  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 $G^F$.
