Induction automorphe: représentations unitaires et spectre résiduel

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 $m\in {\mathbb{N}}^{\ast }$, 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 $\mathbf{E}/\mathbf{F}$ be a finite cyclic extension of number fields, of degree $d$, and $\mathfrak{K}$ be a character of ${\mathbb{A}}_{\mathbf{F}}^{\times }$ whose kernel is ${\mathbf{F}}^{\times }{\mathrm{N}}_{\mathbf{E}/\mathbf{F}}({\mathbb{A}}_{\mathbf{E}}^{\times })$. We prove that every automorphic discrete representation of ${\mathrm{GL}}_m({\mathbb{A}}_{\mathbf{E}})$ has a (strong) $\mathfrak{K}$-lift to ${\mathrm{GL}}_{md}({\mathbb{A}}_{\mathbf{F}})$, 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.

arXiv comment97 pages, in French language