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

For a fixed parabolic subalgebra p of gl(n,C) we prove that the centre of the principal block O(p) of the parabolic category O is naturally isomorphic to the cohomology ring of the corresponding Springer fibre. We give a diagrammatic description of O(p) for maximal parabolic p and give an explicit isomorphism to Braden's description of the category Perv_B(G(n,n)) of perverse sheaves on Grassmannians. As a consequence Khovanov's algebra H^n is realised as the endomorphism ring of some object from Perv_B(G(n,n)) which corresponds under localisation and the Riemann-Hilbert correspondence to a full projective-injective module in the corresponding category $O(p)$. From there one can deduce that Khovanov's tangle invariants are obtained from the more general functorial invariants involving category O by restriction.