Nilpotent subspaces and nilpotent orbits

TL;DR

This paper investigates the maximal dimensions of subspaces within nilpotent orbits of semisimple algebraic groups, establishing bounds, characterizing special orbits, and exploring properties of associated B-stable subspaces like the Dynkin ideal.

Contribution

It proves an upper bound for the dimension of subspaces in nilpotent orbits, characterizes Richardson orbits, and analyzes properties of the Dynkin ideal related to nilpotent orbit closures.

Findings

Maximal subspace dimension is at most half the orbit dimension.

Richardson orbits attain the maximal subspace dimension.

The Dynkin ideal has special minimality and uniqueness properties for certain orbits.

Abstract

Let $G$ be a semisimple algebraic group with Lie algebra $\mathfrak g$. For a nilpotent $G$-orbit $\mathcal O\subset\mathfrak g$, let $d_\mathcal O$ denote the maximal dimension of a subspace $V\subset \mathfrak g$ that is contained in the closure of $\mathcal O$. In this note, we prove that $d_\mathcal O \le \frac{1}{2}\dim\mathcal O$ and this upper bound is attained if and only if $\mathcal O$ is a Richardson orbit. Furthermore, if $V$ is $B$-stable and $\dim V= \frac{1}{2}\dim\mathcal O$, then $V$ is the nilradical of a polarisation of $\mathcal O$. Every nilpotent orbit closure has a distinguished $B$-stable subspace constructed via an $\mathfrak{sl}_2$-triple, which is called the Dynkin ideal. We then characterise the nilpotent orbits $\mathcal O$ such that the Dynkin ideal (1) has the minimal dimension among all $B$-stable subspaces $\mathfrak c$ such that $\mathfrak c\cap\mathcal O$ is dense in $\mathfrak c$, or (2) is the only $B$-stable subspace $\mathfrak c$ such that $\mathfrak c\cap\mathcal O$ is dense in $\mathfrak c$.