Parabolic category O, perverse sheaves on Grassmannians, Springer fibres and Khovanov homology

Abstract

For a fixed parabolic subalgebra of P of gl(n,c) we prove that the centre of the principal block 0po of the parabolic category O is naturally isomorphic to the cohomology ring H*(Bp) of the corresponding Springer fibre. We give a diagrammatic description of 0po for maximal parabolic and give an explicit isomorphism to Bradens description of the category PervB (G(k,n)) of Schubert-constructible perverse sheaves on Grassmannians. As a consequence Khovanovs algebra Hn is realised as the endomorphism ring of some object from PervB(G(n,n)) which corresponds under localisation and the Riemann-Hilbert correspondence to a full projective-injective module in the corresponding category 0po. From there one can deduce that Khovanovs tangle invariants are obtained from the more general functorial invariants in [C.Stroppel, Categorification of the TemperleyLieb category, tangles, and cobordisms via projective functors, DukeMath.J. 126(3) (2005), 547-596] by restriction. © Foundation Compositio Mathematica 2009.