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

Abstract

We study the category of representations of $\mathfrak{sl}_{m+2n}$ in positive characteristic, whose p-character is a nilpotent whose Jordan type is the two-row partition (m+n,n). In a previous paper with Anno, we used Bezrukavnikov-Mirkovic-Rumynin's theory of positive characteristic localization and exotic t-structures to give a geometric parametrization of the simples using annular crossingless matchings. Building on this, here we give combinatorial dimension formulae for the simple objects, and compute the Jordan-Holder multiplicities of the simples inside the baby Vermas (in special case where n=1, i.e. that a subregular nilpotent, these were known from work of Jantzen). We use Cautis-Kamnitzer's geometric categorification of the tangle calculus to study the images of the simple objects under the [BMR] equivalence. The dimension formulae may be viewed as a positive characteristic analogue of the combinatorial character formulae for simple objects in parabolic category O for $\mathfrak{sl}_{m+2n}$, due to Lascoux and Schutzenberger.