Irreducible objects in the Gaiotto category at roots of unity

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.

💡 Research Summary

The paper investigates whether the celebrated equivalence between the Gaiotto category of $q$‑twisted $D$‑modules on the affine Grassmannian $Gr_{GL_N}$ and the representation category of the quantum supergroup $U_q(\mathfrak{gl}(M|N))$, proved by Travkin and Yang for generic $q$, can be extended to the case where $q$ is a root of unity. Since the quantum supergroup at a root of unity depends on the choice of a Borel subalgebra (different Borel choices may lead to non‑isomorphic Hopf algebras in the super setting), the authors avoid a direct quantum construction and instead work with the classical supergroup $GL(M|N)$ over a field of positive characteristic $p$, where $q^p=1$.

The central observation is that the set of irreducible objects on both sides can be identified via a combinatorial description of highest weights with respect to a particular “mixed” Borel subgroup of $GL(M|N)$. For the standard Borel $B_s$, dominant integral weights are simply pairs $(\lambda,\theta)\in\mathbb Z^M\times\mathbb Z^N$ satisfying the usual non‑increasing conditions. However, the mixed Borel $B_m$—obtained by interleaving even and odd basis vectors as much as possible—is not conjugate to $B_s$, and its dominant weights obey extra congruence relations modulo $p$.

To translate the standard dominant weights into those for $B_m$, the authors employ the Serganova algorithm as reformulated by Brundan and Kujawa (2002). The algorithm processes the set of “extra” positive roots $\Phi^+{st}\setminus\Phi^+{m}$ in a fixed linear order (two possible orders are described, but the final output is independent of the choice). At each step, if the sum $\lambda_i+\theta_j$ is congruent to $0$ modulo $p$, the algorithm does nothing; otherwise it decrements $\lambda_i$ by one and increments $\theta_j$ by one. Iterating through all such roots yields a map $S:A\to\mathbb Z^M\times\mathbb Z^N$, where $A$ is the set of standard dominant weights.

Lemma 3 proves that the image $S(A)$ coincides precisely with the set $M$ of pairs satisfying:

1. $\tilde\lambda_1\ge\cdots\ge\tilde\lambda_M$ and $\tilde\theta_1\ge\cdots\ge\tilde\theta_N$;
2. Whenever $\tilde\lambda_{i-1}=\tilde\lambda_i$ (or $\tilde\theta_i=\tilde\theta_{i+1}$), one has $\tilde\lambda_i+\tilde\theta_i\equiv0\pmod p$. Thus $M$ gives the complete list of highest weights for irreducible $GL(M|N)$‑modules with respect to $B_m$ in characteristic $p$.

The second part of the paper returns to the Gaiotto side. Travkin and Yang’s analysis of $GL_M(\mathcal O)\ltimes U^{-}_{M,N}(K)$‑orbits on $Gr_N$ shows that each orbit contains a unique representative of the form \