# On spherical unitary representations of groups of spheromorphisms of Bruhat--Tits trees

## TL;DR

This paper investigates the structure of spherical unitary representations of the group of spheromorphisms of Bruhat--Tits trees, revealing new representations and their relation to classical groups like Thompson and PSL(2,Z).

## Contribution

It introduces a combinatorial framework for understanding double cosets and constructs new spherical representations of the spheromorphism group.

## Key findings

- Aut(T_n) is spherical in Hie(T_n)
- Constructed a new family of spherical representations
- Thompson group has PSL(2,Z)-spherical unitary representations

## Abstract

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