On the endomorphism algebra of generalised Gelfand-Graev representations

TL;DR

This paper studies the endomorphism algebra of generalized Gelfand-Graev representations of finite reductive groups, showing its dimension is polynomial in q with degree related to the centralizer of a unipotent element, extending previous results.

Contribution

It proves the polynomial nature of the endomorphism algebra's dimension in q for groups with connected center and extends this to disconnected centers under certain conventions.

Findings

Dimension of endomorphism algebra is polynomial in q.

Degree of polynomial equals the dimension of the centralizer of u.

Results extend to disconnected centers with additional conventions.

Abstract

Let $G$ be a connected reductive algebraic group defined over the finite field $\F_q$, where $q$ is a power of a good prime for $G$, and let $F$ denote the corresponding Frobenius endomorphism, so that $G^F$ is a finite reductive group. Let $u \in G^F$ be a unipotent element and let $\Gamma_u$ be the associated generalised Gelfand-Graev representation of $G^F$. Under the assumption that $G$ has a connected centre, we show that the dimension of the endomorphism algebra of $\Gamma_u$ is a polynomial in $q$, with degree given by $\dim C_G(u)$. When the centre of $G$ is disconnected, it is impossible, in general, to parametrise the (isomorphism classes of) generalised Gelfand-Graev representations independently of $q$, unless one adopts a convention of considering separately various congruence classes of $q$. Subject to such a convention we extend our result.