Casimir elements associated with Levi subalgebras of simple Lie algebras and their applications

Abstract

Let $\mathfrak{g}$ be a simple Lie algebra, $\mathfrak{h}$ a Levi subalgebra, and $C_{\mathfrak{h}}\in U(\mathfrak{h})$ the Casimir element defined via the restriction of the Killing form on $\mathfrak{g}$ to $\mathfrak{h}$. We study $C_{\mathfrak{h}}$-eigenvalues in $\mathfrak{g}/\mathfrak{h}$ and related $\mathfrak{h}$-modules. Without loss of generality, one may assume that $\mathfrak{h}$ is a maximal Levi. Then $\mathfrak{g}$ is equipped with the natural $\mathbb{Z}$-grading $\mathfrak{g}={⨁}_{i\in \mathbb{Z}}\mathfrak{g}(i)$ such that $\mathfrak{g}(0)=\mathfrak{h}$ and $\mathfrak{g}(i)$ is a simple $\mathfrak{h}$-module for $i\ne 0$. We give explicit formulae for the $C_{\mathfrak{h}}$-eigenvalues in each $\mathfrak{g}(i)$, $i\ne 0$, and relate eigenvalues of $C_{\mathfrak{h}}$ in ${\bigwedge }^{\bullet }\mathfrak{g}(1)$ to the dimensions of abelian subspaces of $\mathfrak{g}(1)$. We also prove that if $\mathfrak{a}\subset \mathfrak{g}(1)$ is abelian, whereas $\mathfrak{g}(1)$ is not, then $\dim \mathfrak{a}\le \dim \mathfrak{g}(1)/2$. Moreover, if $\dim \mathfrak{a}=(\dim \mathfrak{g}(1))/2$, then $\mathfrak{a}$ has an abelian complement. The $\mathbb{Z}$-gradings of height $\le 2$ are closely related to involutions of $\mathfrak{g}$, and we provide a connection of our theory to (an extension of) the "strange formula" of Freudenthal-de Vries.

arXiv comment30 pages