Hecke category actions via Smith-Treumann theory

TL;DR

This paper constructs a monoidal action of the diagrammatic Hecke category on the principal block of a semisimple algebraic group's representation category, using Smith-Treumann theory to confirm a conjecture by Riche-Williamson.

Contribution

It introduces a novel construction of the Hecke category action on representation blocks via Smith-Treumann theory, confirming a prior conjecture.

Findings

Established a monoidal action of the Hecke category on $ ext{Rep}_0( extbf{G})$

Utilized Smith-Treumann theory in the construction

Confirmed the conjecture by Riche-Williamson

Abstract

Let $\textbf{G}$ be a simply connected semisimple algebraic group over a field of characteristic greater than the Coxeter number. We construct a monoidal action of the diagrammatic Hecke category on the principal block $\text{Rep}_0(\textbf{G})$ of $\text{Rep}(\textbf{G})$ by wall-crossing functors. This action was conjectured to exist by Riche-Williamson. Our method uses constructible sheaves and relies on Smith-Treumann theory.