Generalised Gelfand--Graev representations in bad characteristic?

TL;DR

This paper explores the extension of Kawanaka's generalized Gelfand-Graev representations to cases where the prime characteristic is bad for the group, revealing new insights and conjectures about unipotent classes.

Contribution

It investigates the possibility of constructing Gelfand-Graev representations without the good prime restriction, proposing a new conjectural characterization of special unipotent classes.

Findings

Extended Gelfand-Graev representations to bad characteristic cases

Proposed a new conjectural characterization of special unipotent classes

Connected representation theory with weighted Dynkin diagrams

Abstract

Let $G$ be a connected reductive algebraic group defined over a finite field with $q$ elements. In the 1980's, Kawanaka introduced generalised Gelfand-Graev representations of the finite group $G(F_q)$, assuming that $q$ is a power of a good prime for $G$. These representations have turned out to be extremely useful in various contexts. Here we investigate to what extent Kawanaka's construction can be carried out when we drop the assumptions on~$q$. As a curious by-product, we obtain a new, conjectural characterisation of Lusztig's concept of special unipotent classes of $G$ in terms of weighted Dynkin diagrams.