Nilpotent subspaces and nilpotent 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 $\mathfrak{c}$.

23 pages, to appear in the "Journal of the Australian Math. Soc."