2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/130836We 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.Representation Theory16G10, 17B20, 18E30, 16E20, 18F25text