Irreducible representations of tree automorphism groups into Pontryagin spaces

TL;DR

This paper classifies all continuous irreducible representations of the automorphism group of a regular tree into Pontryagin spaces, showing that no such representations exist for index greater than one, completing the classification.

Contribution

The paper extends the classification of irreducible representations of tree automorphism groups into Pontryagin spaces to all indices, proving nonexistence for indices greater than one.

Findings

Classified irreducible representations for index 0 and 1.

Proved nonexistence of such representations for index > 1.

Completed the full classification of these representations.

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.