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Irreducible representations of tree automorphism groups into Pontryagin spaces

Federico Viola

Abstract

Let G = Aut(T) be the automorphism group of a regular tree T. We study continuous irreducible representations of G that preserve a continuous strongly nondegenerate sesquilinear form of finite index on a Hilbert space. These are already classified for index 0 (unitary case) and for index 1. We show that there are no more representations for index > 1, which completes the classification.

Irreducible representations of tree automorphism groups into Pontryagin spaces

Abstract

Let G = Aut(T) be the automorphism group of a regular tree T. We study continuous irreducible representations of G that preserve a continuous strongly nondegenerate sesquilinear form of finite index on a Hilbert space. These are already classified for index 0 (unitary case) and for index 1. We show that there are no more representations for index > 1, which completes the classification.

Paper Structure

This paper contains 7 sections, 25 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Restricting to spherical representations
  4. An explicit form for spherical representations
  5. Preserved sesquilinear forms
  6. The non-transitive case
  7. Appendix

Key Result

Theorem 1.1

Let $G=\mathrm{Aut}(T)$, where $T$ is a locally finite regular tree. Let $\pi$ be a continuous irreducible representation of $G$ into a Hilbert space $H$. If $\pi$ preserves a continuous strongly nondegenerate sesquilinear form of finite index $p\geq 0$, then $p=0$ or $p=1$.

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 39 more