The centralisers of nilpotent elements in classical Lie algebras

TL;DR

This paper proves Elashvili's conjecture that the index of the centraliser of a nilpotent element in classical Lie algebras equals the algebra's rank, and explores the existence of generic stabilisers in these cases.

Contribution

It confirms Elashvili's conjecture for classical reductive Lie algebras and investigates the structure of coadjoint actions of centralisers.

Findings

Proved the conjecture for classical types $gl_n$ and $sp_{2n}$.

Showed the existence of generic stabilisers in these cases.

Provided a counterexample in $so_8$ where no generic stabiliser exists.

Abstract

The index of a finite-dimensional Lie algebra $g$ is the minimum of dimensions of stabilisers $g_\alpha$ of elements $\alpha\in g^*$. Let $g$ be a reductive Lie algebra and $z(x)$ a centraliser of a nilpotent element $x\in g$. Elashvili has conjectured that the index of the centraliser $z(x)$ equals the index of $g$, i.e., the rank of $g$. Here Elashvili's conjecture is proved for reductive Lie algebras of classical type. It is shown that in cases $g=gl_n$ and $g=sp_{2n}$ the coadjoint action of $z(x)$ has a generic stabiliser. Also, we give an example of a nilpotent element $x\in so_8$ such that the coadjoint action of $z(x)$ has no generic stabiliser.