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Drinfeld suggested the definition of a certain endo-functor, called thepseudo-identity functor, on the category of D-modules on an algebraic stack. Weextend this definition to an arbitrary DG category, and show that if certainfiniteness conditions are satisfied, this functor is the inverse of the Serrefunctor. We apply this to the category of (g,K)-modules, and we stipulate thatthe pseudo-identity functor is the analog of the Deligne-Lusztig functor. Inorder to support this, we show that the pseudo-identity functor for(g,K)-modules is isomorphic to the composition of cohomological andcontragredient dualities, which is parallel to the corresponding assertion forp-adic groups in [BBK].

An analog of the Deligne-Lusztig duality for $(\mathfrak{g},K)$-modules