2-block Springer fibers: convolution algebras and coherent sheaves

TL;DR

This paper analyzes the structure of 2-block Springer fibers, describing their irreducible components, fixed points, and intersection properties, and introduces a convolution algebra related to link homology and coherent sheaves.

Contribution

It provides a detailed description of 2-block Springer fibers, proves Fung's conjecture on intersections, and constructs a convolution algebra linking to Khovanov's arc algebra and link homology.

Findings

Description of irreducible components and fixed points

Proof of Fung's intersection conjecture

Construction of a convolution algebra related to link homology

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.