Ichino–Ikeda conjecture for Gan–Gross–Prasad periods

Let $F$ be a number field, let $F'=F$ in the orthogonal case and a quadratic extension of $F$ in the Hermitian case, and let $W_{n-1}\subset W_n$ be as in the Gan–Gross–Prasad setting. Put ${\mathrm {G}}={\mathrm {G}}_{n-1}\times {\mathrm {G}}_n$ and ${\mathrm {H}}={\mathrm {G}}_{n-1}$, with the diagonal embedding, and let $\pi=\bigotimes_v\pi_v$ be a tempered cuspidal automorphic representation. Normalize the local forms $\alpha_v$ by

$$\alpha_v(\phi_v,\varphi_v)=\frac{1}{{\mathscr{L}}(\frac{1}{2},\pi_v)}\int_{{\mathrm {H}}(F_v)}\langle \pi_v(h)\phi_v,\varphi_v\rangle_v\,dh.$$

Ichino–Ikeda conjecture. For every pure tensor $\phi=\bigotimes_v\phi_v\in\pi$,

$$\left|{\mathscr{P}}_{\mathrm {H}}(\phi)\right|^2=2^{-\beta_\pi}{\mathscr{L}}(1/2,\pi)\prod_v\alpha_v(\phi_v,\phi_v),$$

where $\beta_\pi$ is the rank of the finite elementary $2$-group associated to the $L$-parameter of $\pi$. This refines the global Gan–Gross–Prasad conjecture; it is known in several cases, including the stated Hermitian case under the hypotheses given later in the paper, but is not resolved in general.