Irreducible objects in the Gaiotto category at roots of unity

TL;DR

This work investigates whether the Travkin–Yang equivalence for q-twisted D-modules on Gr_N can extend to roots of unity by examining irreducible objects. It builds a bridge between the Gaiotto category and positive-characteristic representations of GL(M|N) via Serganova's algorithm to translate highest-weight data between standard and mixed Borels in the modular setting. The authors show a natural bijection between irreducibles in the Gaiotto category and irreducibles in Rep_p(GL(M|N)) with respect to the mixed Borel, by classifying relevant orbits and incorporating the mod p constraints that arise at roots of unity. This provides evidence for the root-of-unity extension of the correspondence and links modular representation theory of supergroups to geometric categories of D-modules on the affine Grassmannian. The results rely on the Brundan–Kujawa reformulation of Serganova's algorithm and clarify how the choice of unipotent subgroup affects the combinatorics of relevant weights while preserving the overall equivalence.

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.

Theorems & Definitions (13)

• Theorem 1
• Theorem 2
• Definition 2.1
• Lemma 3
• proof
• Corollary 4
• Lemma 5
• Proposition 6
• Proposition 7
• proof