  We describe Serre functors for (generalisations of) the category O associated
with a semi-simple complex Lie algebra. In our approach, projective-injective
modules play an important role. They control the Serre functor in the case of a
quasi-hereditary algebra having a double centraliser property with respect to a
symmetric algebra. As an application of the double centraliser property and our
description of Serre functors, we prove three conjectures of Khovanov about the
projective-injective modules in the parabolic category O for sl_n.
