The structure of the nilpotent cone, the Kazhdan-Lusztig map and   algebraic group analogues of the Slodowy slices
  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. 
A. Sevostyanov