In this paper, we consider Kawanaka's generalized Gelfand-Greav characters associated with the various unipotent classes of a finite group of Lie type $G(q)$ (where $q$ is a power of a prime~$p$). Under the assumption that $p$ is large enough, Lusztig [{\em Advances in Math.} 94 (1992) 139-179] has expressed these characters in terms of characteristic functions of certain character sheaves on $G$, where the resulting formulae contain as unknown quantities certain fourth roots of unity. It is one purpose of this paper to determine these roots of unity explicitly, in the case where the centre of $G$ is connected. We then use these results to study the restriction of character sheaves to the unipotent variety of~$G$. Our motivation was to find a more conceptual approach to the properties listed by Lusztig at the end of the introduction of [{\em J. Algebra} 104 (1986) 146-194].