Modular representations of Hecke algebras

TL;DR

This paper surveys the modular representation theory of Hecke algebras, focusing on non-semisimple cases, and discusses recent classification results of simple modules using Kazhdan--Lusztig theory and quantum groups.

Contribution

It provides a comprehensive overview of the role of Hecke algebras in modular representation theory and summarizes recent advances in classifying simple modules.

Findings

Complete classification of simple modules in non-semisimple Hecke algebras

Connection between Kazhdan--Lusztig cells and module classification

Application of quantum group canonical bases to representation theory

Abstract

These notes are based on a course given at the EPFL in May 2005. It is concerned with the representation theory of Hecke algebras in the non-semisimple case. We explain the role that these algebras play in the modular representation theory of finite groups of Lie type and survey the recent results which complete the classification of the simple modules. These results rely on the theory of Kazhdan--Lusztig cells in finite Weyl groups (with respect to possibly unequal parameters) and the theory of canonical bases for representations of quantum groups.