Nilpotent orbits and their secant varieties

TL;DR

This paper characterizes the secant varieties of nilpotent orbits in simple algebraic groups using doubled actions, computes their dimensions, and relates them to moment map images, providing a comprehensive geometric analysis.

Contribution

It introduces a description of secant varieties of nilpotent orbits via doubled actions and establishes their relation to moment map images, with explicit dimension calculations.

Findings

${f CS}( ext{orbit})$ is the closure of a $G$-orbit of an abelian subalgebra.

Dimensions of ${f CS}( ext{orbit})$ are computed using complexity and rank.

${f CS}( ext{orbit})$ coincides with the closure of the moment map image.

Abstract

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.