Nilpotent orbits and their secant varieties

Abstract

Let G be a simple algebraic group, $g=\mathrm{Lie}G$, and $O$ a nilpotent orbit in $g$. Let $\text{CS}\left(O\right)$ denote the affine cone over the secant variety of $\overline{PO}\subset Pg$. Using the theory of doubled G-actions, we describe $\text{CS}\left(O\right)$ for all $O$. Let $c\left(O\right)$ and $r\left(O\right)$ denote the complexity and rank of the G-variety $O$. It is proved that $\dim \text{CS}\left(O\right)=2\dim O-2c\left(O\right)-r\left(O\right)$ and there is a subspace $t_O$ of a Cartan subalgebra of $g$ such that $\text{CS}\left(O\right)$ is the closure of $G\cdot t_O$. We compute $c\left(O\right)$ and $r\left(O\right)$ for all nilpotent orbits and show that $\text{CS}\left(O\right)$ is the closure of $\mathrm{Im}\left(\mu \right)$, where $\mu :T^{⁎}\left(O\right)\to g^{⁎}\simeq g$ is the moment map. It is also shown that the secant variety of $\overline{PO}$ is defective if and only if $r\left(O\right)<\mathrm{rk}G$ if and only if $\text{CS}\left(O\right)\ne g$.