Let

be a character sheaf on a connected reductive group

over an
algebraically closed field. Assuming that the characteristic is not bad we show
that for certain conjugacy classes

in

, the restriction of

to

is a
local system up to shift. We also give a parametrization of unipotent cuspidal
character sheaves of

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.