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Let $G$ be a connected reductive group over $F_q$, where $q$ is large enoughand the center of $G$ is connected. We are concerned with Lusztig's theory of{\em character sheaves}, a geometric version of the classical character theoryof 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 definedby 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'sconjecture).

On the unipotent support of character sheaves