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. © 2013 Elsevier Inc.