On the unipotent support of character sheaves

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 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 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).