Let $E/F$ be a quadratic extension of $p$-adic fields and let $d$, $m$ be nonnegative integers of distinct parities. Fix admissible irreducible tempered representations $\pi$ and $\sigma$ of $GL_d(E)$ and $GL_m(E)$ respectively. We assume that $\pi$ and $\sigma$ are conjugate-dual. That is to say $\pi\simeq \pi^{ \vee,c}$ and $\sigma\simeq \sigma^{ \vee,c}$) where $c$ is the non trivial $F$-automorphism of $E$. This implies, we can extend $\pi$ to an unitary representation $\tilde{ \pi}$ of a nonconnected group $GL_d(E)\rtimes { 1,\theta}$. Define $\tilde{ \sigma}$ the same way. We state and prove an integral formula for $\epsilon(1/2,\pi\times \sigma,\psi_E)$ involving the characters of $\tilde{ \pi}$ and $\tilde{ \sigma}$. This formula is related to the local Gan-Gross-Prasad conjecture for unitary groups.
Categories: math.RT math.NT
