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.

arXiv comment23 pages