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

Published online by Cambridge University Press:  02 February 2016

Pierre Clare
Dartmouth College, Department of Mathematics, HB 6188, Hanover, NH 03755, USA email
Tyrone Crisp
Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany email
Nigel Higson
Pennsylvania State University, Department of Mathematics, University Park, PA 16802, USA email
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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.

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