# Séminaires : Séminaire Groupes Réductifs et Formes Automorphes

 Equipe(s) : fa, tn, Responsables : Alexis Bouthier, Benoît Stroh Email des responsables : alexis.bouthier@imj-prg.fr, benoit.stroh@imj-prg.fr Salle : Adresse : Description

 Orateur(s) D. GROSS - Harvard University et Université Paris XI, Titre On the restriction of irreducible representations of the group $U_n(k)$ to the subgroup $U_{n-1}(k)$ Date 02/10/2008 Horaire 14:00 à 15:00 Diffusion Résume (L'exposé sera suivi d'une réunion sur le contenu du séminaire. Les participants seront invités à faire part de leur souhait pour cette année.)\\ Let $k$ be a local field, and let $K$ be a separable quadratic field extension of $k$. It is known that an irreducible complex representation $\pi_1$ of the unitary group $G_1=U_n(k)$ has a multiplicity free restriction to the subgroup $G_2=U_{n-1}(k)$ fixing a non-isotropic line in the corresponding Hermitian space over $K$. More precisely, if $\pi_2$ is an irreducible representation of $G_2$, then $\pi:=\pi_1\otimes\pi_2$ is an irreducible representation of the product $G:=G_1\times G_2$ which we can restrict to the subgroup $H=G_2$, diagonally embedded in $G$. The space of $H$-invariant linear forms on $\pi$ has dimension $\leq 1$. In this talk, I will use the local Langlands correspondence and some number theoretic invariants of the Langlands parameter of $\pi$ to predict when the dimension of $H$-invariant forms is equal to $1$, i.e. when the dual of $\pi_2$ occurs in the restriction of $\pi_1$. I will also illustrate this prediction with several examples, including the classical branching formula for representations of compact unitary groups. This is joint work with Wee Teck Gan and Dipendra Prasad. Salle Adresse