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

Abstract

Let g be a simple Lie algebra, h a Levi subalgebra, and Ch∈U(h) the Casimir element defined via the restriction of the Killing form on g to h. We study Ch-eigenvalues in g/h and related h-modules. Without loss of generality, one may assume that h is a maximal Levi. Then g is equipped with the natural Z-grading g=⨁i∈Zg(i) such that g(0)=h and g(i) is a simple h-module for i≠0. We give explicit formulae for the Ch-eigenvalues in each g(i), i≠0, and relate eigenvalues of Ch in ⋀•g(1) to the dimensions of abelian subspaces of g(1). Then we prove that if a⊂g(1) is abelian, whereas g(1) is not, then dim⁡a⩽dim⁡g(1)/2. Moreover, if dim⁡α=(dim⁡g(1))/2, then a has an abelian complement. The Z-gradings of height ⩽2 are closely related to involutions of g, and we provide a connection of our theory to (an extension of) the “strange formula” of Freudenthal–de Vries.