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.