The action of component groups on irreducible components of Springer fibers

TL;DR

This paper studies how component groups act on the irreducible components of Springer fibers in simple Lie groups, providing explicit descriptions for exceptional types and stabilizer information for classical types, confirming a conjecture.

Contribution

It offers explicit descriptions of the action of component groups on Springer fiber components for all simple Lie types, proving a conjecture of Lusztig and Sommers.

Findings

Explicit description of irreducible components for exceptional Lie groups.

Determination of stabilizers in classical Lie groups.

Proof of Lusztig and Sommers' conjecture relating Springer fibers and Weyl group cells.

Abstract

Let $G$ be a simple Lie group. Consider a nilpotent element $e\in \mathfrak{g}$. Let $Z_G(e)$ be the centralizer of $e$ in $G$, and let $A_e:= Z_G(e)/Z_G(e)^{o}$ be its component group. Write $\text{Irr}(\mathcal{B}_e)$ for the set of irreducible components of the Springer fiber $\mathcal{B}_e$. We have an action of $A_e$ on $\text{Irr}(\mathcal{B}_e)$. When $\mathfrak{g}$ is exceptional, we give an explicit description of $\text{Irr}(\mathcal{B}_e)$ as an $A_e$-set. For $\mathfrak{g}$ of classical type, we describe the stabilizers for the $A_e$-action. With this description, we prove a conjecture of Lusztig and Sommers. These results suggest relations (first proposed by Lusztig) between Springer fibers and cells in Weyl groups.