Nilpotent orbits and their secant varieties

Let $G$ be a simple algebraic group and $\mathcal{O}$ a nilpotent orbit in $\mathfrak{g}$. Let $\mathbf{C}\mathbf{S}(\mathcal{O})$ 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 $\mathbf{C}\mathbf{S}(\mathcal{O})$ for all $\mathcal{O}$. We compute $\dim \mathbf{C}\mathbf{S}(\mathcal{O})$ 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 $\mathbf{C}\mathbf{S}(\mathcal{O})$ is the closure of $G\cdot {\mathfrak{t}}_{\mathcal{O}}$. Another observation is that $\mathbf{C}\mathbf{S}(\mathcal{O})$ 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.