Gan–Gross–Prasad conjecture

Let $E/k$ be a quadratic extension of local fields with $k$ not of characteristic $2$, and let $\varepsilon \in \{+1,-1\}$. Let $V$ be a finite-dimensional $E$-vector space equipped with a non-degenerate sesquilinear form that is $\varepsilon$-Hermitian (that is, Hermitian if $\varepsilon = 1$ and skew-Hermitian if $\varepsilon = -1$), and let $W \subseteq V$ be a non-degenerate subspace such that

$$V = W \oplus W^{\perp}, \qquad \dim W^{\perp} = \tfrac{\varepsilon+1}{2}.$$

Write $G(V)$ and $G(W)$ for the unitary groups preserving the forms on $V$ and on $W$, put

$$G = G(V) \times G(W), \qquad H = \Delta G(W) \subseteq G,$$

the diagonal subgroup, and let $\nu$ be the representation of $H$ given by the trivial representation when $\varepsilon = 1$ (the Bessel case) and by the Weil representation when $\varepsilon = -1$ (the Fourier–Jacobi case).

Let $\varphi = \varphi_1 \times \varphi_2$ be a generic local $L$-parameter for $G$ and let $\Pi_{\varphi}$ be the associated Vogan $L$-packet, whose members are the irreducible smooth representations $\pi = \pi_1 \boxtimes \pi_2$ of the groups $G(V') \times G(W')$ attached to the pure inner forms of $G$, indexed as $\pi(\varphi,\eta)$ by the characters $\eta$ of the component group of $\varphi$. Call $\pi \in \Pi_{\varphi}$ relevant when the pair $(V',W')$ on which it is realized satisfies $V' = W' \oplus W'^{\perp}$ with $W'^{\perp} \cong W^{\perp}$, so that the corresponding subgroup $H' = \Delta G(W')$ and the representation $\nu$ of $H'$ are defined. Then

$$\sum_{\text{relevant }\pi \in \Pi_{\varphi}} \dim \operatorname{Hom}_{H}\!\left(\pi \otimes \overline{\nu}, \mathbb{C}\right) = 1,$$

and, with $\eta_{\mathrm{GP}}$ the distinguished character of the component group of $\varphi$ defined in terms of Langlands–Deligne local constants,

$$\operatorname{Hom}_{H}\!\left(\pi(\varphi,\eta) \otimes \overline{\nu}, \mathbb{C}\right) \neq 0 \quad \text{if and only if} \quad \eta = \eta_{\mathrm{GP}}.$$

Globally, let $E/F$ be a quadratic extension of number fields, let $V \supseteq W$ and $G = G(V) \times G(W)$, $H = \Delta G(W)$, $\nu$ be as above over $E/F$, with $\dim_E V = n$ and $\dim_E W = n - \tfrac{\varepsilon+1}{2}$, and let $\pi = \pi_1 \boxtimes \pi_2$ be an irreducible cuspidal automorphic representation of $G(F)\backslash G(\mathbb{A}_F)$ with generic global $L$-parameter, with local components $\pi_v$. Let

$$L_E(s,\pi_1 \times \pi_2) := L_E\!\left(s, \pi_1 \boxtimes \pi_2, \mathrm{std}_{n} \boxtimes \mathrm{std}_{n-1}\right)$$

be the global $L$-function given by the product of the local $L$-factors attached by the local Langlands correspondence, and let $P_H$ be the period functional on $\pi$ obtained by integrating against $\overline{\nu}$ over $[H] = H(F)\backslash H(\mathbb{A}_F)$. Then the following two conditions are equivalent:

1. $P_H$ is not identically zero on $\pi$; 2. $\operatorname{Hom}_{H(F_v)}(\pi_v, \nu_v) \neq 0$ for every place $v$ of $F$, and $L_E(1/2, \pi_1 \times \pi_2) \neq 0$.