Accessibility of Nilpotent Orbits in Classical Algebras

TL;DR

This paper investigates the accessibility order on nilpotent orbits in classical Lie algebras, revealing it aligns with the dominance order in some cases but differs in others, thus deepening understanding of orbit structures.

Contribution

It introduces and analyzes the accessibility order on nilpotent orbits, showing its equivalence to the dominance order in general and special linear cases, but not in symplectic and orthogonal cases.

Findings

Accessibility order coincides with dominance order in general and special linear algebras.

Accessibility order differs from dominance order in symplectic and orthogonal algebras.

Provides new insights into the structure of nilpotent orbits in classical Lie algebras.

Abstract

Let $G$ be a classical linear algebraic group over an algebraically closed field, and let $\mathfrak{n}$ denote the subset of nilpotent elements in its Lie algebra. In this paper we study a partial order on the $G$-orbits in $\mathfrak{n}$ given by taking limits along cocharacters of $G$. This gives rise to the so-called accessibility order on the nilpotent orbits. Our main results show that for general and special linear algebras, this new order coincides with the usual dominance order on nilpotent orbits, but for symplectic and orthogonal algebras this is not the case.