Ad-nilpotent ideals and equivalence relations

In this paper we study ad-nilpotent ideals of a complex simple Lie algebra g 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 corresponds to Lusztig's star operation for simply laced algebraic group G.