Parabolic induction and restriction via -algebras and Hilbert -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 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.

Adjoint Functors Between Categories of Hilbert -Modules

“…Our main example of parabolic induction is considered in Section 5. As a particular consequence we show that the parabolic restriction functor constructed in [CCH16] is a two-sided adjoint to the parabolic induction functor from tempered unitary representations of a Levi factor of a reductive group G to tempered unitary representations of G. This adjoint functor appears to be new.…”

“…Associated to such a correspondence F there are functors B H −→ A H and H A −→ H B that are constructed using Hilbert module tensor products. The functors of parabolic induction and restriction from [Cla13] and [CCH16] are of this type.…”

“…Here L is a Levi subgroup of a parabolic subgroup P ⊆ G. We shall examine the relationship between parabolic induction and the functor of parabolic restriction Res G P : C * r (G) H −→ C * r (L) H that we introduced in [CCH16]. We shall prove that parabolic induction and restriction are left and right adjoints of one another.…”

“…In this paper we shall study the categories A H and H A with a particular view to the case of the reduced C * -algebra of a real reductive Lie group G. We shall further develop the analysis of the parabolic induction functor Ind G P : C * r (L) H −→ C * r (G) H from the perspective of Hilbert modules that was begun in [Cla13] and [CCH16]. Here L is a Levi subgroup of a parabolic subgroup P ⊆ G. We shall examine the relationship between parabolic induction and the functor of parabolic restriction Res G P : C * r (G) H −→ C * r (L) H that we introduced in [CCH16].…”

“…Each local adjunction gives rise to an ordinary adjunction of functors between categories of Hilbert space representations. In this way we shall show that the parabolic induction functor has a simultaneous left and right adjoint, namely the parabolic restriction functor constructed in [CCH16]. …”