Mirabolic Satake equivalence and supergroups

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 $GL(N-1,\mathbb{C}[\text{\hspace{0.17em}⁣}[t]\text{\hspace{0.17em}⁣}])$-equivariant perverse sheaves on the affine Grassmannian of $G{L}_N$. We explain how our equivalences fit into a more general framework of conjectures due to Gaiotto and to Ben-Zvi, Sakellaridis and Venkatesh.