Positive definite functions on semi-homogeneous trees and spherical representations

Abstract

We consider the group $\mathrm{Aut}(T)$ of isometries of a semi-homogeneous tree $T=T_{q_{+},q_{-}}$ with valencies $q_{+}+1$ and $q_{-}+1$ and its two orbits $V_{+}$, $V_{-}$ respectively. We make use of the action of $\mathrm{Aut}(T)$ to equip the spaces of finitely supported radial functions on each of $V_{\pm }$ with convolution products, hence with a notion of positive definite functions. The ${\ell }^1$-functions radial around a root vertex $v_0\in V_{+}$ form an abelian convolution algebra. We study its multiplicative functionals, called spherical functions, given by 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 $\mathrm{Aut}(T)$; in this way, we produce a series of unitary spherical representations. For $q_{+}<q_{-}$, the representation whose spherical function has eigenvalue 0 is square-integrable.

arXiv admin note: text overlap with arXiv:2208.00910