On multiplicity-free induced representations of a homogeneous tree

Abstract

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