Duality in category O vs. Duality of D-modules - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T00:31:07Z http://mathoverflow.net/feeds/question/115880 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/115880/duality-in-category-o-vs-duality-of-d-modules Duality in category O vs. Duality of D-modules Sasha 2012-12-09T11:12:14Z 2012-12-09T11:12:14Z <p>Hello,</p> <p>I omit in the following all the words "derived, twisted, holonomic, finitely-generated...".</p> <p>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.</p> <p>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).</p> <p>Thank you, Sasha</p>