Untwisted Gaiotto equivalence

TL;DR

This paper establishes an equivalence between categories of representations of degenerate supergroups and certain equivariant D-modules on affine Grassmannians and mirabolic subgroups, extending previous results.

Contribution

It proves a new equivalence between supergroup representations and D-modules on affine Grassmannians and mirabolic subgroups, expanding the understanding of their categorical relationships.

Findings

Proves an equivalence between supergroup representations and equivariant D-modules on affine Grassmannians.

Realizes supergroup representation categories as D-modules on mirabolic subgroups with equivariance.

Extends previous work to include more general subgroup conditions.

Abstract

This is a successive paper of arXiv:1909.11492. We prove an equivalence between the category of finite-dimensional representations of degenerate supergroup $\underline{GL}(M|N)$ and the category of $(GL_M(O) \ltimes U_{M, N}(F), \chi_{M, N})$-equivariant D-modules on $Gr_{N}$. We also prove that we can realize the category of finite-dimensional representations of degenerate supergroup $\underline{GL}(M|N)$ as a category of D-modules on the mirabolic subgroup $Mir_L(F)$ with certain equivariant conditions for any $L$ bigger than $N$ and $M$.