Parabolic induction, categories of representations and operator spaces

TL;DR

This paper explores the operator algebraic aspects of parabolic induction and Frobenius reciprocity in representation theory, aiming to develop an operator algebraic framework for Bernstein's reciprocity theorem.

Contribution

It introduces an operator algebraic perspective on parabolic induction and investigates the potential for formulating Bernstein's reciprocity theorem within this framework.

Findings

Frobenius reciprocity is naturally interpreted via operator modules.

Examines the possibility of an operator algebraic formulation of Bernstein's second adjoint theorem.

Provides insights into the structure of reduced group C*-algebras in the context of representation theory.

Abstract

We study some aspects of the functor of parabolic induction within the context of reduced group C*-algebras and related operator algebras. We explain how Frobenius reciprocity fits naturally within the context of operator modules, and examine the prospects for an operator algebraic formulation of Bernstein's reciprocity theorem (his second adjoint theorem).