On the endomorphism algebra of generalised Gelfand-Graev representations

Abstract

Let G be a connected reductive algebraic group defined over the finite field Fq, 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 ∈ G F be a unipotent element and let Γu 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 Γu is a polynomial in q, with degree given by dim CG(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.