Langlands Correspondence for Loop Groups

Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 11
Preface......Page 13
Contents......Page 16
Acknowledgments......Page 18
1.1.1 Local non-archimedian fields......Page 19
1.1.2 Smooth representations of GLn (F)......Page 20
1.1.3 The Galois group......Page 21
1.1.4 The local Langlands correspondence for GLn......Page 22
1.1.5 Generalization to other reductive groups......Page 23
1.1.6 On the global Langlands correspondence......Page 24
1.2.1 The Galois group and the fundamental group......Page 27
1.2.2 Flat bundles......Page 29
1.2.3 Flat bundles in the holomorphic setting......Page 31
1.2.4 Flat G-bundles......Page 32
1.2.5 Regular vs. irregular singularities......Page 34
1.2.6 Connections as the Langlands parameters......Page 35
1.3.1 Smooth representations......Page 36
1.3.2 From functions to sheaves......Page 39
1.3.3 A toy model......Page 41
1.3.4 Back to loop groups......Page 44
1.3.5 From the loop algebra to its central extension......Page 45
1.3.6 Afine Kac–Moody algebras and their representations......Page 46
2.1.1 The case of simple Lie algebras......Page 49
2.1.2 The case of afine Lie algebras......Page 51
2.1.3 The afine Casimir element......Page 53
2.2 Basics of vertex algebras......Page 56
2.2.1 Fields......Page 57
2.2.2 Definition......Page 58
2.2.3 More on locality......Page 60
2.2.4 Vertex algebra associated to…......Page 62
2.2.5 Defining vertex operators......Page 64
2.2.6 Proof of the Theorem......Page 68
2.3 Associativity in vertex algebras......Page 70
2.3.2 Some useful facts......Page 71
2.3.3 Proof of associativity......Page 73
2.3.4 Corollaries of associativity......Page 75
3 Constructing central elements......Page 79
3.1.1 Commutation relations with…......Page 80
3.1.2 Relations between Segal–Sugawara operators......Page 83
3.1.3 The Virasoro algebra......Page 84
3.1.4 Conformal vertex algebras......Page 86
3.1.5 Digression: Why do central extensions keep appearing?......Page 87
3.2.1 Lie algebra of Fourier coefficients......Page 88
3.2.2 From vertex operators to the enveloping algebra......Page 91
3.2.3 Enveloping algebra associated to a vertex algebra......Page 92
3.3.1 Definition of the center......Page 94
3.3.2 The center of the affine Kac–Moody vertex algebra......Page 95
3.3.3 The associated graded algebra......Page 96
3.3.4 Symbols of central elements......Page 98
3.4.1 Generalities on schemes......Page 99
3.4.2 Definition of jet schemes......Page 100
3.4.3 Description of invariant functions on…......Page 103
3.5.1 First description......Page 105
3.5.2 Coordinate-independence......Page 107
3.5.3 The group of coordinate changes......Page 108
3.5.4 Action of coordinate changes......Page 109
3.5.5 Warm-up: Kac–Moody fields as one-forms......Page 111
3.5.6 The transformation formula for the central elements......Page 112
3.5.7 Projective connections......Page 114
3.5.8 Back to the center......Page 116
4.1.1 Projective structures......Page 119
4.1.2 PGL2-opers......Page 121
4.1.3 More on the oper condition......Page 123
4.2 Opers for a general simple Lie algebra......Page 125
4.2.1 Definition of opers......Page 126
4.2.2 Realization as gauge equivalence classes......Page 127
4.2.3 Action of coordinate changes......Page 128
4.2.4 Canonical representatives......Page 130
4.2.5 Alternative choice of representatives for…......Page 134
4.3.1 The center of the vertex algebra......Page 135
4.3.2 The center of the enveloping algebra......Page 138
4.3.3 The center away from the critical level......Page 142
5.1 Overview......Page 144
5.2 Finite-dimensional case......Page 146
5.2.1 Flag variety......Page 147
5.2.3 Verma modules and contragredient Verma modules......Page 149
5.2.4 Identification of…......Page 151
5.3.1 The infinite-dimensional Weyl algebra......Page 153
5.3.2 Action of…......Page 155
5.3.3 The Weyl algebra......Page 156
5.4 Vertex algebra interpretation......Page 158
5.4.1 Heisenberg vertex algebra......Page 159
5.4.2 More canonical definition of…......Page 160
5.4.3 Local extension......Page 162
5.5 Computation of the two-cocycle......Page 163
5.5.1 Wick formula......Page 164
5.5.2 Double contractions......Page 166
5.5.3 The extension is non-split......Page 167
5.5.4 A reminder on cohomology......Page 168
5.6.1 Clifford algebras......Page 169
5.6.2 The local Chevalley complex......Page 170
5.6.3 Another complex......Page 172
5.6.4 Restricting the cocycles......Page 175
5.6.5 Obstruction for other flag varieties......Page 177
6 Wakimoto modules......Page 179
6.1.1 Homomorphism of vertex algebras......Page 180
6.1.2 Other g-module structures on…......Page 182
6.2.1 Homomorphism of vertex algebras......Page 185
6.2.3 Conformal structures at non-critical levels......Page 188
6.2.4 Quasi-conformal structures at the critical level......Page 191
6.2.5 Transformation formulas for the fields......Page 192
6.2.6 Coordinate-independent version......Page 194
6.3 Semi-infinite parabolic induction......Page 196
6.3.2 The main result......Page 197
6.3.3 General parabolic subalgebras......Page 202
6.4 Appendix: Proof of the Kac–Kazhdan conjecture......Page 207
7.1 Strategy of the proof......Page 210
7.1.1 Finite-dimensional case......Page 211
7.1.2 Injectivity......Page 213
7.2.1 Vertex operators associated to a module over a vertex algebra......Page 215
7.2.2 The screening operator......Page 216
7.2.3 Friedan–Martinec–Shenker bosonization......Page 219
7.3.1 Parabolic induction......Page 223
7.3.2 Screening operators of the first kind......Page 224
7.3.3 Screening operators of the second kind......Page 225
7.3.4 Screening operators of second kind at the critical level......Page 226
7.3.5 Other Fourier components of the screening curents......Page 230
8 Identification of the center with functions on opers......Page 232
8.1.1 Computation of the character of......Page 233
8.1.2 The center and the classical W-algebra......Page 237
8.1.3 The appearance of the Langlands dual Lie algebra......Page 238
8.1.4 The vertex Poisson algebra structures......Page 239
8.1.5 module structures......Page 242
8.2.1 Miura opers......Page 244
8.2.2 Explicit realization of the Miura transformation......Page 247
8.2.3 Screening operators......Page 248
8.2.4 Back to the W-algebras......Page 250
8.2.5 Completion of the proof......Page 251
8.2.6 The associated graded algebras......Page 252
8.3 The center of the completed universal enveloping algebra......Page 255
8.3.1 Isomorphism between…......Page 256
8.3.2 The Poisson structure on…......Page 257
8.3.3 The Miura transformation as the Harish–Chandra homomorphism......Page 258
9 Structure of g-modules of critical level......Page 261
9.1.2 Residue......Page 262
9.1.3 Canonical representatives......Page 263
9.2 Nilpotent opers......Page 265
9.2.1 Definition......Page 266
9.2.2 Nilpotent opers and opers with regular singularity......Page 267
9.2.3 Residue......Page 271
9.2.4 More general forms for opers with regular singularity......Page 272
9.2.5 A characterization of nilpotent opers using the Deligne extension......Page 273
9.3.1 Connections and opers with regular singularities......Page 274
9.3.2 The case of integral dominant coweights......Page 276
9.3.3 Connections and Miura opers......Page 279
9.4 Categories of representations of at the critical level......Page 283
9.4.1 Compatibility with the finite-dimensional Harish–Chandra homomorphism......Page 284
9.4.2 Induction functor......Page 286
9.4.3 Wakimoto modules and categories of…......Page 288
9.5.1 Families of Wakimoto modules......Page 291
9.5.2 Verma modules and Wakimoto modules......Page 294
9.5.3 Description of the endomorphisms of Verma modules......Page 295
9.6 Endomorphisms of the Weyl modules......Page 297
9.6.2 Weyl modules and Wakimoto modules......Page 298
9.6.3 Nilpotent opers and Miura transformation......Page 300
9.6.4 Computations with jet schemes......Page 304
9.6.5 The invariant subspace in the associated graded space......Page 309
9.6.6 Completion of the proof......Page 311
10 Constructing the Langlands correspondence......Page 313
10.1 Opers vs. local systems......Page 315
10.2.1 Spaces of K-invariant vectors......Page 319
10.2.2 Equivariant modules......Page 320
10.2.3 Categorical Hecke algebras......Page 321
10.3.1 Unramified representations of......Page 323
10.3.2 Unramified categories of g -modulesb......Page 325
10.3.3 Categories of… equivariant modules......Page 328
10.3.4 Proof of Theorem 10.3.4......Page 329
10.3.5 The action of the spherical Hecke algebra......Page 334
10.3.6 Categories of representations and D-modules......Page 336
10.3.7 Equivalences between categories of modules......Page 341
10.3.8 Generalization to other dominant integral weights......Page 343
10.4 The tamely ramified case......Page 344
10.4.1 Tamely ramified representations......Page 345
10.4.2 Categories admitting …Harish–Chandra modules......Page 349
10.4.3 Conjectural description of the categories of… Harish–Chandra modules......Page 351
10.4.4 Connection between the classical and the geometric settings......Page 355
10.4.5 Evidence for the conjecture......Page 361
10.4.6 The case of…......Page 366
10.5 From local to global Langlands correspondence......Page 371
10.5.1 The unramified case......Page 372
10.5.2 Global Langlands correspondence with tame ramification......Page 377
10.5.3 Connections with regular singularities......Page 381
A.2 Universal enveloping algebras......Page 384
A.3 Simple Lie algebras......Page 386
A.4 Central extensions......Page 388
A.5 Affine Kac–Moody algebras......Page 389
References......Page 391
Index......Page 395