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

Abstract

Let $G$ be a simple algebraic group with $\mathfrak{g}=\mathsf{Lie}G$ and ${\mathcal{O}}_{\mathsf{min}}\subset \mathfrak{g}$ 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 $\phi :\overline{{\mathcal{O}}_{\mathsf{min}}}\to {\mathfrak{g}}_0$ and $\psi :\overline{{\mathcal{O}}_{\mathsf{min}}}\to {\mathfrak{g}}_1$. It is shown that the properties of $\overline{\phi ({\mathcal{O}}_{\mathsf{min}})}$ and $\overline{\psi ({\mathcal{O}}_{\mathsf{min}})}$ essentially depend on whether the intersection ${\mathcal{O}}_{\mathsf{min}}\cap {\mathfrak{g}}_1$ is empty or not. If ${\mathcal{O}}_{\mathsf{min}}\cap {\mathfrak{g}}_1\ne \varnothing$, then both $\overline{\phi ({\mathcal{O}}_{\mathsf{min}})}$ and $\overline{\psi ({\mathcal{O}}_{\mathsf{min}})}$ contain a 1-parameter family of closed $G_0$-orbits, while if ${\mathcal{O}}_{\mathsf{min}}\cap {\mathfrak{g}}_1=\varnothing$, then both are $G_0$-prehomogeneous. We prove that $\overline{G\cdot \phi ({\mathcal{O}}_{\mathsf{min}})}=\overline{G\cdot \psi ({\mathcal{O}}_{\mathsf{min}})}$. Moreover, if ${\mathcal{O}}_{\mathsf{min}}\cap {\mathfrak{g}}_1\ne \varnothing$, then this common variety is the affine cone over the secant variety of $\mathbb{P}({\mathcal{O}}_{\mathsf{min}})\subset \mathbb{P}(\mathfrak{g})$. 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 ${\mathcal{O}}_{\mathsf{min}}$.

arXiv comment32 pages