On Unipotent Supports of Reductive Groups With a Disconnected Centre

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 GF 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 GF. Throughout this article G will be a connected reductive algebraic group over Fp, an algebraic closure of the finite field Fp, where p is a good prime for G. Furthermore, we assume G is defined over Fq ⊂ Fp (where q is a power of a p) and F : G → G is the associated Frobenius endomorphism. Throughout we will denote an algebraic group in bold and its corresponding rational structure in roman, for instance G := GF.