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Abstract. Let G be a connected reductive algebraic group over an algebraic closure Fp of the finite field of prime order p and let F : G → G be a Frobenius endomorphism with G = G F the corresponding Fq-rational structure. One of the strongest links we have between the representation theory of G and the geometry of the unipotent conjugacy classes of G is a formula, due to Lusztig (Adv. Math. 94(2) (1992), 139-179), which decomposes Kawanaka's Generalized Gelfand-Graev Representations (GGGRs) in terms of characteristic functions of intersection cohomology complexes defined on the closure of a unipotent class. Unfortunately, the formula given in Lusztig (Adv. Math. 94(2) (1992), 139-179) is only valid under the assumption that p is large enough.In this article, we show that Lusztig's formula for GGGRs holds under the much milder assumption that p is an acceptable prime for G (p very good is sufficient but not necessary). As an application we show that every irreducible character of G, respectively, character sheaf of G, has a unique wave front set, respectively, unipotent support, whenever p is good for G.

Let G be a simple algebraic group defined over a finite field of good characteristic, with associated Frobenius endomorphism F. In this article we extend an observation of Lusztig, (which gives a numerical relationship between an ordinary character of G F and its unipotent support), to the case where Z(G) is disconnected. We then use this observation in some applications to the ordinary character theory of G F .

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Assume G is a finite symplectic group Sp 2n (q) over a finite field F q of odd characteristic. We describe the action of the automorphism group Aut(G) on the set Irr(G) of ordinary irreducible characters of G. This description relies on the equivariance of Deligne-Lusztig induction with respect to automorphisms. We state a version of this equivariance which gives a precise way to compute the automorphism on the corresponding Levi subgroup; this may be of independent interest. As an application we prove that the global condition in Späth's criterion for the inductive McKay condition holds for the irreducible characters of Sp 2n (q).