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.
Reviewer: Volodymyr Mazorchuk (Uppsala)
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) |