Lie groupoids, the Satake compactification and the tempered dual, II: The Harish-Chandra principle
Jacob Bradd, Nigel Higson, Robert Yuncken
TL;DR
This work provides a noncommutative geometric proof of Harish-Chandra's principle for real reductive groups by leveraging the Satake compactification and the associated Satake groupoid. It constructs ideal-theoretic fillings I and J inside the reduced group C*-algebra to separate discrete-series components from those embeddable in parabolically induced representations, using the exact sequence of the Satake groupoid and a morphism from C*-algebras that connects G with its groupoid. The main achievement is showing C^*_cmc(G)=C^*_cusp(G), which translates the representation-theoretic statement into an equality of C*-algebraic ideals, yielding the dichotomy and its corollaries about admissibility and inducibility. The framework further provides a canonical description of the tempered dual via tensor products with characters of the central torus A_Σ and clarifies the role of parabolic induction through Hilbert C*-modules and Fell bundles on the Satake groupoid, with potential applications to index theory and geometric representation theory.
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.