Lie groupoids, the Satake compactification and the tempered dual, II: The Harish-Chandra principle

TL;DR

This paper provides a geometric perspective on Harish-Chandra's principle for real reductive groups, using the Satake compactification and associated groupoid to understand the structure of tempered irreducible representations.

Contribution

It introduces a geometric approach employing the Satake compactification and groupoid $C^*$-algebra to analyze the embedding properties of tempered representations.

Findings

Characterization of tempered irreducible representations via geometric methods

Connection between parabolic induction and the Satake compactification

Use of groupoid $C^*$-algebra to study representation structure

Abstract

We give a geometric account of Harish-Chandra's principle that a tempered irreducible representation of a real reductive group is either square-integrable modulo center, or embeddable in a representation that is parabolically induced from such a representation. Our approach uses the Satake compactification, an associated groupoid that was constructed in the first paper of this series, and its $C^*$-algebra.