Category O and sl(k) link invariants

Abstract

We construct a functor valued invariant of oriented tangles on certain singular blocks of category O. Parabolic subcategories of these blocks categorify tensor products of various fundamental sl(k) representations. Projective functors restricted to these categories give rise to a functorial action of the Lie algebra. On the derived category, Zuckerman functors categorify sl(k)- homomorphisms. Cones of natural transformations between the identity functor and Zuckerman functors are assigned to crossings and this assignment satisfies the appropriate relations. On the Grothendieck group, the functors assigned to the crossings satisfy the sl(k)- specialization of the two variable HOMFLYPT polynomial. For the special case of links, we get a homological invariant.