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

Pierre Clare, Tyrone Crisp, Nigel Higson

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11 Scopus citations

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 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.

Original languageEnglish (US)
Pages (from-to)1286-1318
Number of pages33
JournalCompositio Mathematica
Volume152
Issue number6
DOIs
StatePublished - Jun 1 2016

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

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