Irreducible Koopman representations for nonsingular actions on boundaries of rooted trees

Abstract

Let $G$ be a countable branch group of automorphisms of a spherically homogeneous rooted tree. Under some assumption on finitarity of $G$, we construct, for each sequence $ω\in\{0,1\}^\Bbb N$, an irreducible unitary representation $κ_ω$ of $G$. Every two representations $κ_ω$ and $κ_{ω'}$ are weakly equivalent. They are unitarily equivalent if and only if $ω$ and $ω'$ are tail equivalent. Each $κ_ω$ appears as the Koopman representation associated with some ergodic $G$-quasiinvariant measure (of infinite product type) on the boundary of the tree.