 76
Congruence subgroups of the modular group	p. 77
Automorphic functions of integral weight k	p. 78
Fourier expansion at ∞ of holomorphic modular forms	p. 80
Maass forms	p. 81
Whittaker functions	p. 84
Fourier-Whittaker expansions of Maass forms	p. 87
Eisenstein series	p. 89
Maass raising and lowering operators	p. 90
The bottom of the spectrum	p. 92
Hecke operators, oldforms, and newforms	p. 94
Finite dimensionality of the eigenspaces	p. 97
Exercises for Chapter 3	p. 98
Automorphic forms for GL (2,AQ)	p. 103
Iwasawa and Cartan decompositions for GL(2,R)	p. 103
Iwasawa and Cartan decompositions for GL (2,QP)	p. 105
The adele group GL(2,AQ)	p. 107
The action of GL (2,Q) on GL (2,AQ)	p. 108
The universal enveloping algebra of gl(2,C)	p. 112
The center of the universal enveloping algebra of gI(2,C)	p. 117
Automorphic forms for GL (2,AQ)	p. 117
Adelic lifts of weight zero, level one, Maass forms	p. 119
The Fourier expansion of adelic automorphic forms	p. 126
Global Whittaker functions for GL(2,AQ)	p. 128
Strong approximation for congruence subgroups	p. 134
Adelic lifts with arbitrary weight, level, and character	p. 136
Global Whittaker functions for adelic lifts with arbitrary weight, level, and character	p. 141
Exercises for Chapter 4	p. 147
Automorphic representations for GL(2,AQ)	p. 152
Adelic automorphic representations for GL (2,AQ)	p. 152
Explicit realization of actions defining a (g,K8)-module	p. 161
Explicit realization of the action of GL (2,Alinite)	p. 168
Examples of cuspidal automorphic representations	p. 172
Admissible (g,K8) x GL (2,Alinite)-modules	p. 173
Exercises for Chapter 5	p. 178
Theory of admissible representations of GL (2,QP)	p. 183
Short roadmap to chapter 6	p. 183
Admissible representations of GL(2,Qp)	p. 183
Ramified versus unramified	p. 192
Local representation coming from a level 1 Maass form	p. 193
Jacquet's local Whittaker function	p. 195
Principal series representations	p. 200
Jacquet's map: Principal series-->Whittaker functions	p. 205
The Kirillov model	p. 214
The Kirillov model of the principal series representation	p. 221
Haar measure on GL(2,Qp)	p. 228
The special representations	p. 232
Jacquet modules	p. 236
Induced representations and parabolic induction	p. 238
The supercuspidal representations of GL (2,QP)	p. 240
The uniqueness of the Kirillov model	p. 243
The Kirillov model of a supercuspidal representation	p. 252
The classification of the irreducible and admissible representations of GL (2,Qp)	p. 252
Exercises for Chapter 6	p. 253
Theory of admissible (g,K8) modules for GL (2,R)	p. 259
Admissible (g,K8)-modules	p. 259
Ramified versus unramified	p. 260
Jacquet's local Whittaker function	p. 260
Principal series representations	p. 263
Classification of irreducible admissible (g,K8)-modules	p. 269
Exercises for Chapter 7	p. 275
The contragredient representation for GL(2)	p. 277
The contragredient representation for GL(2,Qp)	p. 277
The contragredient representation of a principal series representation of GL(2,Qp)	p. 281
Contragredient of a special representation of GL (2,Qp)	p. 283
Contragredient of a supercuspidal representation	p. 285
The contragredient representation for GL (2,R)	p. 289
The contragredient representation of a principal series representation of GL(2,R)	p. 294
Global contragredients for GL (2,AQ)	p. 303
Integration on GL(2,AQ)	p. 306
The contragredient representation of a cuspidal automorphic representation of GL(2,AQ)	p. 311
Growth of matrix coefficients	p. 316
Asymptotics of matrix coefficients of (g,K8)-modules	p. 330
Matrix coefficients of GL (2,Qp) via the Jacquet module	p. 343
Exercises for Chapter 8	p. 353
Unitary representations of GL (2)	p. 358
Unitary representations of GL (2,Qp)	p. 358
Unitary principal series representations of GL (2,Qp)	p. 360
Unitary and irreducible special or supercuspidal representations of GL(2,Qp)	p. 364
Unitary (g,K8)-modules	p. 365
Unitary (g,K8)x GL(2,Afinite)modules	p. 368
Exercises for Chapter 9	p. 374
Tensor products of local representations	p. 378
Euler products	p. 378
Tensor product of (g,K8)-modules and representations	p. 379
Infinite tensor products of local representations	p. 381
The factorization of unramified irreducible admissible cuspidal automorphic representations	p. 383
Decomposition of representations of locally compact groups into finite tensor products	p. 388
The spherical Hecke algebra for GL (2,Qp)	p. 396
Initial decomposition of admissible (g,K8)x GL(2,Afinite)-modules	p. 403
The tensor product theorem	p. 406
The Ramanujan and Selberg conjectures for GL(2,AQ)	p. 413
Exercises for Chapter 10	p. 415
The Godenent-Jacquet L-function for GL (2,AQ)	p. 418
Historical remarks	p. 418
The Poisson summation formula for GL(2,AQ)	p. 419
Haar measure	p. 423
The global zeta integral for GL (2,AQ)	p. 425
Factorization of the global zeta integral	p. 430
The local functional equation	p. 432
The local L-function for GL(2, Qp) (unramified case)	p. 434
The local L-function for irreducible supercuspidal representations of GL(2,Qp)	p. 440
The local L-function for irreducible principal series representations of GZ(2,Qp)	p. 441
Local L-function for unitary special representations of GL(2,Qp)	p. 444
Proof of the local functional equation for principal series representations of GL(2,Qp)	p. 446
The local functional equation for the unitary special representations of GZ(2,Qp)	p. 450
Proof of the local functional equation for the supercuspidal representations of GL(2,Qp)	p. 452
The local L-function for irreducible principal series representations of GL(2,R)	p. 463
Proof of the local functional equation for principal series representations of GL(2,R)	p. 467
The local L-function for irreducible discrete series representations of GL(2,R)	p. 471
Exercises for Chapter 11	p. 474
Solutions to Selected Exercises	p. 478
References	p. 531
Symbols Index	p. 537
Index	p. 541
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Automorphic Representations and L-Functions for the General Linear Group

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