Gross–Prasad conjecture for orthogonal groups

Let $V_{m+1}$ be an orthogonal space of dimension $m+1$, and let $V_m$ be a non-degenerate codimension-one subspace. Assume that $\operatorname{O}(V_m)\times\operatorname{O}(V_{m+1})$ is quasi-split. Put $c=-\operatorname{disc}(V_{\rm odd})/\operatorname{disc}(V_{\rm even})\in F^\times/F^{\times2}$, so that $V_{\rm even}$ is associated to $(\operatorname{disc}(V_{\rm even}),c)$. Let $\phi\in\Phi_{\rm temp}(\operatorname{O}(V_{\rm even}))$ and $(\phi',b)\in\Phi_{\rm temp}(\operatorname{O}(V_{\rm odd}))$. Gross–Prasad conjecture. There exists a unique pair $(\sigma,\tau)\in\Pi_\phi\times\Pi_{\phi',b}$ such that $\sigma\boxtimes\tau$ is a representation of $\operatorname{O}(V_{\rm even}^\bullet)\times\operatorname{O}(V_{\rm odd}^\bullet)$ for a relevant pair of companion spaces and

$$\operatorname{Hom}_{\Delta\operatorname{O}(V_m^\bullet)}(\sigma\boxtimes\tau,\mathbb C)\ne0.$$

Moreover,

$$\iota_c(\sigma)\times\iota(\tau)=(\chi_{\phi'}\cdot d_{\phi,b})\times\chi_\phi.$$

This refines the multiplicity-one result for the orthogonal Gross–Prasad period by determining when the multiplicity is nonzero and identifying the corresponding component-group characters. The analogous special-orthogonal statement is stated as a theorem in the source, but this orthogonal-group formulation remains presented as a conjecture.