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.

中文翻译：

---

关于 Bruhat-Tits 树的球态群的球面酉表示

考虑一个价数为 $n+1$ 的无限齐次树 $\mathcal{T}_n$、它的自同构群 Aut$(\mathcal{T}_n)$ 和群 Hier$(\mathcal{T}_n) $ 其球同胚（层同胚），即 $\mathcal{T}_n$ 边界的同胚群，局部与自同构定义的变换重合。我们证明子群 Aut$(\mathcal{T}_n)$ 在 Hier$(\mathcal{T}_n)$ 中是球形的，即 Hier$(\mathcal{T}_n)$ 的任何不可约酉表示包含至多一个 Aut$(\mathcal{T}_n)$ 固定向量。我们提出了 Hier$(\mathcal{T}_n)$ 关于 Aut$(\mathcal{T}_n)$ 的双陪集空间的组合描述，并构建了 Hier$ 的“新”球面表示系列(\mathcal{T}_n)$。我们还表明 Thompson 群 Th 具有 PSL$(2,