Global Gan–Gross–Prasad conjecture for discrete global A-parameters

Let $M \times N$ be a discrete global $A$-parameter of $G_n \times G_m$, and let $\pi_1 \boxtimes \pi_2$ be an irreducible discrete automorphic representation of $G_n(\mathbb{A}) \times G_m(\mathbb{A})$ with $A$-parameter $M \times N$.

Global Gan–Gross–Prasad conjecture. If $n-m$ is odd, then a nonzero Bessel period $\mathcal{B}$ on $\pi_1 \boxtimes \pi_2$ implies that $(M,N)$ is a relevant pair. When $(M,N)$ is relevant, there exists an irreducible discrete automorphic representation $\pi_1' \boxtimes \pi_2' \in \Pi_{M \times N}^{\mathrm{rel}}$ with nonzero Bessel period if and only if

$$L(s,M,N)\big|_{s=0} \neq 0.$$

If $n-m$ is even, then a nonzero Fourier–Jacobi period $\mathcal{FJ}$ on $\pi_1 \boxtimes \pi_2 \boxtimes \nu_{\psi^{-1},Z_m}$ implies that $(M,N)$ is a relevant pair. When $(M,N)$ is relevant, there exists an irreducible discrete automorphic representation $\pi_1' \boxtimes \pi_2' \in \Pi_{M \times N}^{\mathrm{rel}}$ with nonzero Fourier–Jacobi period if and only if

$$L(s,M,N)\big|_{s=0} \neq 0.$$

This is the global form of the Gan–Gross–Prasad conjecture in the framework of Arthur's global $A$-parameters. It relates nonvanishing of Bessel and Fourier–Jacobi periods across relevant pure inner forms to the nonvanishing of the corresponding central extended $L$-value.