Twin categories in representation of Lie algebra 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?
 A: The equivalence (or something very close, I haven't checked carefully what is written) follows from Beilinson-Bernstein localization. The two categories can be realized roughly speaking as D-modules on B\G/N and N\G/B, and the equivalence comes from the interchange of the two sides. Slightly more precisely, Beilinson-Bernstein tells us that (assuming we ignore singular infinitesimal characters, where things need to be slightly modified) if we want representations on which Z acts semisimply, we look at twisted D-modules on G/B with twisting given by (a lift from h^/W to h^ of) the eigenvalues of the Z action. 
Equivalently these are D-modules on G/N which are weakly H-equivariant - meaning locally constant along the fibers G/N-->G/B, and h acts with strictly prescribed semisimple monodromy-- ie we presecribe monodromy along these fibers. If we want representations with locally finite Z action, we really just look at D-modules on G/N and ask for them to be locally constant along the fibers but don't strictly presecribe monodromies.
Now the two conditions you give for representations correspond to asking for these D-modules to be N-equivariant in the strict case or B-equivariant in the locally finite case. 
In any case the whole picture is symmetric under exchanging left and right, hence the equivalence.
(There are two other categories which are symmetric under exchange of left and right -- if we impose the weak/locally finite conditions on both sides we get the category of Harish-Chandra bimodules -- ie (g+g,G)-Harish Chandra modules -- which correspond to representations of G considered as a real Lie group. If we impose strict conditions on both sides we get the Hecke category, which appears as intertwining functors acting on categories of representations and is the subject of Kazhdan-Lusztig theory. Category O, in these two forms, is some kind of intermediate form -- both of the above are monoidal categories and Category O is a bimodule for them, with your involution exchanging the two actions..)
As for a reference, this is standard but I don't know the proper reference. Similar things are discussed in the Beilinson-Ginzburg-Soergel JAMS paper on Koszul duality patterns in representation theory or the Beilinson-Ginzburg paper on wall-crossing functors (available on the arxiv). I presume Ben Webster will let us know..
A: 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.
