Expression d'un facteur epsilon de paire par une formule intégrale

Abstract

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 $π$ and $σ$ of $GL_d(E)$ and $GL_m(E)$ respectively. We assume that $π$ and $σ$ are conjugate-dual. That is to say $π\simeq π^{\vee,c}$ and $σ\simeq σ^{\vee,c}$) where $c$ is the non trivial $F$-automorphism of $E$. This implies, we can extend $π$ to an unitary representation $\tildeπ$ of a nonconnected group $GL_d(E)\rtimes {1,θ}$. Define $\tildeσ$ the same way. We state and prove an integral formula for $ε(1/2,π\times σ,ψ_E)$ involving the characters of $\tildeπ$ and $\tildeσ$. This formula is related to the local Gan-Gross-Prasad conjecture for unitary groups.