Refined Gan–Gross–Prasad-type norm formula for Miyawaki lifts

Let $\sigma=\otimes_v\sigma_v$, $\Sigma=\otimes_v\Sigma_v$, and $\pi=\otimes_v\pi_v$ be the cuspidal representations corresponding to $\mathbf{f}$, $\mathbf{F}$, and $\mathbf{g}$, respectively. Let $\varphi\in\Sigma$ and $\psi\in\pi$ be their adelizations, and define

$$\varphi_\psi(h)=(\varphi|_{\operatorname{Sp}_{4n+2r}\times\operatorname{Sp}_{2r}(\mathbb{A}_{\mathbb{Q}})}(h,-),\psi).$$

For each place $v$, let $\Phi_v$ and $\Psi_v$ be the normalized matrix coefficients, and set

$$I(\Phi_v,\Psi_v)=\int_{\operatorname{Sp}_{2r}(\mathbb{Q}_v)}\Phi_v(g_v)\overline{\Psi_v(g_v)}\,dg_v.$$

Assume that $r>0$, $n<k$, and $\pi_p$ is tempered for every finite place $p$ of $\mathbb{Q}$. Then, for every finite set $S$ of places of $\mathbb{Q}$ containing $\infty$, one has

$$\frac{(\varphi_\psi,\varphi_\psi)}{(\varphi,\varphi)(\psi,\psi)}=C\mathcal{L}^{S}\prod_{v\in S}I(\Phi_v,\Psi_v),$$

where

$$C=\begin{cases}1 & n>0,\\ 1/2 & n=0,\end{cases}$$

and $\mathcal{L}^{S}$ is the explicit partial Euler product specified in the paper.