Induction automorphe: représentations unitaires et spectre résiduel

Let $E\mathrm{/}F$ 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 ${\mathrm{N}}_{E\mathrm{/}F}(E^{\times })$. For $m\in {\mathbb{N}}^{\ast }$, we prove that every irreducible unitary representation of ${\mathrm{G}\mathrm{L}}_m(E)$ has a $\kappa$-lift to ${\mathrm{G}\mathrm{L}}_{md}(F)$, given by a character identity as in Henniart-Herb [HH]. Let $\mathbf{E}\mathrm{/}\mathbf{F}$ 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 ${\mathbf{F}}^{\times }{\mathrm{N}}_{\mathbf{E}\mathrm{/}\mathbf{F}}({\mathbb{A}}_{\mathbf{E}}^{\times })$. We prove that every automorphic discrete representation of ${\mathrm{G}\mathrm{L}}_m({\mathbb{A}}_{\mathbf{E}})$ has a (strong) $\mathfrak{K}$-lift to ${\mathrm{G}\mathrm{L}}_{md}({\mathbb{A}}_{\mathbf{F}})$, 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.