  We show that, for a connected reductive algebraic group G over an
algebraically closed field of zero or good characteristic, the parts, called
strata, in the partition of G recently introduced by Lusztig are unions of
sheets of conjugacy classes. For G simple and adjoint we refine the
parametrization of such sheets obtained in previous work with F. Esposito. We
give a simple combinatorial description of strata containing spherical
conjugacy classes, showing that Lusztig's correspondence induces a bijection
between unions of spherical conjugacy classes and unions of classes of
involutions in the Weyl group. Using ideas from the Appendix by M. Bulois, we
show that the closure of a stratum is not necessarily a union of strata.
