The Global Jacquet-Langlands Correspondence via Tensor Products

Technical Summary: The Global Jacquet–Langlands Correspondence via Tensor Products

Author: Jun Yang
Subject Area: Automorphic Representation Theory, Langlands Program, Theta Correspondence

1. The Problem

The Jacquet–Langlands (JL) correspondence is a fundamental instance of Langlands functoriality that establishes a relationship between automorphic representations of a non-split inner form of $GL(2)$ (specifically, the multiplicative group of a quaternion division algebra $D^\times$) and the group $GL(2)$.

Traditionally, the JL correspondence is proven using the Arthur–Selberg Trace Formula, a heavy analytic tool. The problem addressed in this paper is to provide a different, more structural realization of this correspondence. Specifically, the author seeks to show that the JL transfer is not merely an isolated phenomenon but a direct consequence of Howe duality (the theta correspondence) for a specific quaternionic similitude dual pair.

2. Methodology

The paper shifts the perspective from trace formulas to the theory of theta correspondences for similitude groups. The methodology follows these logical steps:

• Similitude Dual Pairs: The author utilizes the framework of similitude groups (groups that preserve a form up to a scalar factor). The specific dual pair identified is $(G, H)$, where $G = D^\times$ and $H \cong (D^\times \times GL(2, F))/F^\times$.
• Induced Weil Representation: Instead of the classical Weil representation (which acts on isometry groups), the author employs the induced Weil representation $\Omega$, which is constructed to act on the similitude groups.
• Hecke Algebra Tensor Products: A central technical innovation is the realization of the "big theta lift" $\Theta(\pi)$ as a tensor product of a representation and a module over a global Hecke algebra. The author defines a global Hecke algebra $\mathcal{H}(G)$ as a restricted tensor product of local Hecke algebras (convolution algebras for non-archimedean places and $(\mathfrak{g}, K)$-modules for archimedean places).
• Local-to-Global Transition: The author proves that if Howe duality holds locally at every place $v$, the global theta lift can be decomposed into a tensor product of local lifts, provided the global lift is non-zero and cuspidal.

3. Key Contributions

• Tensor Product Realization: The paper proves that for an irreducible admissible representation $\pi$ of $G(\mathbb{A})$, the similitude big theta lift $\Theta(\pi)$ can be expressed as:
  $$\Theta(\pi) \cong \pi^\vee \otimes_{\mathcal{H}(G)} \Omega_{\mathbb{A}}$$
  This provides a "universal tensor construction" for the transfer.
• Unified Treatment of Places: Unlike many classical approaches that treat archimedean and non-archimedean places differently, this tensor-product formalism treats them uniformly through the language of Hecke modules.
• Explicit Model for $\Omega$: The author provides a concrete realization of the underlying space of the induced Weil representation as a Schwartz–Bruhat space $\mathcal{S}(D(\mathbb{A}) \times \mathbb{A}^\times)$, making the correspondence computationally and conceptually transparent.

4. Main Results

The culmination of the paper is the proof of the Global Jacquet–Langlands Correspondence via Tensor Products:

1. Representation Level: For an irreducible automorphic representation $\pi$ of $D^\times(\mathbb{A})$ with central character $\chi$, the correspondence is realized as:
   $$\pi^\vee \otimes_{\mathcal{H}(G)} \mathcal{S}(D(\mathbb{A}) \times \mathbb{A}^\times) \cong \pi \otimes JL(\pi)$$
   where $JL(\pi)$ is the Jacquet–Langlands transfer to $GL(2, \mathbb{A})$.
2. Spectral Level: The entire automorphic spectrum of the quaternion algebra, when tensored with the $L^2$-space of the similitude group, decomposes into the sum of pairs of corresponding representations:
   $$L^2(G(F)\backslash G(\mathbb{A}), \chi) \otimes_{\mathcal{H}(G)} L^2(D(\mathbb{A}) \times \mathbb{A}^\times) \cong \bigoplus_{\pi \in \text{Irrep}_{\mathbb{A}}(G, \chi^{-1})} \pi \otimes JL(\pi)$$

5. Significance

This paper is significant for several reasons:

• Conceptual Reinterpretation: It reclassifies the Jacquet–Langlands correspondence as an instance of theta functoriality. This places it within the broader, more unified context of the Langlands Program.
• Methodological Alternative: It offers a way to view functorial transfers through the lens of representation-theoretic structures (Hecke modules and Howe duality) rather than purely analytic identities (trace formulas).
• Generalizability: The framework developed here (realizing lifts via Hecke-module tensor products) suggests a general strategy for proving other instances of functorial transfer arising from theta correspondences.