On spherical unitary representations of groups of spheromorphisms of Bruhat–Tits trees

Abstract

Consider an infinite homogeneous tree ${\mathcal{T}}_n$ of valence $n+1$, its group Aut$({\mathcal{T}}_n)$ of automorphisms, and the group Hier$({\mathcal{T}}_n)$ of its spheromorphisms (hierarchomorphisms), i.e., the group of homeomorphisms of the boundary of ${\mathcal{T}}_n$ that locally coincide with transformations defined by automorphisms. We show that the subgroup Aut$({\mathcal{T}}_n)$ is spherical in Hier$({\mathcal{T}}_n)$, i.e., any irreducible unitary representation of Hier$({\mathcal{T}}_n)$ contains at most one Aut$({\mathcal{T}}_n)$-fixed vector. We present a combinatorial description of the space of double cosets of Hier$({\mathcal{T}}_n)$ with respect to Aut$({\mathcal{T}}_n)$ and construct a “new” family of spherical representations of Hier$({\mathcal{T}}_n)$. We also show that the Thompson group Th has PSL$(2,\mathbb{Z})$-spherical unitary representations.