The Local Langlands Conjecture for GL(2)

Hardcover | 2 August 2006

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If F is a non-Archimedean local field, local class field theory can be viewed as giving a canonical bijection between the characters of the multiplicative group GL(1, F) of F and the characters of the Weil group of F. If n is a positive integer, the n-dimensional analogue of a character of the multiplicative group of F is an irreducible smooth representation of the general linear group GL(n, F). The local Langlands Conjecture for GL(n) postulates the existence of a canonical bijection between such objects and n-dimensional representations of the Weil group, generalizing class field theory.

This conjecture has now been proved for all F and n, but the arguments are long and rely on many deep ideas and techniques. This book gives a complete and self-contained proof of the Langlands conjecture in the case n=2. It is aimed at graduate students and at researchers in related fields. It presupposes no special knowledge beyond the beginnings of the representation theory of finite groupsand the structure theory of local fields. It uses only local methods, with no appeal to harmonic analysis on adele groups.

Industry Reviews

From the reviews:

"In this book the authors present a complete proof of the Langlands conjecture for GL (2) over a non-archimedean local field, which uses local methods and is accessible to students. ... The book is very well written and easy to read." (J. G. M. Mars, Zentralblatt MATH, Vol. 1100 (2), 2007)

"The book under review gives a complete and self-contained insight into the theory of representations of G. ... We highly recommend this book to Ph.D. students as well as to specialists. The book contains a huge amount of information, definition and facts ... . The book has a Bibliography containing 91 references ... ." (Alexandru Ioan Badulescu, Mathematical Reviews, Issue 2007 m)

"The aim of this monograph is to present a complete and self-contained proof of the Langlands conjecture for GL(2) over a non-archimedean local field. ... This volume presents a large amount of difficult material in a clear and readable manner. It can be recommended to anyone interested in representations of linear algebraic groups." (Ch. Baxa, Monatshefte fuer Mathematik, Vol. 154 (4), August, 2008)

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Introduction	p. 1
Notation	p. 3
Notes for the reader	p. 4
Smooth Representations	p. 7
Locally Profinite Groups	p. 8
Smooth Representations of Locally Profinite Groups	p. 13
Measures and Duality	p. 25
The Hecke Algebra	p. 33
Finite Fields	p. 43
Linear Groups	p. 43
Representations of Finite Linear Groups	p. 45
Induced Representations of Linear Groups	p. 49
Linear Groups over Local Fields	p. 50
Representations of the Mirabolic Group	p. 56
Jacquet Modules and Induced Representations	p. 61
Cuspidal Representations and Coefficients	p. 69
Appendix: Projectivity Theorem	p. 73
Intertwining, Compact Induction and Cuspidal Representations	p. 76
Cuspidal Representations	p. 85
Chain Orders and Fundamental Strata	p. 86
Classification of Fundamental Strata	p. 95
Strata and the Principal Series	p. 100
Classification of Cuspidal Representations	p. 105
Intertwining of Simple Strata	p. 111
Representations with Iwahori-Fixed Vector	p. 115
Parametrization of Tame Cuspidals	p. 123
Admissible Pairs	p. 123
Construction of Representations	p. 125
The Parametrization Theorem	p. 129
Tame Intertwining Properties	p. 131
A Certain Group Extension	p. 134
Functional Equation	p. 137
Functional Equation for GL(1)	p. 138
Functional Equation for GL(2)	p. 147
Cuspidal Local Constants	p. 155
Functional Equation for Non-Cuspidal Representations	p. 162
Converse Theorem	p. 170
Representations of Weil Groups	p. 179
Weil Groups and Representations	p. 180
Local Class Field Theory	p. 186
Existence of the Local Constant	p. 190
Deligne Representations	p. 200
Relation with [ell]-adic Representations	p. 201
The Langlands Correspondence	p. 211
The Langlands Correspondence	p. 212
The Tame Correspondence	p. 214
The [ell]-adic Correspondence	p. 221
The Weil Representation	p. 225
Whittaker and Kirillov Models	p. 226
Manifestation of the Local Constant	p. 230
A Metaplectic Representation	p. 236
The Weil Representation	p. 245
A Partial Correspondence	p. 249
Arithmetic of Dyadic Fields	p. 251
Imprimitive Representations	p. 251
Primitive Representations	p. 257
A Converse Theorem	p. 262
Ordinary Representations	p. 267
Ordinary Representations and Strata	p. 267
Exceptional Representations and Strata	p. 279
The Dyadic Langlands Correspondence	p. 285
Tame Lifting	p. 286
Interior Actions	p. 295
The Langlands-Deligne Local Constant modulo Roots of Unity	p. 297
The Godement-Jacquet Local Constant and Lifting	p. 304
The Existence Theorem	p. 307
Some Special Cases	p. 313
Octahedral Representations	p. 316
The Jacquet-Langlands Correspondence	p. 325
Division Algebras	p. 326
Representations	p. 328
Functional Equation	p. 331
Jacquet-Langlands Correspondence	p. 334
References	p. 339
Index	p. 345
Some Common Symbols	p. 349
Some Common Abbreviations	p. 351
Table of Contents provided by Ingram. All Rights Reserved.

The Local Langlands Conjecture for GL(2)

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