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.
Categories: math.RT hep-th math.NT