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

Published:2014-06-11
Printed: Oct 2014
• Raphaël Beuzart-Plessis,
Institut de Mathématiques de Jussieu, 2 Place Jussieu 75005 Paris, France
 Format: LaTeX MathJax PDF

## 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 $GL_d(E)$ 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.
 Keywords: epsilon factor, twisted groups
 MSC Classifications: 22E50 - Representations of Lie and linear algebraic groups over local fields [See also 20G05] 11F85 - $p$-adic theory, local fields [See also 14G20, 22E50]

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