On multiplicity-free induced representations of a homogeneous tree

Let Tq, n be a homogeneous rooted tree, let K be a subgroup of Aut(Tq, n), the group of automorphisms of Tq, n, and let τq, ℓ be a unitary irreducible representation of K. In this paper, we construct some multiplicity-free irreducible representations of Aut(Tq, n). We also give a necessary condition for (Aut(Tq, n), K) to be a strong Gelfand pair.