Nilpotent orbits and mixed gradings of semisimple Lie algebras

Abstract

Let $\sigma$ be an involution of a complex semisimple Lie algebra $\mathfrak g$ and $\mathfrak g=\mathfrak g_0\oplus\mathfrak g_1$ the related $\mathbb Z_2$-grading. We study relations between nilpotent $G_0$-orbits in $\mathfrak g_0$ and the respective $G$-orbits in $\mathfrak g$. If $e\in\mathfrak g_0$ is nilpotent and $\{e,h,f\}\subset\mathfrak g_0$ is an $\mathfrak{sl}_2$-triple, then the semisimple element $h$ yields a $\mathbb Z$-grading of $\mathfrak g$. Our main tool is the combined $\mathbb Z\times\mathbb Z_2$-grading of $\mathfrak g$, which is called a mixed grading. We prove, in particular, that if $e_\sigma$ is a regular nilpotent element of $\mathfrak g_0$, then the weighted Dynkin diagram of $e_\sigma$, $\mathcal D(e_\sigma)$, has only isolated zeros. It is also shown that if $G{\cdot}e_\sigma\cap\mathfrak g_1\ne\varnothing$, then the Satake diagram of $\sigma$ has only isolated black nodes and these black nodes occur among the zeros of $\mathcal D(e_\sigma)$. Using mixed gradings related to $e_\sigma$, we define an inner involution $\check\sigma$ such that $\sigma$ and $\check\sigma$ commute. Here we prove that the Satake diagrams for both $\check\sigma$ and $\sigma\check\sigma$ have isolated black nodes.