ON THE ENDOMORPHISM ALGEBRA OF GENERALISED GELFAND-GRAEV REPRESENTATIONS

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 Probenius endomorphism, so that G(F) is a finite reductive group. Let u is an element of 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.