Ichino–Ikeda refinement of the global Gross–Prasad conjecture

Let $F$, $\mathbb{A}$, $V_n$, $V_{n+1}$, $\operatorname{SO}_n$, and $\operatorname{SO}_{n+1}$ be as above, and let $\mathcal{P}(\phi,f)=\int_{\operatorname{SO}_n(F)\backslash\operatorname{SO}_n(\mathbb{A})}\phi(g)f(g)\,dg$. For factorizable vectors, write $\phi=\otimes_v\phi_v$ and $f=\otimes_v f_v$, and let $\alpha_v(\phi_v,f_v)$ denote the normalized local period. Ichino–Ikeda refinement. Assume that $\pi_{n+1}$ and $\pi_n$ are tempered cuspidal automorphic representations of $\operatorname{SO}_{n+1}(\mathbb{A})$ and $\operatorname{SO}_n(\mathbb{A})$, respectively, and appear with multiplicity one in the discrete spectrum. Then

$$|\mathcal{P}(\phi,f)|^2=\frac{\Delta_{\operatorname{SO}_{n+1}}}{2^\beta}\frac{L(1/2,\pi_n\times\pi_{n+1})}{L(1,\pi_n,\operatorname{Ad})L(1,\pi_{n+1},\operatorname{Ad})}\prod_v\alpha_v(\phi_v,f_v),$$

where $2^\beta$ is the product of the cardinalities of the component groups attached to the $L$-packets, and $\Delta_{\operatorname{SO}_{n+1}}$ is defined by the parity of $\dim V_{n+1}$ as in the source. The conjecture refines the nonvanishing criterion by predicting the exact normalized period formula.