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A geometric model for blocks of Frobenius kernels

Pramod N. Achar, Simon Riche

Abstract

Building on a geometric counterpart of Steinberg's tensor product formula for simple representations of a connected reductive algebraic group $G$ over a field of positive characteristic, and following an idea of Arkhipov--Bezrukavnikov--Braverman--Gaitsgory--Mirković, we define and initiate the study of some categories of perverse sheaves on the affine Grassmannian of the Langlands dual group to $G$ that should provide geometric models for blocks of representations of the Frobenius kernel $G_1$ of $G$. In particular, we show that these categories admit enough projective and injective objects, which are closely related to some tilting perverse sheaves, and that they are highest weight categories in an appropriate generalized sense.

A geometric model for blocks of Frobenius kernels

Abstract

Building on a geometric counterpart of Steinberg's tensor product formula for simple representations of a connected reductive algebraic group over a field of positive characteristic, and following an idea of Arkhipov--Bezrukavnikov--Braverman--Gaitsgory--Mirković, we define and initiate the study of some categories of perverse sheaves on the affine Grassmannian of the Langlands dual group to that should provide geometric models for blocks of representations of the Frobenius kernel of . In particular, we show that these categories admit enough projective and injective objects, which are closely related to some tilting perverse sheaves, and that they are highest weight categories in an appropriate generalized sense.
Paper Structure (57 sections, 81 theorems, 431 equations)

This paper contains 57 sections, 81 theorems, 431 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Main results
  4. Some comments on Theorem \ref{['thm:main-intro']}
  5. A Finkelberg--Mirković conjecture for the Frobenius kernel
  6. Tilting G-modules and projective G1T-modules
  7. Contents of the paper
  8. Conventions on fields
  9. Combinatorics of the affine Weyl group
  10. The extended affine affine Weyl group
  11. Geometry of alcoves and restricted elements
  12. A length computation
  13. Coset representatives
  14. The periodic order
  15. The Hecke algebra and the left spherical module
  16. ...and 42 more sections

Key Result

Theorem 1.1

Theorems & Definitions (176)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 1.3
  • Remark 1.4
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 166 more