Exotic t-structures for two-block Springer fibers

Abstract

We study the exotic t-structure on Dn, 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 on Dn 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 characteristic p analogue of some categorification results using two-block parabolic category O (by Bernstein-Frenkel-Khovanov, Brundan, Stroppel, et al).

Modular representations in type A with a two-row nilpotent central character

“…In [3], we show that the simple objects in Mod fg e,λ (U g) can be parametrized by Cross(m, n). Let us denote the simple objects in Mod fg e,0 (U g) by M α (see Section 2.4 for more details).…”

“…The proof of Lusztig's conjectures in [7] builds upon Bezrukavnikov-Mirkovic-Rumynin's positive characteristic localization theory from [6], which gives a derived equivalence between categories of modular representations, and categories of coherent sheaves on Springer theoretic varieties. In a previous paper joint with Rina Anno, [3], we use [BMR] localization theory, combined with Cautis and Kamnitzer's tangle categorification results from [10], to study the case where g = sl m+2n and the p-character is a nilpotent with Jordan type (m + n, n). Under the equivalences from [6], the irreducible representations will correspond to complexes of coherent sheaves, which are known as "simple exotic sheaves".…”

“…We use our results from [3] to calculate the dimensions of the irreducible representations. [BMR]localization theory allows to express the dimensions in terms of the Euler characteristics of the simple exotic sheaves; we compute the latter quantity on the Grothendieck using Cautis-Kamnitzer's tangle categorification from [10].…”

“…The main result of this paper is a computation of the dimensions of the simple objects. In order to state our results, we briefly recall the combinatorial set-up from [3], and the parametrization of the irreducibles there. Definition 1.1.…”

“…In Section 2, we review some background material on modular representation theory of Lie algebras, BMR localization theory (cf. [6]), and the results from the previous paper [3] constructing the exotic sheaves corresponding to the simple objects.…”