Irreducibly represented Lie groups and Nebbia's CCR conjecture on trees

TL;DR

This thesis explores the algebraic and representation-theoretic properties of locally compact groups, especially automorphism groups of trees, contributing new insights into Nebbia's CCR conjecture and the classification of irreducible representations.

Contribution

It provides an algebraic characterization of irreducibly represented Lie groups and develops an axiomatic framework for irreducible representations of totally disconnected groups, advancing understanding of Nebbia's CCR conjecture.

Findings

Characterization of certain Lie groups as irreducibly represented

Framework for irreducible representations with small isotropy groups

Progress towards Nebbia's CCR conjecture on trees

Abstract

This thesis is devoted to the study of the interactions existing between the algebraic structure of locally compact groups and the properties of their continuous unitary representations, with a special emphasis on the Type I groups. On the one hand, the thesis provides a general overview of the theory of unitary representations of locally compact groups aswell as an overview of the classification of the irreducible unitary representations of the full group of automorphisms of any semi-regular tree. On the other hand, it contains various of our personal contributions to this domain of mathematics. Among other things, we provide an algebraic characterisation of certain Lie groups that are irreducibly represented such as the connected nilpotent Lie groups. Concerning totally disconnected locally compact groups, we expose an axiomatic framework to describe their irreducible representations all of whose isotropy groups are "small". We provide concrete applications of this machinery to groups of automorphisms of trees and right-angled buildings leading to a substantial contribution to Nebbia's CCR conjecture. We also provide a description of the unitary dual of the groups of automorphisms of trees acting 2-transitively on the boundary and whose local action at every vertex contains the alternating group and describe the Fell topology of this space.