Affine flag manifolds are infinite dimensional versions of familiar objects such as Gra?mann varieties. The book features lecture notes, survey articles, and research notes - based on workshops held in Berlin, Essen, and Madrid - explaining the significance of these and related objects (such as double affine Hecke algebras and affine Springer fibers) in representation theory (e.g., the theory of symmetric polynomials), arithmetic geometry (e.g., the fundamental lemma in the Langlands program), and algebraic geometry (e.g., affine flag manifolds as parameter spaces for principal bundles). Novel aspects of the theory of principal bundles on algebraic varieties are also studied in the book.
Author(s): Alexander Schmitt
Series: Trends in Mathematics
Edition: 1st Edition.
Publisher: Springer Basel
Year: 2010
Language: English
Pages: 303
Cover......Page 1
Affine Flag Manifolds
and Principal Bundles......Page 4
ISBN 9783034602877......Page 5
Table of Contents......Page 6
Preface......Page 10
1. Introduction......Page 15
1.1. Notation......Page 16
2.1. Ind-schemes......Page 17
2.2. The loop group......Page 19
2.3. Lattices......Page 20
2.4. The affine Grassmannian for GLn......Page 22
2.5. The affine Grassmannian for an arbitrary linear algebraic group......Page 24
2.6. Decompositions......Page 25
2.8. The Bruhat–Tits building......Page 26
3.1. Springer fibers......Page 29
3.2. Affine Springer fibers......Page 30
3.3. General properties......Page 31
3.4.1. SL2......Page 33
3.5. Purity......Page 34
3.7. Equivariant cohomology......Page 35
3.8. The fundamental lemma......Page 38
4.1. Deligne–Lusztig varieties......Page 39
4.2. σ-conjugacy classes......Page 40
4.3. Affine Deligne–Lusztig varieties: the hyperspecial case......Page 43
4.4. Affine Deligne–Lusztig varieties: the Iwahori case......Page 48
4.5. The rank 2 case......Page 52
4.6. Type A2......Page 53
4.8. Type G2......Page 54
4.9. Relationship to Shimura varieties......Page 58
4.10. Local Shtuka......Page 59
4.11. Cohomology of affine Deligne–Lusztig varieties......Page 60
References......Page 61
Quantization of Hitchin’s Integrable System and the Geometric Langlands Conjecture......Page 65
1. D-modules on stacks......Page 67
2. Chiral algebras......Page 69
3. Geometry of the affine Grassmannian......Page 73
4.1. Convolution product......Page 76
4.2. Hecke stacks and Hecke functors......Page 80
4.3. Statement of Hecke eigenproperty......Page 82
5. Opers......Page 83
6. Constructing D-modules......Page 87
7. Hitchin integrable system I: definition......Page 90
8. Localization functor......Page 92
9. Quantum integrable system h......Page 95
10. Hitchin integrable system II: D-algebras......Page 97
11. Quantization condition......Page 99
12. Proof of the Hecke eigenproperty......Page 100
References......Page 102
1. Outline of the construction......Page 105
2.1. Notation......Page 106
2.2. The Picard torus and the Poincar´e line bundle......Page 107
2.3. Stability......Page 109
2.4. Properties of vector bundles on algebraic curves......Page 110
3. A nice over-parameterizing family......Page 112
4. The generalized Θ-divisor......Page 114
4.2. The invariant sections......Page 115
4.3. The multiplicative structure......Page 117
5.1. The case of genus zero and one......Page 118
5.2. Preparations for the proof of 5.1......Page 119
5.3. A proof for genus two using the rigidity theorem......Page 120
5.4. A proof based on Clifford’s theorem......Page 121
5.5. Generalizations and consequences......Page 122
6.1. Limits of vector bundles......Page 123
Construction: Elementary transformation......Page 124
6.4. Semistable limits exist......Page 125
6.5. Semistable limits are almost uniquely determined......Page 126
7.1. Notation and preliminaries......Page 127
7.2. Positivity – global sections of O(Θ)......Page 129
7.3. The case of deg(OC(ΘC)) = 0......Page 130
8.1. Constructing the moduli space of vector bundles......Page 131
8.3. Generalization to the case of arbitrary rank and degree......Page 132
9. Prospect to higher dimension......Page 133
References......Page 135
Introduction......Page 137
1.1. Motivation and definition......Page 138
1.2. How to make this geometric?......Page 142
2.1. Properties of stacks and morphisms......Page 145
2.2. Sheaves on stacks......Page 149
Lecture 3: Relation with coarse moduli spaces......Page 150
3.1. Coarse moduli spaces......Page 151
Lecture 4: Cohomology of Bund......Page 155
4.1. First step: Independence of the generators......Page 157
4.2. Second step: Why is it the whole ring?......Page 158
Lecture 5: The cohomology of the coarse moduli space (coprime case)......Page 160
References......Page 166
1. Introduction......Page 169
2. Algebraicity......Page 170
3. Lifting principal bundles......Page 171
4. Smoothness......Page 172
5. Connected components......Page 173
References......Page 177
1. Introduction......Page 179
2. Definition of γn and γ'n......Page 183
3. Mercat’s conjecture......Page 185
4. The invariants dr......Page 189
5. Rank two......Page 200
6. Ranks three and four......Page 202
7. Rank five......Page 204
8. Plane curves......Page 209
9. Problems......Page 212
References......Page 214
1. Classical finiteness results: The case of a curve......Page 217
2. Locally free sheaves of modules over Azumaya algebras: The case of a surface......Page 222
3. Elementary modifications and connectivity......Page 225
References......Page 231
1. Introduction......Page 233
2. N=4 supersymmetric gauge theory......Page 234
3. S-duality......Page 235
4. Topological twisting......Page 236
5. Dimensional reduction......Page 237
6. Wilson operators......Page 238
7. Mirror symmetry......Page 240
8. Higher-dimensional operators......Page 242
9. The six-dimensional view......Page 243
10. Conclusion......Page 244
References......Page 245
Introduction......Page 247
1.1.2. Direct and inverse 2-limits......Page 249
1.2.1. Background......Page 250
1.2.2. Remark......Page 251
1.2.6. Proposition......Page 252
1.2.9. Definitions......Page 253
1.2.12. Derived direct image......Page 254
1.2.18. Remarks......Page 255
1.2.23......Page 256
1.3.3. Remarks......Page 257
1.3.6. Remarks......Page 258
1.4.5. Definitions......Page 259
1.5.2. Definition......Page 260
1.5.4. Remarks......Page 261
1.5.6. Proposition......Page 262
1.5.8. Compatibility of the derived functors......Page 263
1.5.9. Lemma......Page 264
1.5.12. Admissible ind-coherent ind-schemes and reduction of the group action......Page 265
1.5.14. Thom isomorphism and pro-finite-dimensional vector bundles over indschemes......Page 266
1.5.15. Descent and torsors over ind-schemes......Page 267
1.5.16. Remark......Page 268
1.5.20......Page 269
2.1.1.......Page 270
2.1.6......Page 271
2.1.9......Page 272
2.2.1. The affine flag manifold......Page 273
2.2.3. Proposition......Page 274
2.2.6. Group actions on flag varieties and related objects......Page 275
2.3.3. Examples......Page 276
2.3.5. Examples......Page 277
2.3.6. Convolution product on KI(N)......Page 278
2.3.7. Proposition......Page 279
2.3.9. Remarks......Page 281
2.4.1. Definition of the concentration map rΣ......Page 282
2.4.3. Proposition......Page 283
2.4.6. Concentration of O-modules supported on Nsα......Page 284
2.4.8. Multiplicativity of rΣ.......Page 285
2.4.9. Proposition......Page 286
2.5.3. Remark......Page 289
2.5.6. Theorem......Page 290
3.1.1. Convolution algebras and schemes which are locally of finite type......Page 293
3.1.4. Admissible modules over the convolution algebra......Page 294
3.2.1. From O(H) to modules over the convolution algebra of M......Page 295
3.2.3. Proposition......Page 296
3.2.6. Lemma......Page 297
3.2.8. Lemma......Page 299
3.2.9. The classification theorem......Page 300
References......Page 301
