Eigenvalues
Given that Ω is central in the enveloping algebra, it acts on simple modules by a scalar. Let 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 where ρ is the weight defined by half the sum of the positive roots.
Read more about this topic: Casimir Invariant