The Local Langlands Conjecture for GL(2) (eBook)

Foreword 6
Contents 7
Introduction 10
1 Smooth Representations 15
1. Locally Pro.nite Groups 16
1.1. 16
2. Smooth Representations of Locally Pro.nite Groups 21
3. Measures and Duality 33
4. The Hecke Algebra 41
2 Finite Fields 50
5. Linear Groups 50
6. Representations of Finite Linear Groups 52
3 Induced Representations of Linear Groups 56
7. Linear Groups over Local Fields 57
8. Representations of the Mirabolic Group 63
9. Jacquet Modules and Induced Representations 68
10. Cuspidal Representations and Coe.cients 76
10a. Appendix: Projectivity Theorem 80
11. Intertwining, Compact Induction and Cuspidal 83
Representations 83
4 Cuspidal Representations 91
12. Chain Orders and Fundamental Strata 92
13. Classi.cation of Fundamental Strata 101
14. Strata and the Principal Series 106
15. Classi.cation of Cuspidal Representations 111
16. Intertwining of Simple Strata 117
17. Representations with Iwahori-Fixed Vector 121
5 Parametrization of Tame Cuspidals 129
18. Admissible Pairs 129
19. Construction of Representations 131
20. The Parametrization Theorem 135
21. Tame Intertwining Properties 137
22. A Certain Group Extension 140
6 Functional Equation 143
23. Functional Equation for GL(1) 144
24. Functional Equation for GL(2) 153
25. Cuspidal Local Constants 161
26. Functional Equation for Non-Cuspidal 168
Representations 168
27. Converse Theorem 176
7 Representations of Weil Groups 184
28. Weil Groups and Representations 185
29. Local Class Field Theory 191
30. Existence of the Local Constant 195
31. Deligne Representations 205
32. Relation with 206
adic 206
Representations 206
8 The Langlands Correspondence 215
33. The Langlands Correspondence 216
34. The Tame Correspondence 218
35. The 225
adic 225
Correspondence 225
9 The Weil Representation 229
36. Whittaker and Kirillov Models 230
37. Manifestation of the Local Constant 234
38. A Metaplectic Representation 240
39. The Weil Representation 249
40. A Partial Correspondence 253
10 Arithmetic of Dyadic Fields 255
41. Imprimitive Representations 255
42. Primitive Representations 261
43. A Converse Theorem 266
11 Ordinary Representations 271
44. Ordinary Representations and Strata 271
45. Exceptional Representations and Strata 283
12 The Dyadic Langlands Correspondence 288
46. Tame Lifting 289
47. Interior Actions 298
48. The Langlands-Deligne Local Constant modulo 300
Roots of Unity 300
49. The Godement-Jacquet Local Constant and Lifting 307
50. The Existence Theorem 310
51. Some Special Cases 316
52. Octahedral Representations 319
13 The Jacquet-Langlands Correspondence 328
53. Division Algebras 329
54. Representations 331
55. Functional Equation 334
56. Jacquet-Langlands Correspondence 337
References 341
Index 346
Some Common Symbols 349
Some Common Abbreviation 350