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 $\pi$ and $\sigma$ of $G{L}_d(E)$ and $G{L}_m(E)$ 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 $G{L}_d(E)⋊1,\theta$. 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.

58p

References in corpus (1)

• Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups