Mirabolic Satake equivalence and supergroups

Abstract

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

Untwisted Gaiotto equivalence

“…The last isomorphism follows from [BFGT,Lemma 3.13] As a corollary, we have Corollary 2.5.2. For M ≤ N , we have…”

“…Convolution. In this section, let us recall the definitions of the convolution products given in [BFGT,Section 3.4].…”

“…Using the method applied in [BFGT,Section 4.3], we can prove that D(T M,N ) can also be regarded as a full cocomplete subcategory…”

“…Accordingly, the action of GL M (O) becomes the original action composed with g → (g t ) −1 . For more details, see [BFGT,Section 3.15].…”

“…In §4, to finish the proof, we only need to compare two tensor products of categories. We only need to show that the equivalence [BFGT,Theorem 4…”