Irreducible objects in the Gaiotto category at roots of unity

TL;DR

This paper investigates the relationship between irreducible objects in the Gaiotto category at roots of unity and supergroup representations, suggesting a natural bijection based on Serganova's algorithm.

Contribution

It proposes a conjectural extension of the Gaiotto category equivalence to roots of unity and establishes a bijection with supergroup representations in positive characteristic.

Findings

Bijection between irreducible objects in Gaiotto category and supergroup representations.

Supports the conjecture that the category equivalence extends to roots of unity.

Utilizes Serganova's algorithm to relate these categories.

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.