Parabolic induction and restriction via C*-algebras and Hilbert C*-modules

Abstract

This paper is about the reduced group C*-algebras of real reductive groups, and about Hilbert C*-modules over these C*-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*-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*-algebra to determine the structure of the Hilbert C*-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 the sequel to this paper that parabolic restriction is adjoint, on both the left and the right, to parabolic induction.

Final version, to appear in Compositio Mathematica

References in corpus (1)

• Adjoint functors between categories of Hilbert C*-modules

Cited by in corpus (10)

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• A fixed point formula and Harish-Chandra's character formula
• Adjoint functors between categories of Hilbert C*-modules
• Connective C*-algebras
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• Orbital integrals and $K$-theory classes
• An index theorem for higher orbital integrals
• A Second Adjoint Theorem for SL(2,R)
• Continuity of the Mackey-Higson bijection
• The ring structure of twisted equivariant $KK$-theory for noncompact Lie groups