Positive definite functions on semi-homogeneous trees and spherical representations

TL;DR

This paper characterizes positive definite functions on semi-homogeneous trees and explores their connection to spherical representations, identifying conditions under which these representations are square-integrable.

Contribution

It provides a detailed analysis of spherical functions on semi-homogeneous trees and characterizes which are positive definite, linking them to unitary representations of the automorphism group.

Findings

Identification of positive definite spherical functions on semi-homogeneous trees

Construction of associated unitary spherical representations

Determination of square-integrability conditions for specific 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.