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

TL;DR

This paper introduces algebraic group analogues of Slodowy slices in complex semisimple Lie algebras, establishing their properties and relating them to Weyl group elements, with results extending Kostant's theorem.

Contribution

It defines new algebraic group slices transversal to conjugacy classes, linked to Weyl group elements, and proves an analogue of Kostant's cross-section theorem for these slices.

Findings

Defined algebraic group analogues of Slodowy slices

Proved an analogue of Kostant's cross-section theorem

Established transversality to conjugacy classes

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 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.