Nilpotent orbits and their secant varieties

Let $G$ be a simple algebraic group and $\mathcal O$ a nilpotent orbit in $\mathfrak g$. Let ${\mathbf{CS}}(\mathcal O)$ denote the affine cone over the secant variety of $\overline{\mathbb P\mathcal O}\subset \mathbb P\mathfrak g$. Using the theory of doubled actions of $G$, we describe ${\mathbf{CS}}(\mathcal O)$ for all $\mathcal O$. We compute $\dim{\mathbf{CS}}(\mathcal O)$ using the complexity and rank of the $G$-variety $\mathcal O$ and show that there is an abelian subalgebra $\mathfrak t_{\mathcal O}\subset\mathfrak g$ such that ${\mathbf{CS}}(\mathcal O)$ is the closure of $G{\cdot}\mathfrak t_\mathcal O$. Another observation is that ${\mathbf{CS}}(\mathcal O)$ coincide with the closure of the image of the moment map associated with the cotangent bundle of $\mathcal O$. We also compute the complexity and rank for all nilpotent orbits.