  Let $G$ be a connected reductive group over $F_q$, where $q$ is large enough
and the center of $G$ is connected. We are concerned with Lusztig's theory of
{\em character sheaves}, a geometric version of the classical character theory
of the finite group $G(F_q)$. We show that under a certain technical condition,
the restriction of a character sheaf to its {\em unipotent support} (as defined
by Lusztig) is either zero or an irreducible local system. As an application,
the generalized Gelfand-Graev characters are shown to form a $\Z$-basis of the
$\Z$-module of unipotently supported virtual characters of $G(F_q)$ (Kawanaka's
conjecture).
