Stratifying Hecke endomorphism algebras using exact categories

TL;DR

This paper develops stratified Hecke endomorphism algebras by constructing exact categories to simplify Ext^1 vanishing conditions, advancing the authors' long-standing conjecture in Kazhdan-Lusztig theory.

Contribution

It introduces a new approach to stratify Hecke endomorphism algebras using exact categories, facilitating the proof of a conjecture related to Kazhdan-Lusztig cell theory.

Findings

Construction of new stratified Hecke endomorphism algebras

Simplification of Ext^1 vanishing conditions via exact categories

Progress towards proving the authors' 1998 conjecture

Abstract

The paper constructs new Hecke endomorphism algebras with a stratified structure. A novel feature of the proof is to approach difficult Ext^1 vanishing conditions by building entire exact category structures in which the analogous vanishing conditions are easier to check. This work is the second in a series aimed at proving a conjecture of the authors published in 1998. The conjecture concerns the enlargement, in a context of Kazhdan-Lusztig cell theory, of Hecke endomorphism algebras related to cross-characteristic representation theory of finite groups of Lie type. This second version corrects some typos and makes other small modifications, some motivated by an anonymous referee and a reader of a prior posting.