A Functorial Refinement of the Franke Filtration and the Jacquet--Langlands Correspondence for Spaces of Automorphic Forms

TL;DR

This paper extends the Jacquet--Langlands correspondence to the entire space of automorphic forms using a functorial refinement of the Franke filtration, thereby advancing the understanding of Langlands functoriality beyond the discrete spectrum.

Contribution

It introduces a functorial refinement of the Franke filtration that enables extending the Jacquet--Langlands correspondence to all automorphic forms, not just the discrete spectrum.

Findings

Extended Jacquet--Langlands correspondence to full automorphic spectrum.

Refined Franke filtration facilitates new functorial lifts.

Contains the full functorial lift predicted by Langlands.

Abstract

The global Jacquet--Langlands correspondence is an instance of Langlands functoriality, namely the expected lifting of the irreducible automorphic representations of an inner form of the general linear group to the split form via the identity morphism of $L$-groups. It is established, by the work of Badulescu, in the case of irreducible components of the discrete spectrum. The purpose of this paper is to extend this correspondence beyond the discrete spectrum. To this end, the point of view of the Franke filtration of spaces of automorphic forms is taken. In fact, our technical key ingredient is a functorial refinement of the Franke filtration, which allows us to establish the Jacquet--Langlands correspondence between consecutive quotients of this refined filtration on the general linear group and its inner form. As a result, our extended Jacquet--Langlands correspondence properly extends Badulescu's correspondence and contains the full functorial lift, predicted by Langlands functoriality.