Nilpotent subspaces and nilpotent orbits

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 $d_\mathcal O \le \frac{1}{2}\dim\mathcal O$andthisupperboundisattainedifandonlyif$\mathcal O$isaRichardsonorbit\mathrm{.}Furthermore,if$V$is$B$-stableand$\dim V= \frac{1}{2}\dim\mathcal O$,then$V$isthenilradicalofapolarisationof$\mathcal O$\mathrm{.}Everynilpotentorbitclosurehasadistinguished$B$-stable subspace constructed via an ${\mathfrak{s}\mathfrak{l}}_2$-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 $\mathfrak c\cap\mathcal O$isdensein$\mathfrak c$.