arXiv: Representation Theory | 2019
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.