Exotic t-structures for two-block Springer fibers

TL;DR

This paper explores the exotic t-structure on the derived category of coherent sheaves on two-block Springer fibers, constructing functors related to affine tangles, describing irreducible objects, and linking to annular Khovanov homology.

Contribution

It introduces a new description of irreducible objects in the exotic t-structure and connects these to annular Khovanov's arc algebras, advancing categorification in positive characteristic.

Findings

Irreducible objects are classified by crossingless matchings.

Ext groups between irreducibles form an annular variant of Khovanov's arc algebras.

Constructed functors relate categories of sheaves on different Springer fibers.

Abstract

We study the exotic t-structure on the derived category of coherent sheaves on two-block Springer fibre (i.e. for a nilpotent matrix of type (m+n,n) in type A). The exotic t-structure has been defined by Bezrukavnikov and Mirkovic for Springer theoretic varieties in order to study representations of Lie algebras in positive characteristic. Using work of Cautis and Kamnitzer, we construct functors indexed by affine tangles, between categories of coherent sheaves on different two-block Springer fibres (i.e. for different values of n). After checking some exactness properties of these functors, we describe the irreducible objects in the heart of the exotic t-structure, and enumerate them by crossingless (m,m+2n) matchings. We compute the Ext's between the irreducible objects, and show that the resulting algebras are an annular variant of Khovanov's arc algebras. In subsequent work we will make a link with annular Khovanov homology, and use these results to give a positive characteristic analogue of some categorification results using two-block parabolic category O (by Bernstein-Frenkel-Khovanov, Brundan, Stroppel, et al).