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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).

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This is proved in Chapter 8 of J. Dixmier, Enveloping algebras (MSN). See also 3.1. in J. C. Jantzen, Einhüllende Algebren halbeinfacher Lie-Algebren (MSN).

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