Hello,

I omit in the following all the words "derived, twisted, holonomic, finitely-generated...".

We have the Bernstein-Beilinson equivalence between the category of $N$-equivariant $D$-modules on the flag variety and $\mathfrak{g}$-modules which are $\mathfrak{n}$ locally finite. Also, we have the equivalence between $K$-equivariant $D$-modules and Harish-Chandra modules.

My question is about interaction of duality with this equivalences. We have duality for $D$-modules (like Verdier duality). In the categories of Lie modules, we also have dualities (where we take some finite vectors in the abstract dual). Is there a reference for the relation of these dualities? I think one should be careful with the twistings and such (duality will take a $D$-modules to a $D$-module with opposite twisting, a module in category $O$ to a module in the category $O$ for the opposite Borel, and so on).

Thank you, Sasha

all$D$-modules (such as $D$-module duality) can't by itself capture them. – Emerton Dec 9 '12 at 13:58