Hecke action on the principal block

TL;DR

This paper constructs an affine Hecke category action on the principal block of representations for certain algebraic groups, confirming a conjecture and offering a new proof of the tilting character formula using antispherical p-Kazhdan-Lusztig polynomials.

Contribution

It introduces a novel action of the affine Hecke category on the principal block, confirming Williamson's conjecture and providing a new proof of the tilting character formula.

Findings

Confirmed Williamson's conjecture.

Provided a new proof of the tilting character formula.

Connected Hecke actions with antispherical p-Kazhdan-Lusztig polynomials.

Abstract

In this paper we construct an action of the affine Hecke category (in its "Soergel bimodules" incarnation) on the principal block of representations of a simply-connected semisimple algebraic group over an algebraically closed field of characteristic bigger than the Coxeter number. This confirms a conjecture of G. Williamson and the second author, and provides a new proof of the tilting character formula in terms of antispherical $p$-Kazhdan-Lusztig polynomials.