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

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