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 GFGF and its unipotent support), to the case where Z(G)Z(G) is disconnected. We then use this observation in some applications to the ordinary character theory of GFGF.