Generalised Gelfand – Graev Characters in Small Characteristics

Let G be a connected reductive algebraic group defined over an algebraic closure Fp of the finite field Fp of characteristic p > 0. We will assume that p is a good prime for G (the assumption p > 5 is sufficient for all cases). Furthermore, let us assume that F : G→ G is a Frobenius endomorphism of G defining an Fq-rational structure so that the fixed point group G = G is a finite reductive group. We will denote by Irr(G) the set of ordinary irreducible characters of G (for example over C). In [4, 5] Kawanaka has constructed for every unipotent element u ∈ G a character Γu of G called a generalised Gelfand–Graev character (GGGC). The definition of Γu is somewhat delicate and depends upon the study of the unipotent conjugacy classes of G. However, here are some of the important properties: • Γu = Γv if u = gvg−1 for some g ∈ G, • Γu = IndV (φu) where V 6 G is a p-subgroup (depending upon the Gconjugacy class containing u) and φu is a linear character of V , • if u = 1 then V = {1} is the trivial subgroup and Γ1 is the character of the regular representation of G, • if u is a regular unipotent element of G (i.e., dimCG(u) is minimal) then V is a Sylow p-subgroup of G and Γu is a Gelfand–Graev character (see [2, 14.29]).