| 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. |
