Let $G$ be a connected reductive group over $\mathbb{F}_{q}$,
where $q$ is large enough and the center of $G$ is connected.
We are concerned with Lusztig's theory of character sheaves,
a geometric version of the classical character theory of the
finite group $G(\mathbb{F}_{q})$. We show that under a certain
technical condition, the restriction of a character sheaf
to its 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 $\mathbb{Z}$-basis of the $\mathbb{Z}$-module of unipotently
supported virtual characters of $G(\mathbb{F}_{q})$ (Kawanaka's
conjecture).