Unitary representations of totally disconnected locally compact groups satisfying Ol'shanskii's factorization

TL;DR

This paper develops an axiomatic framework for understanding specific irreducible unitary representations of certain totally disconnected locally compact groups, with applications to automorphism groups of trees and buildings.

Contribution

It introduces a new axiomatic approach to classify irreducible unitary representations of non-discrete unimodular groups inspired by Ol'shanskii's work.

Findings

Framework applies to automorphism groups of trees

Framework extends to semi-regular right-angled buildings

Provides classification insights for irreducible representations

Abstract

Inspired by Ol'shanskii's work, we provide an axiomatic framework to describe certain irreducible unitary representations of non-discrete unimodular totally disconnected locally compact groups. We then look at the applications to certain groups of automorphisms of locally finite trees and semi-regular right-angled buildings.