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 Białynicki-Birula paving, following work of Fung. That is, we consider the space of complete flags in ${\mathbb{C}}^n$ 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 ${\mathbb{C}}^{\ast }$-attracting sets (that is Białynicki-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 and Kamnitzer’s recent work on link homology via coherent sheaves and suggest directions for future research.