Title:

On unipotent supports of reductive groups with a disconnected centre

Let G be a connected reductive algebraic group defined over an algebraic closure of the finite field of prime order p > 0, which we assume to be good for G. We denote by F : G → G a Frobenius endomorphism of G and by G the corresponding Fqrational structure. If Irr(G) denotes the set of ordinary irreducible characters of G then by work of Lusztig and Geck we have a well defined map ΦG : Irr(G) → {Fstable unipotent conjugacy classes of G} where ΦG(χ) is the unipotent support of χ. Lusztig has given a classification of the irreducible characters of G and obtained their degrees. In particular he has shown that for each χ ∈ Irr(G) there exists an integer nχ such that nχ · χ(1) is a monic polynomial in q. Given a unipotent class O of G with representative u ∈ G we may define AG(u) to be the finite quotient group CG(u)/CG(u)◦. If the centre Z(G) is connected and G/Z(G) is simple then Lusztig and H´ezard have independently shown that for each Fstable unipotent class O of G there exists χ ∈ Irr(G) such that ΦG(χ) = O and nχ = AG(u), (in particular the map ΦG is surjective). The main result of this thesis extends this result to the case where G is any simple algebraic group, (hence removing the assumption that Z(G) is connected). In particular if G is simple we show that for each Fstable unipotent class O of G there exists χ ∈ Irr(G) such that ΦG(χ) = O and nχ = AG(u)F where u ∈ OF is a wellchosen representative. We then apply this result to prove, (for most simple groups), a conjecture of Kawanaka’s on generalised Gelfand–Graev representations (GGGRs). Namely that the GGGRs of G form a Zbasis for the Zmodule of all unipotently supported class functions of G. Finally we obtain an expression for a certain fourth root of unity associated to GGGRs in the case where G is a symplectic or special orthogonal group.
