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

TL;DR

This paper analyzes the structure of reduced group C*-algebras of real reductive groups and their associated Hilbert C*-modules, establishing new functorial relationships in tempered representation theory.

Contribution

It provides a detailed structural analysis of reduced C*-algebras and Hilbert C*-bimodules, and introduces a functor of parabolic restriction as adjoint to induction.

Findings

Determined the structure of the reduced C*-algebra of real reductive groups.

Identified the structure of the parabolic induction bimodule.

Proved the existence of a secondary inner product enabling parabolic restriction.

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.