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Modular affine Hecke category and regular unipotent centralizer

R. Bezrukavnikov, S. Riche, L. Rider

Abstract

In this paper we provide, under some mild explicit assumptions, a geometric description of the category of representations of the centralizer of a regular unipotent element in a reductive algebraic group in terms of perverse sheaves on the Langlands dual affine flag variety. This equivalence is suggested and motivated by the "geometric Langlands" philosophy, and is used in later work to construct equivalences of categories relating various geometric incarnations of the affine Hecke algebra of the given reductive group.

Modular affine Hecke category and regular unipotent centralizer

Abstract

In this paper we provide, under some mild explicit assumptions, a geometric description of the category of representations of the centralizer of a regular unipotent element in a reductive algebraic group in terms of perverse sheaves on the Langlands dual affine flag variety. This equivalence is suggested and motivated by the "geometric Langlands" philosophy, and is used in later work to construct equivalences of categories relating various geometric incarnations of the affine Hecke algebra of the given reductive group.

Paper Structure

This paper contains 49 sections, 63 theorems, 216 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Towards an equivalence between the two modular categorifications of the affine Hecke algebra
  3. The regular quotient and representations of the regular unipotent centralizer
  4. Strategy of proof
  5. Contents
  6. Acknowledgements
  7. Preliminaries on reductive groups
  8. Quotients of group schemes
  9. Centralizers of regular unipotent elements in reductive groups
  10. The unipotent cone and the multiplicative Springer resolution
  11. Tilting modules and subgroups of the centralizer of a regular unipotent element
  12. Fixed points of the centralizer of a regular unipotent element
  13. Fixed points of regular unipotent elements in tilting modules with quasi-minuscule highest weight
  14. Reconstructing a subgroup from the restriction functor
  15. Statement
  16. ...and 34 more sections

Key Result

Lemma 2.1

The group scheme $\mathrm{Z}(\mathbf{G})$ is smooth iff the quotient $X^*(\mathbf{T})/\mathbb{Z}\mathbf{R}$ has no $\ell$-torsion.

Figures (2)

  • Figure 2.1: Conditions on $\ell$
  • Figure 5.1: Bounds on $\ell$

Theorems & Definitions (113)

  • Remark 1.1
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 103 more