Table of Contents

Preface ix

Affine Springer Fibers and Affine Deligne-Lusztig Varieties U. Görtz 1

1 Introduction 1

2 The affine Grassmannian and the affine flag manifold 3

3 Affine Springer fibres 15

4 Affine Deligne-Lusztig varieties 25

References 47

Quantization of Hitchin's Integrable System and the Geometric Langlands Conjecture T. L. Gómez 51

1 D-modules on stacks 53

2 Chiral algebras 55

3 Geometry of the affine Grassmannian 59

4 Hecke eigenproperty 62

5 Opers 69

6 Constructing D-modules 73

7 Hitchin integrable system I: definition 76

8 Localization functor 78

9 Quantum integrable system h 81

10 Hitchin integrable system II: D-algebras 83

11 Quantization condition 85

12 Proof of the Hecke eigenproperty 86

References 88

Falting's Construction of the Moduli Space of Vector Bundles on a Smooth Projective Curve G. Hein 91

1 Outline of the construction 91

2 Background and notation 92

3 A nice over-parameterizing family 98

4 The generalized Θ-divisor 100

5 Raynaud's vanishing result for rank two bundles 104

6 Seraistable limits 109

7 Positivity 113

8 The construction 117

9 Prospect to higher dimension 119

References 121

Lectures on the Moduli Stack of Vector Bundles on a Curve J. Heinloth 123

Introduction 123

Lecture 1 Algebraic stacks 124

Lecture 2 Geometric properties of algebraic stacks 131

Lecture 3 Relation with coarse moduli spaces 136

Lecture 4 Cohomology of Bun^d_n 141

Lecture 5 The cohomology of the coarse moduli space (coprime case) 146

References 152

On Moduli Stacks of G-bundles over a Curve N. Hoffmann 155

1 Introduction 155

2 Algebraicity 156

3 Lifting principal bundles 157

4 Smoothness 158

5 Connected components 159

Reference 163

Clifford Indices for Vector Bundles on Curves H. Lange P.E. Newstead 165

1 Introduction 165

2 Definition of γ_n and γ′_n 169

3 Mercat's conjecture 171

4 The invariants d_r 175

5 Rank two 186

6 Ranks three and four 188

7 Rank five 190

8 Plane curves 195

9 Problems 198

References 200

Division Algebras and Unit Groups on Surfaces F. Reede U. Stuhler 203

Introduction 203

1 Classical finiteness results: The case of a curve 203

2 Locally free sheaves of modules over Azumaya algebras: The case of a surface 208

3 Elementary modifications and connectivity 211

References 217

A Physics Perspective on Geometric Langlands Duality K.-G. Schlesinger 219

1 Introduction 219

2 N = 4 supersymmetric gauge theory 220

3 S-duality 221

4 Topological twisting 222

5 Dimensional reduction 223

6 Wilson operators 224

7 Mirror symmetry 226

8 Higher-dimensional operators 228

9 The six-dimensional view 229

10 Conclusion 230

References 231

Double Affine Hecke Algebras and Affine Flag Manifolds, I M. Varagnolo E. Vasserot 233

Introduction 233

1 Schemes and ind-schemes 235

2 Affine flag manifolds 256

3 Classification of the simple admissible modules of the double affine Hecke algebra 279

References 287