The local Langlands conjecture for GL(2) / Colin J. Bushnell, Guy Henniart.

Format:

Book

Author/Creator:

Bushnell, Colin J. (Colin John), 1947-

Contributor:

Henniart, Guy.

Series:

Grundlehren der mathematischen Wissenschaften 0072-7830 ; 335.

Grundlehren der mathematischen Wissenschaften, 0072-7830 ; 335

Language:

English

Subjects (All):

Representations of groups.

L-functions.

Algebraic number theory.

Physical Description:

xi, 347 pages : illustrations ; 24 cm.

Place of Publication:

Berlin ; New York : Springer, [2006]

Summary:

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 groups and the structure theory of local fields. It uses only local methods, with no appeal to harmonic analysis on adele groups.

Contents:

1 Smooth Representations 7

1 Locally Profinite Groups 8

2 Smooth Representations of Locally Profinite Groups 13

3 Measures and Duality 25

4 The Hecke Algebra 33

2 Finite Fields 43

5 Linear Groups 43

6 Representations of Finite Linear Groups 45

3 Induced Representations of Linear Groups 49

7 Linear Groups over Local Fields 50

8 Representations of the Mirabolic Group 56

9 Jacquet Modules and Induced Representations 61

10 Cuspidal Representations and Coefficients 69

10a Appendix: Projectivity Theorem 73

11 Intertwining, Compact Induction and Cuspidal Representations 76

4 Cuspidal Representations 85

12 Chain Orders and Fundamental Strata 86

13 Classification of Fundamental Strata 95

14 Strata and the Principal Series 100

15 Classification of Cuspidal Representations 105

16 Intertwining of Simple Strata 111

17 Representations with Iwahori-Fixed Vector 115

5 Parametrization of Tame Cuspidals 123

18 Admissible Pairs 123

19 Construction of Representations 125

20 The Parametrization Theorem 129

21 Tame Intertwining Properties 131

22 A Certain Group Extension 134

6 Functional Equation 137

23 Functional Equation for GL(1) 138

24 Functional Equation for GL(2) 147

25 Cuspidal Local Constants 155

26 Functional Equation for Non-Cuspidal Representations 162

27 Converse Theorem 170

7 Representations of Weil Groups 179

28 Weil Groups and Representations 180

29 Local Class Field Theory 186

30 Existence of the Local Constant 190

31 Deligne Representations 200

32 Relation with [ell]-adic Representations 201

8 The Langlands Correspondence 211

33 The Langlands Correspondence 212

34 The Tame Correspondence 214

35 The [ell]-adic Correspondence 221

9 The Weil Representation 225

36 Whittaker and Kirillov Models 226

37 Manifestation of the Local Constant 230

38 A Metaplectic Representation 236

39 The Weil Representation 245

40 A Partial Correspondence 249

10 Arithmetic of Dyadic Fields 251

41 Imprimitive Representations 251

42 Primitive Representations 257

43 A Converse Theorem 262

11 Ordinary Representations 267

44 Ordinary Representations and Strata 267

45 Exceptional Representations and Strata 279

12 The Dyadic Langlands Correspondence 285

46 Tame Lifting 286

47 Interior Actions 295

48 The Langlands-Deligne Local Constant modulo Roots of Unity 297

49 The Godement-Jacquet Local Constant and Lifting 304

50 The Existence Theorem 307

51 Some Special Cases 313

52 Octahedral Representations 316

13 The Jacquet-Langlands Correspondence 325

53 Division Algebras 326

54 Representations 328

55 Functional Equation 331

56 Jacquet-Langlands Correspondence 334

Some Common Symbols 349

Some Common Abbreviations 351.

Notes:

Includes bibliographical references (pages [339]-343) and index.

ISBN:

3540314865

OCLC:

70886058

Publisher Number:

9783540314868