Minimal special degenerations and duality

TL;DR

This paper classifies singularities of Slodowy slices between special nilpotent orbits in simple Lie algebras, revealing dualities and resolving a conjecture related to their intersection cohomology.

Contribution

It provides a classification of singularities in Slodowy slices, explores duality actions, and confirms a conjecture on intersection cohomology for special nilpotent orbit slices.

Findings

Most singularities are simple surface singularities or minimal orbit closures.

Lusztig-Spaltenstein duality interchanges certain singularities and orbit closures.

Confirmed Lusztig's conjecture on intersection cohomology of slices.

Abstract

This paper includes the classification, in a simple Lie algebra, of the singularities of Slodowy slices between special nilpotent orbits that are adjacent in the partial order on nilpotent orbits. The irreducible components of most singularities are (up to normalization) either a simple surface singularity or the closure of a minimal special nilpotent orbit in a smaller rank Lie algebra. Besides those cases, there are some exceptional cases that arise as certain quotients of the closure of a minimal orbit in types $A_2$ and $D_n$. We also consider the action on the slice of the fundamental group of the smaller orbit. With this action, we observe that under Lusztig-Spaltenstein duality, in most cases, a simple surface singularity is interchanged with the closure of a minimal special orbit of Langlands dual type (or a cover of it with action). This empirical observation generalizes an observation of Kraft and Procesi in type $A_n$, where all nilpotent orbits are special. We also resolve a conjecture of Lusztig that concerns the intersection cohomology of slices between special nilpotent orbits.