The secant varieties of nilpotent orbits

Abstract

Let $\mathfrak{g}$ be a complex simple Lie algebra. We have the adjoint representation of the adjoint group $G$ on $\mathfrak{g}$. Then $G$ acts on the projective space $\mathbb{P}_{\mathfrak{g}}$. We consider the closure $X$ of the image of a nilpotent orbit in $\mathbb{P}_{\mathfrak{g}}$. The $i$-secant variety $Sec^{(i)}X$ of a projective variety $X$ is the closure of the union of projective subspaces of dimension $i$ in the ambient space $\mathbb{P}$ spanned by $i+1$ points on $X$. In particular we call the 1-secant variety the secant variety. In this paper we give explicit descriptions of the secant and the higher secant varieties of nilpotent orbits of complex classical simple Lie algebras.