Casimir invariants and infinitesimal characters for semi-simple Lie algebras

Abstract

In this paper a new derivation of Weyl's dimension formula is presented and applied to evaluate the eigenvalues of a certain family of Casimir invariants for a semi-simple Lie algebra. The formula obtained may be used in place of Weyl's character formula and is applied to determine the infinitesimal characters occurring in the tensor product spaces V(λ)⊗ V, where V(λ) is a finite-dimensional irreducible module with highest weight λ and V is an infinite-dimensional module admitting an infinitesimal character. The results obtained generalize some well known properties of finite-dimensional tensor products V(λ)⊗ V(μ).