Hecke endomorphism algebras: stratification, finite groups of Lie type, and i-quantum algebras

Abstract: 

Hecke endomorphism algebras are a natural generalization of q-Schur algebras from symmetric groups to arbitrary Coxeter groups. They appear naturally in the study of representations of finite groups of Lie type, especially finite general linear groups whose representations are connected with those of quantum general linear groups.

Generally speaking, the structure and representations of Hecke endomorphism algebras are difficult to understand if the associated Coxeter group is not the symmetric group. Over twenty years ago, B. Parshall, L. Scott and the speaker investigated some rough stratification structure for those associated with Weyl groups. We conjectured the existence of a finer stratification by Kazhdan--Lusztig two-sided cells for an enlarged endomorphism algebra.

I will report on the progression of ideas in our successful efforts to prove a (very slightly modified) version of the conjecture. Inspired by an Ext1 vanishing condition uncovered in a local case, we use exact categories to formulate by analogy a tractable global version, not mentioning localization and often requiring less vanishing. After constructing many relevant exact category settings, we are eventually able to prove this exact category Ext1 vanishing in one of them that contains all the filtered objects we need. If time permits, I will mention possible applications to representations of finite groups of Lie type and i-quantum algebras.