The local Gan–Gross–Prasad conjecture for the symplectic-metaplectic case

Let $W$ be a symplectic space over $F$, let $\psi$ be a non-trivial additive character of $F$, and let $\omega_\psi$ be the Weil representation of $\widetilde{\mathrm{Sp}}(W)$. Let $\phi_M$ and $\phi_N$ be generic parameters for $\widetilde{\mathrm{Sp}}(W)$ and $\mathrm{Sp}(W)$, respectively, with packets $\Pi_{\phi_M}$ and $\Pi_{\phi_N}$; write $\iota_\psi$ and $\iota_{\mathfrak{w}'_1}$ for the associated character parametrizations, and let $\Delta\widetilde{\mathrm{Sp}}(W)$ be the diagonal image in $\widetilde{\mathrm{Sp}}(W)\times\mathrm{Sp}(W)$. The local Gan–Gross–Prasad conjecture for the symplectic-metaplectic case. For $\widetilde{\pi}\in\Pi_{\phi_M}$ and $\pi\in\Pi_{\phi_N}$,

$$\operatorname{Hom}_{\Delta\widetilde{\mathrm{Sp}}(W)}\bigl((\widetilde{\pi}\boxtimes\pi)\otimes\overline{\omega_\psi},\mathbb{C}\bigr)\ne 0 \Longleftrightarrow \iota_\psi(\widetilde{\pi})\times\iota_{\mathfrak{w}'_1}(\pi)=\chi_{N_1}\times\left.\chi_M\right|_{A_\phi^+}.$$

This predicts the precise member of the two packets contributing to the theta-twisted invariant functional and gives a character-theoretic form of the local Gan–Gross–Prasad conjecture for the symplectic-metaplectic pair.