# rt.representation theory – Action of the Casimir on highest weight modules for Kac-Moody algebra

Let $$g$$ be a Kac-Moody algebra with a symmetrizable Cartan matrix, and let $${u_j}$$ and $${u^j}$$ be bases of $$g$$ dual with respect to a nondegenerate invariant bilinear form $$(cdot|cdot)$$ on $$g$$, and consistent with the triangular decomposition of $$g$$. Let $$L(Lambda)$$ be an integrable representation of $$g$$ with highest weight $$Lambda$$, and let $$v_Lambda$$ be its highest weight vector. Denote the Casimir $$Omega=sum_j u_jotimes u^j$$.

I want to know why $$Omega(v_Lambdaotimes v_Lambda)=(Lambda|Lambda)v_Lambdaotimes v_Lambda$$? Could someone give some explanation or some references?