Irreducible Koopman representations for nonsingular actions on boundaries of rooted trees

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 $\omega \in \{0,1{\}}^{\mathbb{N}}$, an irreducible unitary representation ${\kappa }_{\omega }$ of $G$. Every two representations ${\kappa }_{\omega }$ and ${\kappa }_{{\omega }^{\mathrm{\prime }}}$ are weakly equivalent. They are unitarily equivalent if and only if $\omega$ and ${\omega }^{\mathrm{\prime }}$ are tail equivalent. Each ${\kappa }_{\omega }$ appears as the Koopman representation associated with some ergodic $G$-quasiinvariant measure (of infinite product type) on the boundary of the tree.