Let $\mathfrak{g}$ be a simple Lie algebra with Cartan subalgebra $\mathfrak{h}$, $U(\mathfrak{g})$ the universal enveloping algebra of $\mathfrak g$, $Z(\mathfrak g)$ the center of $U(\mathfrak{g})$, $\lambda$ 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](https://www.emis.de/journals/SIGMA/2020/014) (Prop. 4.4).



Maximal quotients of the enveloping algebra of a simple Lie algebra
This is proved in Chapter 8 of [J. Dixmier, Enveloping algebras](http://www.ams.org/books/gsm/011) ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=1393197)).
See also 3.1. in [J. C. Jantzen, Einhüllende Algebren halbeinfacher Lie-Algebren](https://link.springer.com/book/10.1007%2F978-3-642-68955-0) ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=721170)).


