Projective-injective modules, Serre functors and symmetric algebras

We describe Serre functors for (generalisations of) the category O associated with a semisimple complex Lie algebra. In our approach, projective-injective modules, that is modules which are both, projective and injective, play an important role. They control the Serre functor in the case of a quasi-hereditary algebra having a double centraliser with respect to a projective-injective module whose endomorphism ring is a symmetric algebra. As an application of the double centraliser property together with our description of Serre functors, we prove three conjectures of Khovanov about the projective-injective modules in the parabolic category O-0(mu)(SIn).