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

E-ISSN： 1570-5846|145|4|954-992

ISSN： 0010-437x

Source： Compositio Mathematica, Vol.145, Iss.4, 2009-07, pp. : 954-992

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Abstract

For a fixed parabolic subalgebra of $\mathfrak {gl}(n,\mathbb {C})$ we prove that the centre of the principal block ₀ of the parabolic category is naturally isomorphic to the cohomology ring H^*() of the corresponding Springer fibre. We give a diagrammatic description of ₀ for maximal parabolic and give an explicit isomorphism to Braden’s description of the category Perv_B(G(k,n)) of Schubert-constructible perverse sheaves on Grassmannians. As a consequence Khovanov’s algebra 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 ₀. From there one can deduce that Khovanov’s tangle invariants are obtained from the more general functorial invariants in [C. Stroppel, Categorification of the Temperley Lieb category, tangles, and cobordisms via projective functors, Duke Math. J. 126(3) (2005), 547–596] by restriction.