Local Gan–Gross–Prasad conjecture for tempered representations of real unitary groups

Let $\phi_n \in \Phi_\mathrm{temp}(\mathrm{U}_{n}(\mathbb{R}))$ and $\phi_{n+1} \in \Phi_\mathrm{temp}(\mathrm{U}_{n+1}(\mathbb{R}))$ have decompositions

$$\phi_{n}=(m_1\chi_{2\alpha_1}\oplus\dots\oplus m_u\chi_{2\alpha_u})\oplus(\xi_1\oplus\dots\oplus\xi_v)\oplus({}^c\xi_1^{-1}\oplus\dots\oplus{}^c\xi_v^{-1}),$$ $$\phi_{n+1}=(m'_1\chi_{2\beta_1}\oplus\dots\oplus m'_{u'}\chi_{2\beta_{u'}})\oplus(\xi'_1\oplus\dots\oplus\xi'_{v'})\oplus({}^c{\xi'_1}^{-1}\oplus\dots\oplus{}^c{\xi'_{v'}}^{-1}),$$

where $\alpha_i,\beta_j\in\frac{1}{2}\mathbb{Z}$ satisfy $2\alpha_i\equiv n-1\pmod 2$ and $2\beta_j\equiv n\pmod 2$, the $m_i$ and $m'_j$ are the corresponding positive multiplicities, $m_1+\dots+m_u+2v=n$, $m'_1+\dots+m'_{u'}+2v'=n+1$, and the $\xi_i$ and $\xi'_j$ are unitary characters of $\mathbb{C}^{\times}$ not of the indicated character forms. Local Gan–Gross–Prasad conjecture. There exists a unique pair $(\pi_n,\pi_{n+1})\in\Pi_{\phi_n}\times\Pi_{\phi_{n+1}}$ such that $(\pi_n,\pi_{n+1})$ is a pair of representations of a relevant pair $(\mathrm{U}_n(\mathbb{R}),\mathrm{U}_{n+1}(\mathbb{R}))$ and

$$\operatorname{Hom}_{\Delta\mathrm{U}_{n}(\mathbb{R})}(\pi_n\otimes\pi_{n+1},\mathbb{C})\ne0.$$

Moreover, for $e_{2\alpha_i}\in A_{\phi_n}$ and $e_{2\beta_j}\in A_{\phi_{n+1}}$,

$$J(\pi_n)(e_{2\alpha_i})=\varepsilon(\chi_{2\alpha_i}\otimes\phi_{n+1},\psi_{-2}^{\mathbb{C}}),$$ $$J(\pi_{n+1})(e_{2\beta_j})=\varepsilon(\phi_n\otimes\chi_{2\beta_j},\psi_{-2}^{\mathbb{C}}).$$

Here relevance means $(\mathrm{U}(p,q),\mathrm{U}(p+1,q))$ when $n$ is even and $(\mathrm{U}(p,q),\mathrm{U}(p,q+1))$ when $n$ is odd. This conjecture predicts exactly when the restriction multiplicity is nonzero; Sun–Zhu proved that the multiplicity is at most one, while the asserted existence, uniqueness, and epsilon-factor characterization are the remaining content.