Hilbert modules associated to parabolically induced representations of semisimple Lie groups

TL;DR

This paper constructs a Hilbert module linked to parabolically induced representations of semisimple Lie groups, generalizing previous induction methods and providing new insights into their structure and operators.

Contribution

It introduces a novel Hilbert C*(H)-module framework for parabolic induction in semisimple Lie groups, extending Rieffel's construction and analyzing operator commutation and convergence properties.

Findings

The Hilbert module encodes P-series representations of G.

Bounded operators commuting with the G-action are characterized as central multipliers.

Intertwining integrals converge on a dense subset of the module.

Abstract

Given a measured space X with commuting actions of two groups G and H satisfying certain conditions, we construct a Hilbert C*(H)-module E(X) equipped with a left action of C*(G), which generalises Rieffel's construction of inducing modules. Considering G to be a semisimple Lie group and H to be the Levi component L of a parabolic subgroup P=LN, the Hilbert module associated to X=G/N encodes the P-series representations of G coming from parabolic subgroups associated to P. We provide several descriptions of this Hilbert module, corresponding to the classical pictures of P-series. We then characterise the bounded operators on E(G/N) that commute to the left action of C*(G) as central multipliers of C*(L) and interpret this result as a globalised generic irreducibility theorem. Finally, we establish the convergence of intertwining integrals on a dense subset of E(G/N).