Tilting modules and the p-canonical basis

TL;DR

This paper introduces a new approach to tilting modules for reductive algebraic groups in positive characteristic, linking translation functors to the Hecke category and p-canonical basis, with proofs for GL_n and general Coxeter groups.

Contribution

It proposes a conjecture connecting translation functors with the Hecke category action, and proves it for GL_n and general Coxeter groups, advancing understanding of tilting modules and p-canonical bases.

Findings

Conjecture linking translation functors to the Hecke category.

Proof of conjecture for GL_n using 2-Kac-Moody actions.

Description of the Hecke category via parity complexes.

Abstract

In this paper we propose a new approach to tilting modules for reductive algebraic groups in positive characteristic. We conjecture that translation functors give an action of the (diagrammatic) Hecke category of the affine Weyl group on the principal block. Our conjecture implies character formulas for the simple and tilting modules in terms of the p-canonical basis, as well as a description of the principal block as the anti-spherical quotient of the Hecke category. We prove our conjecture for GL_n using the theory of 2-Kac-Moody actions. Finally, we prove that the diagrammatic Hecke category of a general crystallographic Coxeter group may be described in terms of parity complexes on the flag variety of the corresponding Kac-Moody group.