Modular irreducibility of cuspidal unipotent characters

TL;DR

This paper proves a long-standing conjecture that cuspidal unipotent characters of finite reductive groups remain irreducible after -reduction, using a new construction of a progenerator from generalized Gelfand--Graev representations.

Contribution

It introduces a novel approach to prove irreducibility of cuspidal unipotent characters after -reduction by constructing a progenerator from generalized Gelfand--Graev representations.

Findings

Cuspidal unipotent characters are irreducible after -reduction.

Cuspidal representations appear in the head of generalized Gelfand--Graev representations.

Construction of a progenerator for the category of representations of finite reductive groups.

Abstract

We prove a long-standing conjecture of Geck which predicts that cuspidal unipotent characters remain irreducible after $\ell$-reduction. To this end, we construct a progenerator for the category of representations of a finite reductive group coming from generalised Gelfand--Graev representations. This is achieved by showing that cuspidal representations appear in the head of generalised Gelfand--Graev representations attached to cuspidal unipotent classes, as defined and studied in \cite{GM96}.