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

TL;DR

This paper establishes new lifting results for unitary and automorphic representations between general linear groups over cyclic extensions of local and number fields, extending known character identities and describing the structure of these lifts.

Contribution

It proves the existence of explicit lifts for irreducible unitary and automorphic representations in cyclic extensions, including local and global cases, with detailed descriptions of their images and fibers.

Findings

Every irreducible unitary representation of GL_m(E) lifts to GL_{md}(F) via a character identity.

Automorphic discrete representations of GL_m over E have strong lifts to GL_{md} over F, compatible with local maps.

The paper characterizes the image and fibers of the local and global lifting maps, including elliptic representations.

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.