Fonctions L de paires pour les groupes classiques

Abstract

Let $\pi$ be a square integrable representation of a classical group and let $\rho$ be a cuspidal representation of a general linear group. We can define in two different ways an L-function $L(\rho \times \pi,s)$: first we can use the Langlands parametrization at each places which is now available, thanks to Arthur's work, and secondly we can transfer $\pi$ to a general linear group, using the twisted endoscopy as established by Arthur. In this paper, we compare the two definitions and we prove, as expected, that the first one has less poles that the second one. Assuming that $\pi$ is cuspidal, we link the poles of the first L-function to the poles of the Eisensteins series and when $\rho$ is a quadratic character and when the groupe is a special orthogonal group, we also link theses poles with the theta lifts. We have some hypothesis at the archimedean places.