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Parabolic induction and restriction via $C^{\ast }$-algebras and Hilbert $C^{\ast }$-modules

Part of:Locally compact groups and their algebrasLie groupsSelfadjoint operator algebras

Published online by Cambridge University Press:  02 February 2016

Pierre Clare,
Tyrone Crisp and
Nigel Higson
Pierre Clare
Affiliation:
Dartmouth College, Department of Mathematics, HB 6188, Hanover, NH 03755, USA email clare@math.dartmouth.edu
Tyrone Crisp
Affiliation:
Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany email tyronecrisp@mpim-bonn.mpg.de
Nigel Higson
Affiliation:
Pennsylvania State University, Department of Mathematics, University Park, PA 16802, USA email higson@math.psu.edu
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Abstract

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This paper is about the reduced group $C^{\ast }$-algebras of real reductive groups, and about Hilbert $C^{\ast }$-modules over these $C^{\ast }$-algebras. We shall do three things. First, we shall apply theorems from the tempered representation theory of reductive groups to determine the structure of the reduced $C^{\ast }$-algebra (the result has been known for some time, but it is difficult to assemble a full treatment from the existing literature). Second, we shall use the structure of the reduced $C^{\ast }$-algebra to determine the structure of the Hilbert $C^{\ast }$-bimodule that represents the functor of parabolic induction. Third, we shall prove that the parabolic induction bimodule admits a secondary inner product, using which we can define a functor of parabolic restriction in tempered representation theory. We shall prove in a sequel to this paper that parabolic restriction is adjoint, on both the left and the right, to parabolic induction in the context of tempered unitary Hilbert space representations.

Keywords

tempered representationsparabolic inductionuniversal principal seriesparabolic restrictionreduced group C∗ -algebras

MSC classification

Primary:22E45: Representations of Lie and linear algebraic groups over real fields22D25: $C^*$-algebras and $W^*$-algebras in relation to group representations46L08: $C^*$-modules

Information

Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 6 , June 2016, pp. 1286 - 1318
Copyright
© The Authors 2016 

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