Answer by S. Carnahan for Does an element in the center of universal enveloping algebra becomes a scalar in irreducible representations?

No. Let $G$ be $O_2(\mathbb{R})$, so the Lie algebra is one dimensional, and the center of the universal enveloping algebra is the symmetric algebra of the Lie algebra. Then the usual 2-dimensional representation (which is irreducible) does not have elements of the Lie algebra acting by scalars.

You need $\Omega$ to be $Ad$ invariant.