ALT

Abstract

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 with Lie algebra g. These slices are associated to the pairs (p,s), where p is a parabolic subalgebra in g and s is an element of the Weyl group W of g. In the algebraic group framework simple Kleinian singularities are realized as the singularities of the fibers of the restriction of the conjugation quotient map to the slices associated to pairs (b,s), where b is a Borel subalgebra in g and s is an element of W whose representative in G is subregular. We also define some Poisson structures on the slices associated to the pairs (p,s). These structures are analogous to the Poisson structures introduced by DeBoer, Tjin and Premet on the Slodowy slices in complex simple Lie algebras. The quantum deformations of these Poisson structures are known as W-algebras of finite type. One of applications of our construction gives rise to new Poisson structures on the coordinate rings of simple Kleinian singularities.