Harmonic functions on the branching graph associated with the infinite wreath product of a compact group

Abstract

A detailed study of the characters of ${\mathfrak{S}}_{\infty }\left(T\right)$, the wreath product of compact group $T$ with the infinite symmetric group ${\mathfrak{S}}_{\infty }$, is indispensable for harmonic analysis on this big group. In preceding works, we investigated limiting behavior of characters of the finite wreath product ${\mathfrak{S}}_n\left(T\right)$ as $n\to \infty$ and its connection with characters of ${\mathfrak{S}}_{\infty }\left(T\right)$. This paper takes a dual approach to these problems. We study harmonic functions on $\mathbb{Y}\left(\widehat{T}\right)$, the branching graph of the inductive system of ${\mathfrak{S}}_n\left(T\right)$’s, and give a classification of the minimal nonnegative harmonic functions on it. This immediately implies a classification of the characters of ${\mathfrak{S}}_{\infty }\left(T\right)$, which is a logically independent proof of the one obtained in earlier works. We obtain explicit formulas for minimal nonnegative harmonic functions on $\mathbb{Y}\left(\widehat{T}\right)$ and Martin integral expressions for harmonic functions.