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

Part of: Low-dimensional topology Special varieties Linear algebraic groups and related topics Representation theory of groups Rings and algebras arising under various constructions Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  01 July 2009

Catharina Stroppel
Catharina Stroppel*
Affiliation:
Department of Mathematics, University of Glasgow, University Gardens, Glasgow G12 8QW, UK (email: c.stroppel@maths.gla.ac.uk)
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Abstract

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For a fixed parabolic subalgebra 𝔭 of we prove that the centre of the principal block 𝒪0𝔭 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 𝒪0𝔭 for maximal parabolic 𝔭 and give an explicit isomorphism to Braden’s description of the category PervB(G(k,n)) of Schubert-constructible perverse sheaves on Grassmannians. As a consequence Khovanov’s 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 𝒪0𝔭. 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.

Keywords

Khovanov homologySpringer fibreknot invariantsperverse sheavesKazhdan–Lusztig

MSC classification

Secondary: 16S99: None of the above, but in this section 17B10: Representations, algebraic theory (weights) 14M15: Grassmannians, Schubert varieties, flag manifolds 57M27: Invariants of knots and 3-manifolds 20C30: Representations of finite symmetric groups 20G05: Representation theory 14M17: Homogeneous spaces and generalizations
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 4 , July 2009 , pp. 954 - 992
Copyright
Copyright © Foundation Compositio Mathematica 2009

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