Projections of the minimal nilpotent orbit in a simple Lie algebra and secant varieties

TL;DR

This paper investigates the projections of the minimal nilpotent orbit in a simple Lie algebra under certain subgroup actions, revealing their geometric structure and relation to secant varieties, with implications for invariant theory.

Contribution

It characterizes the images of minimal nilpotent orbits under projections associated with $Z_2$-gradings, linking their geometry to secant varieties and extending to invariant-theoretic results.

Findings

If the intersection with $rak g_1$ is non-empty, the projections contain a family of closed orbits.

If the intersection is empty, the projections are $G_0$-prehomogeneous.

The common orbit closure is the affine cone over the secant variety of the minimal orbit.

Abstract

Let $G$ be a simple algebraic group with $\mathfrak g=\mathsf{Lie} G$ and $\mathcal O_{\sf min}\subset\mathfrak g$ the minimal nilpotent orbit. For a $\mathbb Z_2$-grading $\mathfrak g=\mathfrak g_0\oplus\mathfrak g_1$, let $G_0$ be a connected subgroup of $G$ with $\mathsf{Lie} G_0=\mathfrak g_0$. We study the $G_0$-equivariant projections $\varphi:\overline{\mathcal O_{\sf min}}\to \mathfrak g_0$ and $\psi:\overline{\mathcal O_{\sf min}}\to\mathfrak g_1$. It is shown that the properties of $\overline{\varphi(\mathcal O_{\sf min})}$ and $\overline{\psi(\mathcal O_{\sf min})}$ essentially depend on whether the intersection $\mathcal O_{\sf min}\cap\mathfrak g_1$ is empty or not. If $\mathcal O_{\sf min}\cap\mathfrak g_1\ne\varnothing$, then both $\overline{\varphi(\mathcal O_{\sf min})}$ and $\overline{\psi(\mathcal O_{\sf min})}$ contain a 1-parameter family of closed $G_0$-orbits, while if $\mathcal O_{\sf min}\cap\mathfrak g_1=\varnothing$, then both are $G_0$-prehomogeneous. We prove that $\overline{G{\cdot}\varphi(\mathcal O_{\sf min})}=\overline{G{\cdot}\psi(\mathcal O_{\sf min})}$. Moreover, if $\mathcal O_{\sf min}\cap\mathfrak g_1\ne\varnothing$, then this common variety is the affine cone over the secant variety of $\mathbb P(\mathcal O_{\sf min})\subset\mathbb P(\mathfrak g)$. As a digression, we obtain some invariant-theoretic results on the affine cone over the secant variety of the minimal orbit in an arbitrary simple $G$-module. In conclusion, we discuss more general projections that are related to either arbitrary reductive subalgebras of $\mathfrak g$ in place of $\mathfrak g_0$ or spherical nilpotent $G$-orbits in place of $\mathcal O_{\sf min}$.