  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.
