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

Dmitri I. Panyushev

Published 2021-05-06Version 1

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 $\mathcal O_{\sf min}$.