The Local Langlands Conjecture for GL(2) (Grundlehren der mathematischen Wissenschaften Book 335)

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. 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

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.

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. We work with a non-Archimedean local field F which, we always assume, has finite residue field of characteristic p.