# Sur Les Conjectures de Gross et Prasad II

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*Colette Mœglin; Jean-Loup Waldspurger*

A publication of the Société Mathématique de France

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 Astérisque 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 Mœglin, 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.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

#### Readership

Graduate students and research mathematicians interested in Gross-Prasad conjectures.