Liu's refined global Gross–Prasad conjecture for special Bessel periods

Let $F$ be the number field, let $G\bigrin\mathcal G$, and let $\pi=\otimes_v\pi_v$ be an irreducible cuspidal automorphic representation of $G(\mathbb A)$. Suppose that $\pi$ is almost locally generic, meaning that $\pi_v$ is generic at almost all places $v$ of $F$. For each place $v$, let $R_{\lambda,v}$ and $\chi_{\lambda,v}$ define the local Bessel model, let $V_{\pi_v}$ be the representation space, let $K_{G,v}$ be a maximal compact subgroup, and let $\alpha_v$ and $\alpha_v^\natural$ be the local and normalized local integrals. Let $B_{\lambda,\psi}$ be the global Bessel period, let $\Psi(\pi)$ be the elliptic Arthur parameter of $\pi$, let $\mathcal S(\Psi(\pi))$ be the centralizer of its image in the dual group, let $C_\lambda$ be the stated constant, and let $\chi_E$ be the quadratic character associated with the extension $E/F$. Then:

Liu's refined global Gross–Prasad conjecture. One has

$$\dim_{\mathbb C}\operatorname{Hom}_{R_{\lambda,v}}(\pi_v,\chi_{\lambda,v})=1$$

if and only if $\alpha_v(\phi_v')\ne0$ for some $K_{G,v}$-finite vector $\phi_v'\in V_{\pi_v}$. Moreover, for every nonzero decomposable cusp form $\phi=\otimes_v\phi_v\in V_\pi$,

$$\frac{|B_{\lambda,\psi}(\phi)|^2}{\langle\phi,\phi\rangle}=\frac{C_\lambda}{|\mathcal S(\Psi(\pi))|}\left(\prod_{j=1}^n\xi_F(2j)\right)\frac{L(1/2,\pi)L(1/2,\pi\times\chi_E)}{L(1,\pi,\operatorname{Ad})L(1,\chi_E)}\prod_v\frac{\alpha_v^\natural(\phi_v)}{\langle\phi_v,\phi_v\rangle_v},$$

where the product is over the finite set of places that are not good, and all $L$-functions are completed.

This refines the global Gross–Prasad formula by relating the Bessel period to central $L$-values, the Arthur component group, and normalized local periods. The conjecture is attributed in the source to Liu; its resolution status is not specified here.