Affine Flag Manifolds and Principal Bundles (eBook)

Alexander Schmitt (Herausgeber)

eBook Download: PDF

2011 | 2010
290 Seiten
Springer Basel (Verlag)
978-3-0346-0288-4 (ISBN)

Infos

Affine flag manifolds are infinite dimensional versions of familiar objects such as Graßmann 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.

Inhalt

Table of Contents 6
Preface 9
Affine Springer Fibers and Affine Deligne–Lusztig Varieties 13
1. Introduction 13
1.1. Notation 14
2. The affine Grassmannian and the affine flag manifold 15
2.1. Ind-schemes 15
2.2. The loop group 17
2.3. Lattices 18
2.4. The affine Grassmannian for GLn 20
2.5. The affine Grassmannian for an arbitrary linear algebraic group 22
2.6. Decompositions 23
2.7. Connected components 24
2.8. The Bruhat–Tits building 24
3. Affine Springer fibers 27
3.1. Springer fibers 27
3.2. Affine Springer fibers 28
3.3. General properties 29
3.4. Examples 31
3.4.1. SL2 31
3.4.2. The example of Bernstein and Kazhdan 32
3.5. Purity 32
3.6. (Co-)Homology of Ind-schemes 33
3.7. Equivariant cohomology 33
3.8. The fundamental lemma 36
4. Affine Deligne–Lusztig varieties 37
4.1. Deligne–Lusztig varieties 37
4.2. s-conjugacy classes 38
4.3. Affine Deligne–Lusztig varieties: the hyperspecial case 41
4.4. Affine Deligne–Lusztig varieties: the Iwahori case 46
4.5. The rank 2 case 50
4.6. Type A2 51
4.7. Type C2 52
4.8. Type G2 52
4.9. Relationship to Shimura varieties 56
4.10. Local Shtuka 57
4.11. Cohomology of affine Deligne–Lusztig varieties 58
References 59
Quantization of Hitchin’s Integrable System and the Geometric Langlands Conjecture 63
1. D-modules on stacks 65
2. Chiral algebras 67
3. Geometry of the affine Grassmannian 71
4. Hecke eigenproperty 74
4.1. Convolution product 74
4.2. Hecke stacks and Hecke functors 78
4.3. Statement of Hecke eigenproperty 80
5. Opers 81
6. Constructing D-modules 85
7. Hitchin integrable system I: definition 88
8. Localization functor 90
9. Quantum integrable system h 93
10. Hitchin integrable system II: D-algebras 95
11. Quantization condition 97
12. Proof of the Hecke eigenproperty 98
References 100
Faltings’ Construction of the Moduli Space of Vector Bundles on a Smooth Projective Curve 103
1. Outline of the construction 103
2. Background and notation 104
2.1. Notation 104
2.2. The Picard torus and the Poincar´e line bundle 105
2.3. Stability 107
2.4. Properties of vector bundles on algebraic curves 108
3. A nice over-parameterizing family 110
4. The generalized T-divisor 112
4.1. The line bundle OP(V )(R · T) 113
4.2. The invariant sections 113
4.3. The multiplicative structure 115
5. Raynaud’s vanishing result for rank two bundles 116
5.1. The case of genus zero and one 116
5.2. Preparations for the proof of 5.1 117
5.3. A proof for genus two using the rigidity theorem 118
5.4. A proof based on Clifford’s theorem 119
5.5. Generalizations and consequences 120
6. Semistable limits 121
6.1. Limits of vector bundles 121
6.2. Changing limits – elementary transformations 122
Construction: Elementary transformation 122
6.3. Example: Limits are not uniquely determined 123
6.4. Semistable limits exist 123
6.5. Semistable limits are almost uniquely determined 124
7. Positivity 125
7.1. Notation and preliminaries 125
7.2. Positivity – global sections of O(T) 127
7.3. The case of deg(OC(TC)) = 0 128
8. The construction 129
8.1. Constructing the moduli space of vector bundles 129
Points of MX 130
Functoriality 130
8.2. Consequences from the construction 130
8.3. Generalization to the case of arbitrary rank and degree 130
9. Prospect to higher dimension 131
References 133
Lectures on the Moduli Stack of Vector Bundles on a Curve 135
Introduction 135
Lecture 1: Algebraic stacks 136
1.1. Motivation and definition 136
1.2. How to make this geometric? 140
Lecture 2: Geometric properties of algebraic stacks 143
2.1. Properties of stacks and morphisms 143
2.2. Sheaves on stacks 147
Lecture 3: Relation with coarse moduli spaces 148
3.1. Coarse moduli spaces 149
Lecture 4: Cohomology of Bund 153
4.1. First step: Independence of the generators 155
4.2. Second step: Why is it the whole ring? 156
Lecture 5: The cohomology of the coarse moduli space (coprime case) 158
References 164
On Moduli Stacks of G-bundles over a Curve 166
1. Introduction 166
2. Algebraicity 167
3. Lifting principal bundles 168
4. Smoothness 169
5. Connected components 170
References 174
Clifford Indices for Vector Bundles on Curves 175
1. Introduction 175
2. Definition of .n and .'n 179
3. Mercat’s conjecture 181
4. The invariants dr 185
5. Rank two 196
6. Ranks three and four 198
7. Rank five 200
8. Plane curves 205
9. Problems 208
References 210
Division Algebras and Unit Groups on Surfaces 213
Introduction 213
1. Classical finiteness results: The case of a curve 213
2. Locally free sheaves of modules over Azumaya algebras: The case of a surface 218
3. Elementary modifications and connectivity 221
References 227
A Physics Perspective on Geometric Langlands Duality 228
1. Introduction 228
2. N=4 supersymmetric gauge theory 229
3. S-duality 230
4. Topological twisting 231
5. Dimensional reduction 232
6. Wilson operators 233
7. Mirror symmetry 235
8. Higher-dimensional operators 237
9. The six-dimensional view 238
10. Conclusion 239
References 240
Double Affine Hecke Algebras and Affine Flag Manifolds, I 242
Introduction 242
1. Schemes and ind-schemes 244
1.1. Categories and Grothendieck groups 244
1.1.1. Ind-objects and pro-objects in a category 244
1.1.2. Direct and inverse 2-limits 244
1.1.3. Grothendieck groups and derived categories 245
1.1.4. Proposition 245
1.2. K-theory of schemes 245
1.2.1. Background 245
1.2.2. Remark 246
1.2.3. Definitions 247
1.2.4. K-theory of a quasi-compact coherent scheme 247
1.2.5. Definition-Lemma 247
1.2.6. Proposition 247
1.2.7. Remark 248
1.2.8. Basic properties of the K-theory of a coherent quasi-compact scheme 248
1.2.9. Definitions 248
1.2.10. Derived inverse image 249
1.2.11. Derived tensor product 249
1.2.12. Derived direct image 249
1.2.13. Example 250
1.2.14. Projection formula 250
1.2.15. Base change 250
1.2.16. Compact schemes 250
1.2.17. Lemma-Definition 250
1.2.18. Remarks 250
1.2.19. Definition 251
1.2.20. Proposition 251
1.2.21. Remarks 251
1.2.22 251
1.2.23 251
1.2.24. Proposition 252
1.3. K-theory of ind-coherent ind-schemes 252
1.3.1. Spaces and ind-schemes 252
1.3.2. Definitions 252
1.3.3. Remarks 252
1.3.4. Definitions 253
1.3.5. Coherent and quasi-coherent O-modules over ind-coherent ind-schemes 253
1.3.6. Remarks 253
1.3.7. Definition 254
1.4. Group actions on ind-schemes 254
1.4.1. Ind-groups and group-schemes 254
1.4.2. Definition 254
1.4.3. Examples 254
1.4.4. Group actions on an ind-scheme 254
1.4.5. Definitions 254
1.4.6. Remarks 255
1.5. Equivariant K-theory of ind-schemes 255
1.5.1. Equivariant quasi-coherent O-modules over a scheme 255
1.5.2. Definition 255
1.5.3. Definitions 256
1.5.4. Remarks 256
1.5.5. Lemma 257
1.5.6. Proposition 257
1.5.7. Proposition 258
1.5.8. Compatibility of the derived functors 258
1.5.9. Lemma 259
1.5.10. Equivariant coherent sheaves over an ind-coherent ind-scheme 260
1.5.11. Proposition 260
1.5.12. Admissible ind-coherent ind-schemes and reduction of the group action 260
1.5.13. Proposition 261
1.5.14. Thom isomorphism and pro-finite-dimensional vector bundles over indschemes 261
1.5.15. Descent and torsors over ind-schemes 262
1.5.16. Remark 263
1.5.17. Complements on the concentration map 264
1.5.18. 264
1.5.19. 264
1.5.20 264
2. Affine flag manifolds 265
2.1. Notation relative to the loop group 265
2.1.1. 265
2.1.2. 266
2.1.3 266
2.1.4 266
2.1.5 266
2.1.6 266
2.1.7 267
2.1.8 267
2.1.9 267
2.1.10 268
2.1.11 268
2.2. Reminder on the affine flag manifold 268
2.2.1. The affine flag manifold 268
2.2.2. The Kashiwara affine flag manifold 269
2.2.3. Proposition 269
2.2.4. Remarks 270
2.2.5. Pro-finite-dimensional vector-bundles over F. 270
2.2.6. Group actions on flag varieties and related objects 270
2.3. K-theory and the affine flag manifold 271
2.3.1. Induction of ind-schemes 271
2.3.2. Proposition 271
2.3.3. Examples 271
2.3.4. Induction of I-equivariant sheaves 272
2.3.5. Examples 272
2.3.6. Convolution product on KI(N) 273
2.3.7. Proposition 274
2.3.8. Proposition 276
2.3.9. Remarks 276
2.4. Complements on the concentration in K-theory 277
2.4.1. Definition of the concentration map rS 277
2.4.2. Remark 278
2.4.3. Proposition 278
2.4.4. Concentration of O-modules supported on Ne 279
2.4.5. Proposition 279
2.4.6. Concentration of O-modules supported on Nsa 279
2.4.7. Proposition 280
2.4.8. Multiplicativity of rS. 280
2.4.9. Proposition 281
2.5. Double affine Hecke algebras 284
2.5.1. Definitions 284
2.5.2. Proposition 284
2.5.3. Remark 284
2.5.4. Geometric construction of the DAHA 285
2.5.5. Lemma 285
2.5.6. Theorem 285
2.5.7. Remark 288
3. Classification of the simple admissible modules of the double affine Hecke algebra 288
3.1. Constructible sheaves and convolution algebras 288
3.1.1. Convolution algebras and schemes which are locally of finite type 288
3.1.2. Lemma 289
3.1.3. Remark 289
3.1.4. Admissible modules over the convolution algebra 289
3.1.5. Definitions 290
3.1.6. Lemma 290
3.1.7. Proposition 290
3.2. Simple modules in the category O 290
3.2.1. From O(H) to modules over the convolution algebra of M 290
3.2.2. Definition 291
3.2.3. Proposition 291
3.2.4. Proposition 292
3.2.5. The regular representation of H 292
3.2.6. Lemma 292
3.2.7. Proof of Proposition 3.2.4 294
3.2.8. Lemma 294
3.2.9. The classification theorem 295
3.2.10. Theorem 296
References 296

Produkt-Details

• Erscheint lt. Verlag: 28.1.2011
• Reihe/Serie: Trends in Mathematics
• Reihe/Serie: Trends in Mathematics
• Zusatzinfo: 290 p.
• Verlagsort: Basel
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Technik
• Schlagworte: Algebraic Geometry • Algebraic Varieties • Differential Geometry • Dimension • flag manifold • manifold • moduli space
• ISBN-10: 3-0346-0288-X / 303460288X
• ISBN-13: 978-3-0346-0288-4 / 9783034602884