Nilpotent orbits and mixed gradings of semisimple Lie algebras

TL;DR

This paper investigates the relationship between nilpotent orbits in complex semisimple Lie algebras under involutions, introducing mixed gradings to analyze orbit structures and Satake diagrams, revealing new structural properties.

Contribution

It introduces the concept of mixed gradings combining b6 d7 b6_2-gradings to study nilpotent orbits and Satake diagrams, establishing new links between orbit properties and diagram features.

Findings

Regular nilpotent elements have weighted Dynkin diagrams with only isolated zeros.

If an orbit intersects b6_1, the Satake diagram has only isolated black nodes among zeros.

Constructed an involution  that commutes with b6, with Satake diagrams having isolated black nodes.

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.