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

TL;DR

This paper investigates Casimir elements associated with Levi subalgebras of simple Lie algebras, providing explicit eigenvalue formulas, analyzing abelian subspaces, and connecting the results to involutions and the Freudenthal-de Vries formula.

Contribution

It introduces explicit formulas for Casimir eigenvalues in graded components and explores the structure of abelian subspaces within these components.

Findings

Explicit eigenvalue formulas for Casimir elements in each graded component.

Bound on the dimension of abelian subspaces in the first graded component.

Connection between gradings, involutions, and the Freudenthal-de Vries formula.

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.