  Let G be a connected complex reductive group and let K be a symmetric
subgroup of G. We prove a formula for the Drinfeld-Gaitsgory functor for the
dg-category of K-equivariant sheaves on the flag manifold of G in terms of the
Matsuki duality functor. As byproducts, we obtain a description of the Serre
functor and the Deligne-Lusztig duality for (g,K)-modules.
