Adelic C*-correspondences and parabolic induction

TL;DR

The paper develops a comprehensive framework for restricted tensor products of Hilbert C*-modules and C*-correspondences, allowing global objects to be built from compatible local data via projections $p_v$ and distinguished vectors $x_v$. It then applies this framework to adelic representation theory, showing that induction functors, including parabolic induction, commute with restricted tensor products and that global adelic correspondences arise as restricted products of local pieces. A key technical achievement is the demonstration that induction and compact/Type I properties are preserved under these restricted constructions, enabling a C*-algebraic realization of local-global compatibility for adelic induction. The results provide a robust analytic model for adelic representation theory and offer a path toward integrating local correspondence techniques (Rieffel induction) into the global, adele-driven setting, with potential implications for broader theta correspondences and Langlands-type programmatic structures.

Abstract

In analogy with the construction of representations of adelic groups as restricted products of representations of local groups, we study restricted tensor products of Hilbert C*-modules and of C*-correspondences. The construction produces global C*-correspondences from compatible collections of local C*-correspondences. When applied to the collection of C*-correspondences capturing local parabolic induction, the construction produces a global C*-correspondence that captures adelic parabolic induction.

Theorems & Definitions (44)

• Definition 2.1: Definition 12 in Guichardet
• Theorem 2.2
• Definition 3.1
• Proposition 3.2
• Theorem 3.3
• Proposition 3.4
• Proposition 3.5
• proof
• Lemma 3.6
• Lemma 3.7