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 $\mathfrak g=\bigoplus_{i\in\mathbb Z}\mathfrak g(i)$ 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.