  We define algebraic group analogues of the Slodowy transversal slices to
adjoint orbits in a complex semisimple Lie algebra g. The new slices are
transversal to the conjugacy classes in an algebraic group G with Lie algebra
g. These slices are associated to (the conjugacy classes of) elements s of the
Weyl group W of g. For such slices we prove an analogue of the Kostant
cross-section theorem for the action of a unipotent group.
