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Nilpotent orbits and mixed gradings of semisimple Lie algebras

Let $σ$ 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 $e_σ$ is a regular nilpotent element of $\mathfrak g_0$, then the weighted Dynkin diagram of $e_σ$, $\mathcal D(e_σ)$, has only isolated zeros. It is also shown that if $G{\cdot}e_σ\cap\mathfrak g_1\ne\varnothing$, then the Satake diagram of $σ$ has only isolated black nodes and these black nodes occur among the zeros of $\mathcal D(e_σ)$. Using mixed gradings related to $e_σ$, we define an inner involution $\checkσ$ such that $σ$ and $\checkσ$ commute. Here we prove that the Satake diagrams for both $\checkσ$ and $σ\checkσ$ have isolated black nodes.