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Perverse sheaves on a Loop group and Langlands' duality

Victor Ginzburg

Abstract

An intrinsic construction of the tensor category of finite dimensional representations of the Langlands dual group of G in terms of a tensor category of perverse sheaves on the loop group, LG, is given. The construction is applied to the study of the topology of the affine Grassmannian of G and to establishing a Langlands type correspondence for "automorphic" sheaves on the moduli space of G-bundles.

Perverse sheaves on a Loop group and Langlands' duality

Abstract

An intrinsic construction of the tensor category of finite dimensional representations of the Langlands dual group of G in terms of a tensor category of perverse sheaves on the loop group, LG, is given. The construction is applied to the study of the topology of the affine Grassmannian of G and to establishing a Langlands type correspondence for "automorphic" sheaves on the moduli space of G-bundles.

Paper Structure

This paper contains 31 sections, 74 theorems, 186 equations.

Table of Contents

  1. Main results
  2. Duality
  3. Infinite Grassmannian
  4. Tensor category $P(Gr)$
  5. Equivalence of categories
  6. A conjecture on automorphic sheaves
  7. "Topological" gradation
  8. The $H^\bullet({\rm Gr},{{\fam\msbfam C}})$-action
  9. Kostant theorem and the generalized exponents
  10. Integrals over Schubert cycles
  11. Computation of Ext-groups
  12. Tensor category of Perverse Sheaves
  13. Application of the equivariant cohomology.
  14. Convolution construction
  15. Loop group cohomology and the Principal nilpotent
  16. ...and 16 more sections

Key Result

Proposition 1.2.1

Theorems & Definitions (79)

  • Proposition 1.2.1
  • Proposition 1.2.2
  • Theorem 1.3.1
  • Theorem 1.4.1
  • Lemma 1.5.1
  • Theorem 1.5.2
  • Conjecture 1
  • Theorem 1.6.1
  • Remark 1.6.4
  • Proposition 1.7.1
  • ...and 69 more