Towards the Relative Langlands Duality for Orthosymplectic Pairs

Abstract

In this paper we prove a conjectured equivalence of categories, showing that the S-dual of ${\mathrm{SO}}_{2n}\times {\mathrm{Sp}}_{2n}$ acting on ${\mathbb{C}}_{+}^{2n}\otimes {\mathbb{C}}_{-}^{2n}$ is equal to ${\mathrm{SO}}_{2n+1}\times {\mathrm{SO}}_{2n}\circlearrowright T^{\ast }{\mathrm{SO}}_{2n+1}$. This result is a particular case of a non-polarized version of the (local) relative Langlands duality of Ben Zvi, Sakellaridis and Venkatesh. Similar results for the pairs $({\mathrm{SO}}_{2n+1},{\mathrm{Sp}}_{2n})$ and $({\mathrm{GL}}_n,{\mathrm{GL}}_m)$ were proved by Braverman, Finkelberg, Kazhdan and Travkin and by Fu respectively, whereas the converse result was proved by Braverman, Finkelberg, and Travkin. As a consequence of our main result, we prove that Langlands functoriality of the Derived Satake isomorphism for the pair ${\mathrm{Sp}}_{2n},{\mathrm{SO}}_{2n}$ is given by the theta correspondence. Our approach works (with appropriate modifications) in the general even orthosymplectic case of $\mathfrak{osp}(2m|2n)$.