Author Description

A note to readers: This book is in French. The conjecture of Gross and Prasad determines, under some assumptions, the restriction of an irreducible admissible representation of a group $G = SO(n)$ over a local field to a subgroup of the form $G' = SO(n - 1)$. For two generic $L$-paquets (more precisely two generic Vogan's $L$-packets), the first for $G$, the second for $G'$, the conjecture states that there is a unique pair $(\pi,\pi')$ in the product of the two packets such that $\pi'$ appears in the restriction of $\pi$. Moreover, the parametrization of $\pi$ and $\pi'$ (in the usual parametrization of $L$-packets) is given by an explicit formula involving some $\epsilon$-factors. In this second volume of Asterisque devoted to the conjecture, the authors give its proof when the base field is non-archimedean. In the first paper, they consider an irreducible admissible and self-dual representation of a group $GL(N)$. They prove that the value at the center of symmetry of its $\epsilon$-factor is given by an integral formula in which the character of an extension of the representation to the twisted $GL(N)$ appears. The second paper proves the conjecture for tempered representations. It is a consequence of the stabilization, in the sense of endoscopy theory, of the two integral formulas proved in the first paper above and in volume 346. Here the authors use some properties of $L$-packets that are still conjectural, but were probably proved by Arthur. In the last paper with Moeglin, they extend the result to non-tempered generic $L$-packets. It follows from the following fact that they prove that the elements in these $L$-packets are irreducible induced representations from tempered representations.