Inadmissible representations of the tree automorphism group

TL;DR

This paper demonstrates that the automorphism group of a regular locally finite tree has irreducible Banach representations that are not admissible, with a dense subspace of smooth vectors lacking algebraically irreducible components.

Contribution

It reveals the existence of non-admissible irreducible Banach representations for tree automorphism groups, challenging previous assumptions.

Findings

Existence of non-admissible irreducible Banach representations.

Dense subspace of smooth vectors contains no algebraically irreducible component.

Highlights new representation-theoretic properties of tree automorphism groups.

Abstract

The automorphism group of a regular locally finite tree is shown to admit irreducible Banach representations that are not admissible. The dense subspace of smooth vectors contains no algebraically irreducible component.