Spherical Representations of the Group of Isometries of Semi-homogeneous Trees

We consider the group G $$\mathcal G$$ of isometries of a semi-homogeneous tree T = T q + , q − $$T=T_{q_+,q_-}$$ with valencies q + + 1 $$q_+ +1$$ and q − + 1 $$q_- +1$$ and its two orbits, respectively V + $$V_+$$ and V − $$V_-$$ , on the set of vertices. We make use of the action of G $$\mathcal G$$ to equip each of V ± $$V_\pm $$ with a convolution product, hence with a notion of positive definite functions. The ℓ 1 $$\ell ^1$$ -functions radial around a root vertex v 0 $$v_{ 0}$$ in, say, V + $$V_+$$ form an abelian convolution algebra. We study its multiplicative functionals, called spherical functions, that are eigenfunctions of the nearest-neighbor isotropic transition operator (the Laplace operator on T), and determine which of them are positive definite. Each positive definite function gives rise to a unitary representation of G $$\mathcal G$$ ; in this way, we produce a series of unitary spherical representations. For q +