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

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