Irreducible objects in the Gaiotto category at roots of unity

Abstract

A theorem of R. Travkin and R. Yang, initially conjectured by D. Gaiotto, states that for a generic (not a root of unity) $q$ the category of $q$-twisted D-modules on the affine Grassmannian $Gr_{GL_N}$ which are equivariant with respect to a certain subgroup (defined by a choice of $0 \le M <N$) of $GL_N$ is equivalent to the category of representations of the quantum supergroup $U_q(\mathfrak{gl}(M|N))$. We aim to see whether this equivalence should hold when $q$ is a root of unity. We begin by asking if there is a natural bijection between the sets of irreducible objects. In this note we make an observation that suggests this should be the case: we show that there is a natural bijection between irreducible objects in the Gaiotto category and in the category of representations of a supergroup $GL(M|N)$ in positive characteristic. The proof is based on the version of the Serganova's algorithm formulated by J. Brundan and J. Kujawa in arXiv:math/0210108.