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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 (Prop. 4.4).

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  • $\begingroup$ @JoseBrox, I believe that it is best practice when proposing substantive changes, even obviously correct ones such as yours, to leave a comment with the clarification and let the original asker make the change themselves. $\endgroup$
    – LSpice
    Dec 16, 2020 at 17:47
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    $\begingroup$ @LSpice In this very case the change is very focussed and helps the reader (and is precisely justified in the edit history), and the OP technically has the possibly to roll back to the previous version, so I think this change was good practice. In addition the OP has left MO since June, so would most likely have ignored the comments. $\endgroup$
    – YCor
    Dec 17, 2020 at 9:54

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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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