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Journal of Operator Theory

Volume 69, Issue 2, Spring 2013  pp. 483-509.

Hilbert modules associated to parabolically induced representations

Authors Pierre Clare
Author institution: Mathematics Department, The Pennsylvania State University, University Park, PA, 16802, U.S.A.

Summary:  To a measured space carrying two group actions, we associate a Hilbert $C^*$-module in a way that generalises Rieffel's construction of induction modules. This construction is then applied to describe the generalised principal series of a semisimple Lie group. We provide several realisations of this module, corresponding to the classical pictures for the principal series. We also characterise a class of bounded operators on the module which satisfy some commutation relation, and interpret the result as a generic irreducibility theorem. Finally, we establish the convergence of standard intertwining integrals on a dense subset of this module.

Keywords:  Hilbert modules, group $C^*$-algebras, induced representations, semisimple Lie groups, parabolic induction, principal series representations

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