A product formula for spherical representations of a group of automorphisms of a homogeneous tree. I

Let G = Aut(T) be the group of automorphisms of a homogeneous tree T, and let Γ be a lattice subgroup of G. Let π be the tensor product of two spherical irreducible unitary representations of G. We give an explicit decomposition of the restriction of π to Γ. We also describe the spherical component of π explicitly, and this decomposition is interpreted as a multiplication formula for associated orthogonal polynomials.