Gradings on the Hecke category, and categorification with unequal parameters

TL;DR

This paper classifies gradings and autoequivalences of the Hecke category, introduces a bigrading linked to Frobenius automorphism, and provides a new categorification framework for Hecke algebras with unequal parameters.

Contribution

It introduces a natural bigrading and an exotic grading on the Hecke category, enabling categorification of Hecke algebras with unequal parameters across various Weyl groups.

Findings

Classified gradings refining the standard integer grading.

Established a bigrading related to Frobenius automorphism.

Developed a categorification method for Hecke algebras with unequal parameters.

Abstract

We classify gradings on the Hecke category that refine the standard integer grading. We also classify object-preserving autoequivalences of the Hecke category. We obtain a natural bigrading on the Hecke category which is related to the Frobenius automorphism. We also obtain an exotic grading in special characteristic that can be used to categorify many Hecke algebras with unequal parameters, including all Hecke algebras with unequal parameters for all finite and affine Weyl groups. This paper is a replacement for arXiv:2305.08278, which is now obsolete.