简介
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.
目录
Introduction ...................................... ........... 1
Notation .............................................. 3
Notes for the reader ........................................ 4
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. Well Groups and Representations ................... ....... 180
29. Local Class Field Theory ............... ............... 186
30. Existence of the Local Constant ....................... .. .190
31. Deligne Representations ............... ............... 200
32. Relation with i-adic Representations ....................... 201
8 The Langlands Correspondence ............................ 211
33. The Langlands Correspondence .................. ...... ..212
34. The Tame Correspondence ................................ 214
35. The ?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
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