Bounding Harish-Chandra series

TL;DR

This paper develops criteria for identifying cuspidal modules in finite reductive groups, analyzes Harish-Chandra series, and provides new insights into unipotent characters and decomposition matrices.

Contribution

It introduces a necessary condition for cuspidal modules, proves irreducibility of a key unipotent character, and approximates parts of the unipotent decomposition matrix for orthogonal groups.

Findings

Irreducibility of the smallest unipotent character in any Harish-Chandra series

Unitriangular approximation to unipotent decomposition matrices

Gap result on certain Brauer character degrees

Abstract

We use the progenerator constructed in our previous paper to give a necessary condition for a simple module of a finite reductive group to be cuspidal, or more generally to obtain information on which Harish-Chandra series it can lie in. As a first application we show the irreducibility of the smallest unipotent character in any Harish-Chandra series. Secondly, we determine a unitriangular approximation to part of the unipotent decomposition matrix of finite orthogonal groups and prove a gap result on certain Brauer character degrees.