The Global Jacquet-Langlands Correspondence via Tensor Products

Abstract

We prove that the global Jacquet--Langlands correspondence $\mathrm{JL}$ for $\mathrm{GL}(2)$ can be realized via tensor products over Hecke algebras. Let $G$ be a non-split inner form of $\mathrm{GL}(2)$ over a number field. Using the similitude theta correspondence, the space $L^2(D(\mathbb{A})\times {\mathbb{A}}^{\times })$ acquires the structure of a $G(\mathbb{A})$-$(G(\mathbb{A})\times \mathrm{GL}(2,\mathbb{A}))$ bimodule such that $L^2(G(F)\backslash G(\mathbb{A}),\ddot{I}‡){\otimes }_{\mathcal{H}(G)}{L}^2(D(\mathbb{A})\times {\mathbb{A}}^{\times })\cong {\oplus }_{\ddot{I}€\in \mathcal{A}(G,\ddot{I}{‡}^{-1})}$ $\ddot{I}€\otimes \mathrm{JL}(\ddot{I}€).$ This decomposition into irreducible representations of $G(\mathbb{A})\times \mathrm{GL}(2,\mathbb{A})$ recovers the full global Jacquet-Langlands correspondence.