The local Gross–Prasad conjecture for GSpin(4) × GSpin(5)

Let $F$ be a number field, let $v$ be a place of $F$, let $\Pi_v$ be an irreducible smooth representation of $G(F_v)$, and let $\Sigma_v=\Sigma_{1,v}\boxtimes\Sigma_{2,v}$ be an irreducible representation of $\operatorname{GL}_2(F_v)\times\operatorname{GL}_2(F_v)$. Assume that the central characters satisfy $\chi_{\Pi,v}\chi_{\Sigma_1,v}\chi_{\Sigma_2,v}=1$, and let $\Phi(\Pi_v)$ be the local $L$-packet containing $\Pi_v$. Local Gross–Prasad conjecture. One has

$$\sum_{\Pi_v^\sharp\in\Phi(\Pi_v)}\dim\operatorname{Hom}_{H_v}(\Pi_v^\sharp\otimes\Sigma_v,\mathbf C)=\begin{cases}1&\text{if }\varepsilon_v(\Pi_v\times\Sigma_v)=1,\\\\0&\text{if }\varepsilon_v(\Pi_v\times\Sigma_v)=-1.\end{cases}$$

Moreover, an explicit recipe specifies the unique member $\Pi_v^\sharp$ of the $L$-packet for which the Hom-space is non-zero. The conjecture is known when $\chi_{\Pi,v}$ is a square in the character group of $F_v^\times$; the general case is attributed to forthcoming work of Emory and Takeda. The full conjecture also predicts the relevant summand of the restriction of $\Sigma_v$ and periods on a specific non-split inner form when the root number is $-1$.