Induced representations arising from a character with finite orbit in a semidirect product

TL;DR

This paper develops a unified spectral theory for induced representations from characters with finite orbits in semidirect products, with applications to harmonic analysis, ergodic theory, and dynamical systems.

Contribution

It introduces a comprehensive approach to spectral decomposition of induced representations in semidirect products, extending analysis in non-commutative harmonic analysis and related fields.

Findings

Spectral decomposition results for specific classes of induced representations

Applications to Bratteli diagrams and their duals

Analysis of wavelet sets and wavelet representations

Abstract

Making use of a unified approach to certain classes of induced representations, we establish here a number of detailed spectral theoretic decomposition results. They apply to specific problems from non-commutative harmonic analysis, ergodic theory, and dynamical systems. Our analysis is in the setting of semidirect products, discrete subgroups, and solenoids. Our applications include analysis and ergodic theory of Bratteli diagrams and their compact duals; of wavelet sets, and wavelet representations.