Expression d'un facteur epsilon de paire par une formule int\'egrale

TL;DR

This paper derives an integral formula for epsilon factors of pairs of conjugate-dual representations of general linear groups over quadratic p-adic extensions, linking to the Gan-Gross-Prasad conjecture for unitary groups.

Contribution

It introduces a new integral expression for epsilon factors involving characters of extended representations, advancing understanding of local factors in the Gan-Gross-Prasad framework.

Findings

Derived an integral formula for epsilon(1/2, π×σ, ψ_E)

Connected the formula to the local Gan-Gross-Prasad conjecture

Extended representations to nonconnected groups for the analysis

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.