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On the unitary representation theory of locally compact contraction groups

Max Carter

TL;DR

This work advances the unitary representation theory of locally compact contraction groups by developing a Mackey-based framework and applying the Kirillov orbit method to contractive scale groups. It proves that torsion-free locally compact contraction groups yield CCR, and, when the horocycle stabiliser is CCR, the semidirect product with $\mathbb{Z}$ is type I with an explicit description of $\widehat{G}$ via induced representations from $N$ and a $\widehat{\mathbb{Z}}$-component. It also exhibits a new, countable family of non-type-I torsion contraction groups arising as central extensions of $\mathbb{F}_p((t))$ by itself, analyzed through multiplier obstructions and Mackey theory. In parallel, the paper proves CCR for unipotent linear algebraic groups over $\mathbb{F}_p((t))$, including Heisenberg groups, broadening the landscape of CCR examples in the torsion setting. Collectively, these results illuminate the type I/CCR landscape for contraction groups, provide concrete classifications in the torsion-free case, and connect representation theory with automorphism groups of regular trees via the tree representation theorem.

Abstract

The unitary representation theory of locally compact contraction groups and their semi-direct products with $\mathbb{Z}$ 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.

On the unitary representation theory of locally compact contraction groups

TL;DR

This work advances the unitary representation theory of locally compact contraction groups by developing a Mackey-based framework and applying the Kirillov orbit method to contractive scale groups. It proves that torsion-free locally compact contraction groups yield CCR, and, when the horocycle stabiliser is CCR, the semidirect product with is type I with an explicit description of via induced representations from and a -component. It also exhibits a new, countable family of non-type-I torsion contraction groups arising as central extensions of by itself, analyzed through multiplier obstructions and Mackey theory. In parallel, the paper proves CCR for unipotent linear algebraic groups over , including Heisenberg groups, broadening the landscape of CCR examples in the torsion setting. Collectively, these results illuminate the type I/CCR landscape for contraction groups, provide concrete classifications in the torsion-free case, and connect representation theory with automorphism groups of regular trees via the tree representation theorem.

Abstract

The unitary representation theory of locally compact contraction groups and their semi-direct products with 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.
Paper Structure (20 sections, 48 theorems, 76 equations)

This paper contains 20 sections, 48 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graphs and groups
  4. Scale groups
  5. Contraction groups
  6. Multipliers and representations
  7. Group C$^*$-algebras and the Fell topology
  8. Induced representations
  9. Mackey little group method
  10. Crossed-product C$^*$-algebras
  11. Abelian contraction groups
  12. Characters on abelian contraction groups
  13. Contractive scale groups with abelian horocycle stabiliser
  14. Torsion-free contraction groups
  15. Contractive scale groups with CCR horocycle stabiliser
  16. ...and 5 more sections

Key Result

Theorem 1.2

Let $(N,\alpha)$ be a locally compact contraction group and $G := N \rtimes_\alpha \mathbb{Z}$. The group $G$ is not CCR. Furthermore, if $N$ is assumed to be CCR, then the following hold:

Theorems & Definitions (94)

  • Theorem 1.2
  • Definition 2.1: Scale group
  • Proposition 2.2
  • Definition 2.3
  • Definition 2.4: Contraction group
  • Example 2.5
  • Theorem 2.6
  • Remark 2.7
  • Theorem 2.8
  • Definition 2.9
  • ...and 84 more