Setup: Let $E/F$ be a CM-extension of global number fields. Let $(V,\phi)$ be an Hermitian space of dimension $n$ over $E$. Let $(V^{\flat}, \phi^{\flat})$ be a subspace of $V$ of dimension $n-1$ on which the restricted Hermitian form $\phi^{\flat}$ from $\phi$ is nondegenerate. Consider the unitary group $G$ associated to $(V,\phi)$ and $H$ the unitary group associate to $(V^{\flat}, \phi^{\flat})$. We view $H$ as a subgroup of $G$.
Let $\mathbf{G} = G \times H$. Let $\pi$ (resp. $\sigma$) be an irreducible automorphic representation of $G(\mathbb{A}_F)$ (resp. $H(\mathbb{A}_F)$), and define $\Pi := \pi \otimes \sigma$. Let $\Phi \in \Pi$ be a cusp form on $\mathbf{G}$. Then we consider the Gan-Gross-Prasad period integral $$ \mathcal{P}(\Phi) = \int_{H(F) \backslash H(\mathbb{A}_F)} \Phi(h) \mathrm{d}h, $$ where $\mathrm{d}h$ is the Tamagawa measures on $H(\mathbb{A}_F)$.
Goal: how to decompose the global integral $\mathcal{P}(\Phi)$ into a product of local Euler factors (local zeta integrals?)?
Attempt 1: The famous Ichino-Ikeda formula provides for tempered $\Pi$, $\Phi \in \Pi$ and $\Phi^{\prime} = \Pi^{\vee}$ an explicit formula $$ \dfrac{\mathcal{P}(\Phi)\mathcal{P}(\Phi^{\prime})}{(\Phi, \Phi^{\prime})} = (\ast) \cdot \mathcal{L}(\pi \times \sigma) \cdot \prod_{v} \dfrac{I(\Phi_{v}, \Phi_v^{\prime})}{\mathcal{L}(\pi_v \times \sigma_v) (\Phi_{v}, \Phi_v^{\prime})}, $$ where $\mathcal{L}(\pi \times \sigma)$ is an appropriate normalization of $L(1/2, \pi \times \sigma)$, and local matrix coefficients $$ I(\Phi_{v}, \Phi_v^{\prime}) := \int_{H(F_v)} (\Pi_v(h_v, h_v)\Phi_{v}, \Phi_v^{\prime}) \mathrm{d} h_v. $$ But this is different from my goal since $\mathcal{P}(\Phi)$ and $\mathcal{P}(\Phi^{\prime})$ appear together in the formula, yet I hope to see only one individual $\mathcal{P}(\Phi)$. Under some additional assumptions, $\mathcal{P}(\Phi)\mathcal{P}(\Phi^{\prime})$ can be $\mathcal{P}(\Phi)^2$. This means that by the Ichino-Ikeda formula, it seems that I can only decompose the square of the period integral into local matrix coefficients, but unable to do it for the period integral itself.
Question (a): So do we have a way out of this issue?
A cheating method? Note that $\Phi^{\prime}$ is auxillary, so we hope to choose $\Phi^{\prime}$ such that $$ \mathcal{P}(\Phi^{\prime}) \neq 0. $$ Then we can simply dividing this period from both sides of the Ikeda-Ichino formula. Does such $\Phi^{\prime}$ exist? As the global Gan-Gross-Prasad for unitary groups is solved (in most of the interesting cases?), this is equivalent to require that the central $L$-value $L(s, \Pi^{\vee}) \neq 0$. But such a requirement is too strong?
Attempt 2: In the context of Jacquet, Piateski-Shapiro and Shalika, we consider the case of general linear groups: we take $G=\mathrm{GL}_{n+1}/E$ and $H=\mathrm{GL}_n/E$. Let $\pi$ (resp. $\sigma$) be a cuspidal automorphic representation of $G$ (resp. $H$) and $\Pi = \pi \otimes \sigma$. For $\phi \in \Pi$, we have the global (Rankin-Selberg) period integral $$ \lambda(\phi) := \int_{H(F) \backslash H(\mathbb{A}_F)} \Phi(h) \mathrm{d}h. $$ Then by the theory of Whittaker models, we have the Whittaker function $W_{\Phi}$ attached to $\Phi$ and $$ \lambda(\phi) = L(1/2, \pi \times \sigma) \prod_w \lambda_{w}^{\natural}(W_{\Phi,w}), $$ where $\lambda_{w}^{\natural}$ is an appropriate normalization of the local integral $$ \lambda_w(W_w) = \int_{N(E_w) \backslash H(E_w)} W_w(h) \mathrm{d}h. $$
Question (b) - a reference request: does this story of Jacquet, Piateski-Shapiro and Shalika generalize to unitary groups setting?
Question (c): Are the considerations in Attempt 2 actually "reproving the Ichino-Ikeda formula"?
