# Maximal quotients of the enveloping algebra of a simple Lie algebra

Let $$\mathfrak{g}$$ be a simple Lie algebra with Cartan subalgebra $$\mathfrak{h}$$ and centre $$Z(\mathfrak g)$$, $$U(\mathfrak{g})$$ the universal enveloping algebra of $$\mathfrak g$$, $$\lambda$$ a complex weight and $$\chi$$ the central character of the Verma module with highest weight $$\lambda$$. Let $$A_\lambda\mathrel{:=}U(\mathfrak{g})/(z-\chi(z), \, z\in Z(\mathfrak{g}))$$. Why is it true that, as $$\mathfrak{g}$$-modules, $$A_\lambda=\bigoplus_\mu {V_\mu\otimes V_\mu^*[0]},$$ where $$\mu$$ runs over the dominant integral weights, $$V_\mu$$ is the irreducible finite-dimensional representation of $$\mathfrak{g}$$ with highest weight $$\mu$$ and $$V_\mu^*[0]$$ is the $$0$$-weight subspace of $$V_\mu^*$$?

For reference see Etingof and Stryker - Short star products for filtered quantizations, I (Prop. 4.4).