The combinatorics of $\mathrm{GL}_n$ generalized Gelfand--Graev characters

TL;DR

This paper provides a new combinatorial approach to understanding generalized Gelfand--Graev characters for finite general linear groups, enabling explicit decompositions and connections to Kostka--Foulkes polynomials.

Contribution

It reinterprets Kawanaka's definition in type A, offering more flexible computations and explicit decompositions of these characters.

Findings

Explicit decomposition of Gelfand--Graev characters into unipotent representations

Recovery of Kostka--Foulkes polynomials as multiplicities

New combinatorial methods for analyzing Gelfand--Graev characters

Abstract

Introduced by Kawanaka in order to find the unipotent representations of finite groups of Lie type, generalized Gelfand--Graev characters have remained somewhat mysterious. Even in the case of the finite general linear groups, the combinatorics of their decompositions has not been worked out. This paper re-interprets Kawanaka's definition in type $A$ in a way that gives far more flexibility in computations. We use these alternate constructions to show how to obtain generalized Gelfand--Graev representations directly from the maximal unipotent subgroups. We also explicitly decompose the corresponding generalized Gelfand--Graev characters in terms of unipotent representations, thereby recovering the Kostka--Foulkes polynomials as multiplicities.