The derived algebra of a stabilizer, families of coadjoint orbits, and sheets

TL;DR

This paper explores the relationship between the derived algebra of stabilizers and coadjoint orbits, as well as the connection between centralizers and sheets in semisimple Lie algebras.

Contribution

It establishes new links between derived algebras of stabilizers and coadjoint orbit structures, and analyzes the relation between centralizers and sheets in semisimple Lie algebras.

Findings

Relation between derived algebra of stabilizer and coadjoint orbit dimensions

Connection between derived algebra of a centralizer and sheets in semisimple Lie algebras

Insights into the structure of coadjoint orbits and sheets

Abstract

Let $\mathfrak{g}$ be a finite-dimensional real or complex Lie algebra, and let $\mu \in \mathfrak{g}^{*}$. In the first part of the paper, the relation is discussed between the derived algebra of the stabilizer of $\mu$ and the set of coadjoint orbits which have the same dimension as the orbit of $\mu$. In the second part, semisimple Lie algebras are considered, and the relation is discussed between the derived algebra of a centralizer and sheets.