TITLE: Twin categories in representation of Lie algebra
QUESTION [9 upvotes]: Let $\mathfrak{g}=\mathfrak{n}^-\oplus\mathfrak{h}\oplus\mathfrak{n^+}$ be a triangular decomposition of semisimple Lie algebra. Let $\mathcal{Z}$ be the central of universal envoloping Lie algebra of $\mathfrak{g}$.
Let $\mathcal{C}$ be the category of representations of $\mathfrak{g}$, on which $\mathfrak{n ^+}$ and $\mathfrak{h}$ acts locally finite, and $\mathcal{Z}$ acts semisimplely. 
Let $\mathcal{D}$ be the another category of representations of $\mathfrak{g}$, on which $\mathfrak{n ^+}$ and $\mathcal{Z}$ acts also locally finitely, and $\mathfrak{h}$ acts semisimply. If I don't make mistake, $\mathcal{D}$ should be called category $\mathcal{O}$.
Claim. $\mathcal{C}$ is equivalent to $\mathcal{D}$.
Do I formulate the problem correctly? 
About the proof of this theorem, where is it written?

REPLY [5 votes]: Following David's suggestion, I'll point out that this is a theorem of Soergel.  That paper is in French, but an English account of a related (and more general) construction is given in the paper of Soergel and Milicic.  While David's explanation is quite nice, the references below show this isn't an intrinsically geometrical result; Soergel gives completely algebraic proofs.