Let
![$A$](img25.png)
be a character sheaf on a connected reductive group
![$G$](img1.png)
over an
algebraically closed field. Assuming that the characteristic is not bad we show
that for certain conjugacy classes
![$D$](img26.png)
in
![$G$](img1.png)
, the restriction of
![$A$](img25.png)
to
![$D$](img26.png)
is a
local system up to shift. We also give a parametrization of unipotent cuspidal
character sheaves of
![$G$](img1.png)
in terms of restriction to conjugacy classes. Without restriction on characteristic we define canonical bijections from the set of
unipotent representations of the corresponding split group over a finite field
to a set combinatorially defined in terms of the Weyl group.