Induction automorphe: repr\'esentations unitaires et spectre r\'esiduel

Abstract

Let $E/F$ be a finite cyclic extension of local fields of characteristic zero, of degree $d$, and $\kappa$ be a character of $F^\times$ whose kernel is $\mathrm{N}_{E/F}(E^\times)$. For $m\in \mathbb{N}^*$, we prove that every irreducible unitary representation of $\mathrm{GL}_m(E)$ has a $\kappa$-lift to $\mathrm{GL}_{md}(F)$, given by a character identity as in Henniart-Herb [HH]. Let ${\bf E}/{\bf F}$ be a finite cyclic extension of number fields, of degree $d$, and $\mathfrak{K}$ be a character of $\mathbb{A}_{\bf F}^\times$ whose kernel is ${\bf F}^\times \mathrm{N}_{{\bf E}/{\bf F}}(\mathbb{A}_{\bf E}^\times)$. We prove that every automorphic discrete representation of $\mathrm{GL}_m(\mathbb{A}_{\bf E})$ has a (strong) $\mathfrak{K}$-lift to $\mathrm{GL}_{md}(\mathbb{A}_{\bf F})$, i.e. compatible with the local lifting maps. We describe the image and the fibres of these local and global lifting maps. Locally, we also treat the elliptic representations.