The local twisted Gan–Gross–Prasad conjecture for unitary groups

Let $F$ be a $p$-adic field, let $E$ and $K$ be quadratic field extensions of $F$, and let $V$ be an $n$-dimensional skew-hermitian space relative to $E/F$. Set $V_K=V\otimes_F K$, $G=\operatorname{Res}_{K/F}U(V_K)$, and $H_V=U(V)$. For a nontrivial additive character $\psi$ of $F$ and a conjugate-symplectic character $\mu$ of $E^{\times}$, let $\omega_{V,\psi,\mu}$ be the Weil representation and define

$$m_V(\pi)=\dim \operatorname{Hom}_{H_V}(\pi,\omega_{V,\psi,\mu}).$$

Let $L=K\otimes_F E$, let $M$ be a generic $L$-parameter for $G$ with associated $L$-packet $\Pi_M\subset \operatorname{Irr}(G)$, and write $A_M=\prod_{i\in I}\mathbb{Z}/2\mathbb{Z}\cdot a_i$. The local twisted Gan–Gross–Prasad conjecture. (1) For every irreducible representation $\pi$ of $G(F)$, $m_V(\pi)\leq 1$. (2) Summing over the two skew-hermitian spaces $V$ over $E$ of dimension $n$ and over $\pi\in\Pi_M$, one has

$$\underset{V}{\sum}\sum_{\pi\in \Pi_M}m_V(\pi)=1.$$

(3) If $V_0$ is the unique space giving a nonzero contribution, then, for $e\in E_0^{\times}$ with $E=F(e)$,

$$\mu(\det V_0)=\varepsilon(1/2,\operatorname{As}_{L/E}(M)\times\mu^{-1},\psi_E)\cdot\det(\operatorname{As}_{L/E}(M))(e)\cdot\omega_{K/F}(e^2)^{n(n-1)/2}.$$

(4) The unique $\pi\in\Pi_M$ contributing nontrivially corresponds to the character satisfying

$$\chi(a_i)=\varepsilon(1/2,[\operatorname{As}(M_i)+\operatorname{As}(M)+\operatorname{As}(M/M_i)]\cdot\mu^{-1},\psi_{E,e}),$$

where $\psi_{E,e}=\psi(\operatorname{Tr}_{E/F}(ex))$.