The Local Langlands Conjecture for GL(2)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 | |
Finite Fields 43 | 42 |
Induced Representations of Linear Groups | 49 |
Cuspidal Representations | 85 |
Parametrization of Tame Cuspidals | 123 |
Functional Equation | 137 |
Representations of Weil Groups 179 | 178 |
The Langlands Correspondence | 211 |
The LanglandsDeligne Local Constant modulo Roots of Unity | 297 |
The GodementJacquet Local Constant and Lifting | 304 |
The Existence Theorem | 307 |
Some Special Cases | 313 |
Octahedral Representations | 316 |
The JacquetLanglands Correspondence | 325 |
Division Algebras | 326 |
Representations | 328 |