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

Neven Grbac, Harald Grobner

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.

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

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 -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.
Paper Structure (4 sections, 13 theorems, 110 equations, 4 figures)

This paper contains 4 sections, 13 theorems, 110 equations, 4 figures.

Table of Contents

  1. Preliminaries and notation
  2. The global Jacquet--Langlands correspondence for the discrete spectrum
  3. The Franke filtration for inner forms of the general linear group
  4. The global Jacquet--Langlands correspondence beyond the discrete spectrum

Key Result

Theorem 1

Let $\mathcal{A}_{\{P'\},\varphi(\pi')}$, respectively, $\mathcal{A}_{\{Q\},\varphi(\sigma)}$, be the spaces of automorphic forms on $G'(\mathbb{A})$, respectively, $G(\mathbb{A})$, with cuspidal support in the associate class $(\{P'\},\varphi(\pi'))$, respectively, $(\{Q\},\varphi(\sigma))$, as abo of the space $\mathcal{A}_{\{P'\},\varphi(\pi')}$, and there exists a functorial refinement of the

Figures (4)

  • Figure 1: Partition of $\mathcal{S}'$ as in Proposition \ref{['prop:partition-of-S-inner-form']}
  • Figure 2: Partition of $\mathcal{S}$ as in Proposition \ref{['prop:partition-of-tilde-S']}
  • Figure 3: Partition $(\clubsuit)$ of $\mathcal{S}$ written also in the new notation
  • Figure 4: Functorially refined partition of $\mathcal{S}$ as required in Theorem \ref{['thm:franke-modified']}

Theorems & Definitions (26)

  • Theorem
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 3.3
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 16 more