Twisted Gan–Gross–Prasad nonvanishing conjecture for unitary theta periods

Let $E/F$ be a quadratic extension of number fields, let $V$ be an $n$-dimensional skew-Hermitian space over $E$, and let $\U(V)$ be its unitary group. Let $\pi$ be an irreducible cuspidal automorphic representation of $\operatorname{GL}_n(\mathbb A_E)$, and let $\P(\varphi,\phi)$ be the global period pairing $\pi\otimes\overline{\omega_{\psi',\mu}}$ defined using the theta series attached to the Weil representation $\omega_{\psi',\mu}$. For each place $v$ of $F$, write $\pi_v$ and $\omega_{\psi'_v,\mu_v}$ for the corresponding local representations. Twisted Gan–Gross–Prasad conjecture. The period integral $\P(\varphi,\phi)$ is not identically zero if and only if

$$\operatorname{Hom}_{\U(V)(F_v)}\bigl(\pi_v\otimes\overline{\omega_{\psi'_v,\mu_v}},\mathbb C\bigr)\neq 0$$

for every place $v$ of $F$ and

$$L\left(\frac12,\pi\times{}^{\sigma}\!\pi^\vee\otimes\mu^{-1}\right)\neq 0.$$

Furthermore, if the central $L$-value is nonzero, then there exists a unique skew-Hermitian space $V$ of dimension $n$ over $E$ such that $\P$ is nonzero. This is the global twisted Gan–Gross–Prasad criterion relating local Hom-space nonvanishing and the central Rankin–Selberg $L$-value; the source presents it as a conjectural statement and does not provide a resolution status.