Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Hecke action on $G_1T$-modules

Noriyuki Abe

Abstract

We give an action of the Hecke category on the principal block $\mathrm{Rep}_0(G_1T)$ of $G_1T$-modules where $G$ is a connected reductive group over an algebraically closed field of characteristic $p > 0$, $T$ a maximal torus of $G$ and $G_1$ the Frobenius kernel of $G$. To define it, we define a new category with a Hecke action which is equivalent to the combinatorial category defined by Andersen-Jantzen-Soergel.

A Hecke action on $G_1T$-modules

Abstract

We give an action of the Hecke category on the principal block of -modules where is a connected reductive group over an algebraically closed field of characteristic , a maximal torus of and the Frobenius kernel of . To define it, we define a new category with a Hecke action which is equivalent to the combinatorial category defined by Andersen-Jantzen-Soergel.

Paper Structure

This paper contains 26 sections, 80 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. The category $\mathcal{S}\mathrm{Bimod}$
  3. Another combinatorial category
  4. A relation with a work of Fiebig-Lanini
  5. Relations with representation theory
  6. Our combinatorial category
  7. Notation
  8. The categories
  9. Standard filtration
  10. Base change
  11. Hecke action
  12. An example
  13. Hecke actions preserve $\widetilde{\mathcal{K}}_{\Delta}$
  14. Indecomposable objects
  15. The categorification
  16. ...and 11 more sections

Key Result

Theorem 1.1

The category $\mathcal{D}$ acts on $\mathop{\mathrm{Rep}}\nolimits_0(G_1T)$ where $B_s\in \mathcal{D}$ acts as the wall-crossing functor for any affine simple reflection $s$.

Theorems & Definitions (141)

  • Theorem 1.1: Theorem \ref{['thm:Hecke action on Representations']}
  • Theorem 1.2: Theorem \ref{['thm:indecomposables in widetilde(K)']}, \ref{['thm:categorification for tilde(K)']}
  • Theorem 1.3: Proposition \ref{['prop:indecomposable is indecomposable']}, Theorem \ref{['thm:categorification for K']}
  • Theorem 1.4: Proposition \ref{['prop:compativility of translation functors']}, \ref{['prop:fully-faithfulness']}
  • Theorem 1.5: Corollary \ref{['cor:multiplicity']}
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 131 more