Twisted Gaiotto equivalence for $\mathrm{GL}(M|N)$

Abstract

In ${\mathrm{GL}}_{\mathrm{N}}$, a series of subgroups indexed by $0\le M\le N-1$ were noticed by H. Jacquet-I. Piatetski Shapiro-J. Shalika, J. Cogdell, and D. Gaiotto. It was conjectured by D. Gaiotto that the categories of twisted D-modules on the affine Grassmannian of ${\mathrm{GL}}_{\mathrm{N}}$ with equivariant structure with respect to these subgroups are equivalent to the categories of representations of quantum supergroups $U_q(\mathfrak{gl}(M|N))$. When $M=0$ and $M=N-1$, this conjecture was proved due to D. Gaitsgory and A. Braverman-M. Finkelberg-R. Travkin, respectively. In this paper, we prove the other cases. We adapt the global method originating from arXiv:math/0611323 and arXiv:0705.4571. In order to compare the global definition of Gaiotto category with the local definition, we generalize the local-global comparison theorem of arXiv:1811.02468 to a general setting.

82 pages, add more details in the local-global comparison, comments are welcome!