Ad-nilpotent Ideals and Equivalence Relations

Abstract

In this paper we study ad-nilpotent ideals of a complex simple Lie algebra and their connections with affine Weyl groups and nilpotent orbits. We define a left equivalence relation for ad-nilpotent ideals based on their normalizer and generators, and prove that the equivalence relation is compatible with the left cell structure of affine Weyl group of and Lusztig's star operator for type $\tilde A_{n-1}$.