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

TL;DR

We verify the relative Langlands duality for the hyperspherical variety $T^*(Sp_{2n}\backslash GL_{2n+1})$ in both global numerical and geometric frameworks. The dual object is predicted to be $T^*(GL_n\times GL_{n+1}\backslash GL_{2n+1})$, and the paper confirms this through explicit period computations, L-function identities, and geometric analogues. The numeric analysis shows vanishing of selected $Sp_{2n}$-periods for cuspidal and certain Eisenstein data, with precise local–global factorization matching the BZSV predictions; the geometric appendix establishes cuspidal vanishing for automorphic sheaves in the nilpotent setting. Collectively, the work strengthens the case for the BZSV duality in the GL–Sp context and illustrates the deep link between automorphic periods, L-values, and geometric Langlands phenomena.

Abstract

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 étale 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.

Theorems & Definitions (53)

• Conjecture 1.1.1
• Remark 1.1.2
• Theorem 1.1.3
• Theorem 1.2.1
• Remark 1.2.2
• Proposition 3.1.1
• proof
• Lemma 3.1.2
• proof
• Corollary 3.1.3