The global Gan–Gross–Prasad conjecture for GSpin groups

Let $W$ be a non-degenerate quadratic space over $F$, and let $W_0\subset W$ be a non-degenerate subspace whose orthogonal complement $W_0^\perp$ is split of dimension $2r+1$. Let $N_r$ be the unipotent radical of the standard parabolic subgroup of $\operatorname{GSpin}(W)$ stabilizing a complete flag of isotropic subspaces determined by $W_0^\perp$, and let $\psi_{r,w_0}$ be the corresponding character depending on an anisotropic vector $w_0$. The associated Bessel period is

$$\mathcal{P}(\varphi, \varphi^\prime)=\int_{\ker(\operatorname{pr})(\mathbb{A})\operatorname{GSpin}(W_0)(F)\backslash\operatorname{GSpin}(W_0)(\mathbb{A})} \varphi^{N_r,\psi_{r,w_0}}(g)\varphi^\prime(g)\\,dg,$$

where

$$\varphi^{N_r,\psi_{r,w_0}}(g)=\int_{N_r(F)\backslash N_r(\mathbb{A})}\varphi(ug)\psi_{r,w_0}^{-1}(u)\\,du.$$

Global Gan–Gross–Prasad conjecture. Let $\pi$ and $\pi^\prime$ be irreducible tempered cuspidal automorphic representations of $\operatorname{GSpin}(W)(\mathbb{A})$ and $\operatorname{GSpin}(W_0)(\mathbb{A})$, respectively, with $\omega_\pi\omega_{\pi^\prime}=1$, occurring with multiplicity one in the discrete spectrum. Then the following are equivalent:

1. The Bessel period is non-zero, namely $\mathcal{P}(\varphi,\varphi^\prime)\ne 0$ for some $\varphi\in V_\pi$ and $\varphi^\prime\in V_{\pi^\prime}$.
2. The central value $L(\tfrac12,\pi\times\pi^\prime)$ is non-zero.

This is the GSpin analogue of the global Gan–Gross–Prasad conjecture for special orthogonal groups. The conjecture relates the non-vanishing of a distinguished automorphic period to the central Rankin–Selberg $L$-value; the general equivalence remains open, although some low-rank cases and particular implications are known.