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

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    $\begingroup$ Dear Sasha, In the Harish-Chandra setting, the description of Harish-Chandra duality in $D$-module terms was worked out by Kashiwara (if I remember correctly), and is surprisingly non-trivial (again, at least in my memory). One thing to remember is that on the $\mathfrak g$-module side, these dualities involve passage to finite vectors of some kind (as you note), and they aren't meaningful without finiteness assumptions on the original $\mathfrak g$-module. So this suggests that an operation which makes sense for all $D$-modules (such as $D$-module duality) can't by itself capture them. $\endgroup$ – Emerton Dec 9 '12 at 13:58
  • $\begingroup$ OK, thank you, I found the paper: kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/duality.pdf $\endgroup$ – Sasha Dec 9 '12 at 14:32
  • $\begingroup$ @Sasha: Duality in the BGG category is an algebraic notion, which in practice helps to streamline the homological algebra there but is essentially an elementary idea. While the work of BGG was partly intended to clarify the composition factor structure of Verma modules, it wasn't as powerful as the combination of methods from analysis and algebraic geometry used in the proof of the Kazhdan-Lusztig conjecture. As Matt observes, the BGG category has a lot of finiteness restrictions which don't fit all D-module situations. $\endgroup$ – Jim Humphreys Dec 13 '12 at 23:21

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