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

Let $G$ be a simple algebraic group with $\mathfrak{g}=\mathsf{L}\mathsf{i}\mathsf{e}G$ and ${\mathcal{O}}_{\mathsf{m}\mathsf{i}\mathsf{n}}\subset \mathfrak{g}$ 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 $\mathsf{L}\mathsf{i}\mathsf{e}{G}_0={\mathfrak{g}}_0$. 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 $\overline{\phi ({\mathcal{O}}_{\mathsf{m}\mathsf{i}\mathsf{n}})}$ and $\overline{\psi ({\mathcal{O}}_{\mathsf{m}\mathsf{i}\mathsf{n}})}$ essentially depend on whether the intersection ${\mathcal{O}}_{\mathsf{m}\mathsf{i}\mathsf{n}}\cap {\mathfrak{g}}_1$ is empty or not. If ${\mathcal{O}}_{\mathsf{m}\mathsf{i}\mathsf{n}}\cap {\mathfrak{g}}_1\mathrm{\ne }\mathrm{\varnothing }$, then both $\overline{\phi ({\mathcal{O}}_{\mathsf{m}\mathsf{i}\mathsf{n}})}$ and $\overline{\psi(\mathcal O_{\sf min})}$containa1-parameterfamilyofclosed$G_0$-orbits,whileif$\mathcal O_{\sf min}\cap\mathfrak g_1=\varnothing$,thenbothare$G_0$-prehomogeneous. We prove that $\overline{G{\cdot}\varphi(\mathcal O_{\sf min})}=\overline{G{\cdot}\psi(\mathcal O_{\sf min})}$\mathrm{.}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 ${\mathcal{O}}_{\mathsf{m}\mathsf{i}\mathsf{n}}$.