Journal of Lie Theory 34 (2024), No. 4, 911--956
Copyright Heldermann Verlag 2024
On the Unitary Representation Theory of Locally Compact Contraction Groups
Max Carter
Institut de Recherche en Mathématique et Physique, Université Catholique de Louvain, Louvain-la-Neuve, Belgique
max.carter@uclouvain.be
The unitary representation theory of locally compact contraction groups and their semi-direct products with Z is studied. We put forward the problem of completely characterising such groups which are type I or CCR and this article provides a stepping stone towards a solution to this problem. In particular, we determine new examples of type I and non-type-I groups in this class, and we completely classify the irreducible unitary representations of the torsion-free groups, which are shown to be type I. When these groups are totally disconnected, they admit a faithful action by automorphisms on an infinite locally-finite regular tree; this work thus provides new examples of automorphism groups of regular trees with interesting representation theory, adding to recent work on this topic.
Keywords: Unitary representation, type I group, CCR group, scale group, contraction group, unipotent linear algebraic group, amenable group, groups acting on trees.
MSC: 20C25, 22D10, 22D12, 22D25, 20G05, 43A65.
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