Algebraic Group Analogues of the Slodowy Slices and Deformations of Poisson W-algebras

We define algebraic group analogues of the Slodowy transversal slices to adjoint orbits in a complex semisimple Lie algebra Graphic. 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. Using this theorem, we equip the new slices with certain Poisson structures obtained by Poisson reduction from a Poisson–Lie group structure on G.