The Local Langlands Conjecture for GL(2) contributes an unprecedented text to the so-called Langlands theory. It is an ambitious research program of already 40 years and 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.
Author(s): Colin J. Bushnell, Guy Henniart,
Edition: 1
Year: 2006
Language: English
Pages: 352
3540314865......Page 1
Contents......Page 7
Introduction......Page 10
Notation......Page 12
Notes for the reader......Page 13
1 Smooth Representations......Page 15
1. Locally Profinite Groups......Page 16
2. Smooth Representations of Locally Profinite Groups......Page 21
3. Measures and Duality......Page 33
4. The Hecke Algebra......Page 41
5. Linear Groups......Page 50
6. Representations of Finite Linear Groups......Page 52
3 Induced Representations of Linear Groups......Page 56
7. Linear Groups over Local Fields......Page 57
8. Representations of the Mirabolic Group......Page 63
9. Jacquet Modules and Induced Representations......Page 68
10. Cuspidal Representations and Coefficients......Page 76
10a. Appendix: Projectivity Theorem......Page 80
11. Intertwining, Compact Induction and Cuspidal Representations......Page 83
4 Cuspidal Representations......Page 91
12. Chain Orders and Fundamental Strata......Page 92
13. Classification of Fundamental Strata......Page 101
14. Strata and the Principal Series......Page 106
15. Classification of Cuspidal Representations......Page 111
16. Intertwining of Simple Strata......Page 117
17. Representations with Iwahori-Fixed Vector......Page 121
18. Admissible Pairs......Page 129
19. Construction of Representations......Page 131
20. The Parametrization Theorem......Page 135
21. Tame Intertwining Properties......Page 137
22. A Certain Group Extension......Page 140
6 Functional Equation......Page 143
23. Functional Equation for GL(1)......Page 144
24. Functional Equation for GL(2)......Page 153
25. Cuspidal Local Constants......Page 161
26. Functional Equation for Non-Cuspidal Representations......Page 168
27. Converse Theorem......Page 176
7 Representations of Weil Groups......Page 184
28. Weil Groups and Representations......Page 185
29. Local Class Field Theory......Page 191
30. Existence of the Local Constant......Page 195
31. Deligne Representations......Page 205
32. Relation with l-adic Representations......Page 206
8 The Langlands Correspondence......Page 215
33. The Langlands Correspondence......Page 216
34. The Tame Correspondence......Page 218
35. The l-adic Correspondence......Page 225
9 The Weil Representation......Page 229
36. Whittaker and Kirillov Models......Page 230
37. Manifestation of the Local Constant......Page 234
38. A Metaplectic Representation......Page 240
39. The Weil Representation......Page 249
40. A Partial Correspondence......Page 253
41. Imprimitive Representations......Page 255
42. Primitive Representations......Page 261
43. A Converse Theorem......Page 266
44. Ordinary Representations and Strata......Page 271
45. Exceptional Representations and Strata......Page 283
12 The Dyadic Langlands Correspondence......Page 288
46. Tame Lifting......Page 289
47. Interior Actions......Page 298
48. The Langlands-Deligne Local Constant modulo Roots of Unity......Page 300
49. The Godement-Jacquet Local Constant and Lifting......Page 307
50. The Existence Theorem......Page 310
51. Some Special Cases......Page 316
52. Octahedral Representations......Page 319
13 The Jacquet-Langlands Correspondence......Page 328
53. Division Algebras......Page 329
54. Representations......Page 331
55. Functional Equation......Page 334
56. Jacquet-Langlands Correspondence......Page 337
References......Page 341
G......Page 346
S......Page 347
W......Page 348
Some Common Symbols......Page 349
Some Common Abbreviations......Page 350
