Relative Langlands duality for $\mathfrak{osp}(2n + 1|2n)$

TL;DR

This paper proves a specific case of twisted relative Langlands duality involving orthogonal and symplectic groups, establishing an $S$-duality correspondence and formulating a related global conjecture.

Contribution

It establishes an $S$-duality for a particular orthogonal-symplectic group action and formulates a global conjecture on categorical theta-correspondence in the Langlands program.

Findings

Proved $S$-duality for $ ext{SO}(2n+1) imes ext{Sp}(2n)$ action.

Identified the dual space as the symplectic mirabolic space and $T^* ext{Sp}(2n)$.

Formulated a global conjecture on categorical theta-correspondence.

Abstract

We establish an $S$-duality converse to the one studied by the 1st, 2nd and 4th authors; this is also a case of a twisted version of the relative Langlands duality of Ben Zvi, Sakellaridis and Venkatesh.. Namely, we prove that the $S$-dual of $\text{SO}(2n+1)\times \text{Sp}(2n)$ acting on the tensor product of their tautological representations is the symplectic mirabolic space $\text{Sp}(2n)\times\text{Sp}(2n)$ acting on the product $T^* \text{Sp}(2n)$ and the tautological representations of $\text{Sp}(2n)$. (Note that due to the anomaly, the dual of the second factor $\text{Sp}(2n)$ is the metaplectic dual, i.e. $\text{Sp}(2n)$). We also formulate the corresponding global conjecture, which describes explicitly the categorical theta-correspondence on the Langlands dual side.