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2-block Springer fibers: convolution algebras and coherent sheaves. (English) Zbl 1241.14009
The main result of this paper asserts that the convolution algebra associated with a $$2$$-block Springer fiber is isomorphic to a certain combinatorially described algebra with basis given by certain arc diagrams (and inspired by diagrammatic description of Khovanov homology). A similar result holds for the extended version of the convolution algebra and the quasi-hereditary cover of the generalized arc algebra. The generalized arc algebra also describes perverse sheaves on a Grassmanian and endomorphims of a basic projective-injective module in a maximal parabolic block of the BGG category $$\mathcal{O}$$. The same algebra has several other geometric incarnations which are also investigated.

MSC:
 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 44A35 Convolution as an integral transform 16G10 Representations of associative Artinian rings 14F25 Classical real and complex (co)homology in algebraic geometry 53D40 Symplectic aspects of Floer homology and cohomology 57M27 Invariants of knots and $$3$$-manifolds (MSC2010)
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