2-block Springer fibers: convolution algebras and coherent sheaves

Abstract

For a fixed 2-block Springer fiber, we describe the structure of its irreducible components and their relation to the Bialynicki-Birula paving, following work of Fung. That is, we consider the space of complete flags in C^n preserved by a fixed nilpotent matrix with 2 Jordan blocks, and study the action of diagonal matrices commuting with our fixed nilpotent. In particular, we describe the structure of each component, its set of torus fixed points, and prove a conjecture of Fung describing the intersection of any pair. Then we define a convolution algebra structure on the direct sum of the cohomologies of pairwise intersections of irreducible components and closures of C^*-attracting sets (that is, Bialynicki-Birula cells), and show this is isomorphic to a generalization of the arc algebra of Khovanov defined by the first author. We investigate the connection of this algebra to Cautis & Kamnitzer's recent work on link homology via coherent sheaves and suggest directions for future research.

v3: final version, to appear in Commentarii Mathematici Helvetici; corrected statement of main result, and made numerous small changes throughout article. 38 pages.

References in corpus (7)

• Quiver varieties and Beilinson-Drinfeld Grassmannians of type A
• Highest weight categories arising from Khovanov's diagram algebra III: category O
• A Localization Theorem for Finite W-algebras
• An invariant of tangle cobordisms via subquotients of arc rings
• Harish-Chandra bimodules for quantized Slodowy slices
• Singular blocks of parabolic category O and finite W-algebras
• Affine tangles and irreducible exotic sheaves

Cited by in corpus (23)

• The sl_3 web algebra
• An odd categorification of quantum sl(2)
• Highest weight categories arising from Khovanov's diagram algebra III: category O
• 2-row Springer fibres and Khovanov diagram algebras for type D
• sl(N)-Web categories
• Oddification of the cohomology of type A Springer varieties
• ${\mathfrak{gl}}_n$-webs, categorification and Khovanov-Rozansky homologies
• The sl(N)-web algebras and dual canonical bases
• A web basis of invariant polynomials from noncrossing partitions
• Springer fiber components in the two columns case for types A and D are normal
• DG structures on odd categorified quantum sl(2)
• ${\mathfrak{sl}}_3$-web bases, intermediate crystal bases and categorification
• Cuspidal ${\mathfrak{sl}}_n$-modules and deformations of certain Brauer tree algebras
• Odd Khovanov homology for tangles
• Irreducible components of exotic Springer fibres
• Irreducible components of two-row Springer fibers and Nakajima quiver varieties
• Transitioning between tableaux and spider bases for Specht modules
• Two-block Springer fibers of types C and D: a diagrammatic approach to Springer theory
• Euler Characteristic of Springer fibers
• Real Springer fibers and odd arc algebras
• An explicit bijection between semistandard tableaux and non-elliptic sl_3 webs
• A study of irreducible components of Springer fibers using quiver varieties
• Exotic Springer fibers for orbits corresponding to one-row bipartitions