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

We verify the relative Langlands duality conjecture proposed by Ben-Zvi, Sakellaridis, Venkatesh for the hyperspherical Hamiltonian variety $T^{\ast }({\mathrm{Sp}}_{2n}\mathrm{\backslash }{\mathrm{GL}}_{2n+1})$. We provide numerical (over number fields and function fields) and geometric (in the étale setting) evidence that its dual Hamiltonian variety should be $T^{\ast }({\mathrm{GL}}_n\times {\mathrm{GL}}_{n+1}\mathrm{\backslash }{\mathrm{GL}}_{2n+1})$ as is predicted by Ben-Zvi, Sakellaridis, Venkatesh.