On special Bessel periods and the Gross–Prasad conjecture for $$\mathrm {SO}(2n+1) \times \mathrm {SO}(2)$$

Abstract

In this paper we investigate one instance of the global Gross–Prasad conjecture. Our main result proves that if an irreducible cuspidal automorphic representation \(\pi \) of an odd dimensional special orthogonal group, whose local component \(\pi _{w}\) at some finite place w is generic, admits the special Bessel model corresponding to a quadratic extension E over a base field F, then the central L-value \(L(1/2,\pi )L(1/2,\pi \times \chi _E)\) does not vanish. Here \(\chi _E\) denotes the quadratic character of \(\mathbb A_F^\times \) corresponding to E. As an application, we obtain the equivalence between the non-vanishing of the special Bessel period and that of the corresponding central L-value when \(\pi \) is associated to a full modular holomorphic Siegel cusp form of degree two, which is a Hecke eigenform, and E is an imaginary quadratic extension of \(\mathbb Q\).