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

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

2006-08-10

arxiv.org/abs/math/0608234v3

Compositio Math. 2008

Mathematics

Representation Theory

39 pages, 9 figures, added a few remarks

Scientific paper

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.

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