Induction automorphe: représentations unitaires et spectre résiduel
Martin Fatou, Bertrand Lemaire
TL;DR
The paper proves that for a finite cyclic extension $E/F$ of local fields with characteristic zero, every irreducible unitary representation of ${GL}_m(E)$ has a κ-lift to ${GL}_{md}(F)$, realized via a character identity in the spirit of Henniart–Herb. It extends these local lifts to the global automorphic setting, showing that every automorphic discrete representation of ${GL}_m({\mathbb A}_E)$ has a strong ${\mathfrak K}$-lift to ${GL}_{md}({\mathbb A}_F)$, compatible with local lifting maps, and describes their images and fibers. The approach combines parabolic induction, transfer factors, and endoscopic stabilization of the trace formula, with a detailed analysis of elliptic representations and Satake parametrizations for spherical cases. For function fields, the paper outlines how, once the global trace-formula comparison is established, the same lifting and endoscopic framework would extend, while noting current obstacles in positive characteristic stabilisation and transfer. Overall, the work provides a robust global–local picture of Langlands functoriality via κ-relèvements and strong base-change lifts in both archimedean and non-archimedean settings, including a clear description of the fibers and the interaction with the endoscopic data.
Abstract
Let $E/F$ be a finite cyclic extension of local fields of characteristic zero, of degree $d$, and $κ$ 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 $κ$-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.