Mirabolic Satake equivalence and supergroups

TL;DR

This paper develops a mirabolic geometric Satake equivalence and relates supergroup representations to equivariant perverse sheaves, advancing the understanding of geometric representation theory and supergroups.

Contribution

It introduces a mirabolic analogue of the geometric Satake equivalence and establishes a new link between supergroup representations and perverse sheaves on the affine Grassmannian.

Findings

Established a mirabolic geometric Satake equivalence.

Proved an equivalence connecting supergroup representations with equivariant perverse sheaves.

Integrated these results into a broader conjectural framework.

Abstract

We construct a mirabolic analogue of the geometric Satake equivalence. We also prove an equivalence that relates representations of a supergroup with the category of $GL(N-1,{\mathbb C}[\![t]\!])$-equivariant perverse sheaves on the affine Grassmannian of $GL_N$. We explain how our equivalences fit into a more general framework of conjectures due to Gaiotto and to Ben-Zvi, Sakellaridis and Venkatesh.