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

Abstract

We provide a new axiomatic framework, inspired by the work of Ol'shanskii, to describe explicitly certain irreducible unitary representations of second-countable non-discrete unimodular totally disconnected locally compact groups. We show that this setup applies to various families of automorphism groups of locally finite semiregular trees and right-angled buildings.

Radu groups acting on trees are CCR

“…The purpose of Section 3 is to recall Radu's classification of Radu groups [Rad17] and the definition of the G (i) (Y 0 , Y 1 ). In Section 4, we recall the notion of Ol'shanskii's factorization developed in [Sem21] and obtain a classification of the cuspidal representations of the G (i) (Y 0 , Y 1 ). The complete classification of the irreducible representations of G (i) (Y 0 , Y 1 ) resulting from Sections 2 and 4 is then used in Section 5 to prove uniform admissibility.…”

“…Our current purpose is to give a description of the cuspidal representations of those groups. As announced in the introduction, our idea is to take advantage of the recent abstraction of Ol'shanskii's framework developed in [Sem21] and the description of those groups provided by Radu. The main concept developed in [Sem21], is the concept of Ol'shanskii's factorization see Definition 4.2 below.…”

“…The purpose of these notes is to go further. Inspired by Ol'shanskii's work and the recent progress achieved in the abstraction of his framework [Sem21], we give a classification of the irreducible representations of simple Radu groups and deduce a description of the irreducible representations of any Radu groups. Among other things, this provides the following contribution to Nebbia's CCR conjecture on trees.…”

“…Furthermore, he showed that those groups are abstractly simple and that they exhaust the list of simple Radu groups when d 0 , d 1 ≥ 6. Our paper takes advantage of the recent abstraction of Ol'shanskii's framework developed in [Sem21] and the description of those groups provided by Radu to obtain a classification their cuspidal representations see Section 4. The author would like to underline that an application of Ol'shanskii's machinery on Radu groups already exists in [Sem21, Section 4] since they satisfy a generalisation of the Tits independence property (the property IP k defined in [BEW15]).…”

“…However, the approach considered in the present paper relies on the independence provided by local conditions rather than the property IP k . In particular, by contrast with the approach developed in [Sem21], Section 4 below leads to a description of every cuspidal representation of the groups…”