Category O and sl(k) link invariants

The program of categorification via category O was introduced by J. Bernstein, I. Frenkel, and M. Khovanov in [BFK] where tensor powers of the standard two dimensional representation of sl2 were recognized as Grothendieck groups of certain subcategories of O for various gln. They had two different constructions. One was based on studying certain blocks with singular generalized central characters. The other was based on examining the trivial regular block but by considering various parabolic subcategories. In the first case the action of sl2 was categorified by projective functors acting on these singular categories. The intertwiners were categorified by derived Zuckerman functors acting on the derived category O. In the latter case, the action of the Lie algebra was lifted to Zuckerman functors and the intertwiners became projective functors. These two constructions are related by Koszul duality where the projective functors get exchanged for Zuckerman functors and visa-versa [Rh], [MOS].