Eigenvalues

Given that Ω is central in the enveloping algebra, it acts on simple modules by a scalar. Let \langle,\rangle be our favorite bilinear symmetric non-degenerate form, by which we define Ω. Let L(λ) be the finite dimensional highest weight module of weight λ. Then the Casimir element Ω acts on L(λ) by the constant \langle \lambda, \lambda + 2 \rho \rangle, where ρ is the weight defined by half the sum of the positive roots.