On the relative Langlands duality for operatorname{Sp}_(2n) backslash operatorname{GL}_(2n+1) (with an appendix by Zeyu Wang)

We verify the relative Langlands duality conjecture proposed by Ben-Zvi, Sakellaridis, Venkatesh for the hyperspherical Hamiltonian variety $T^*(\operatorname{Sp}_{2n}\backslash \operatorname{GL}_{2n+1})$. We provide numerical (over number fields and function fields) and geometric (in the \'{e}tale setting) evidence that its dual Hamiltonian variety should be $T^*(\operatorname{GL}_n \times \operatorname{GL}_{n+1} \backslash \operatorname{GL}_{2n+1})$ as is predicted by Ben-Zvi, Sakellaridis, Venkatesh.