## Kyoto Journal of Mathematics

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

#### Abstract

A detailed study of the characters of $\mathfrak{S}_{\infty}(T)$, 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 $\mathfrak{S}_{n}(T)$ as $n\to\infty$ and its connection with characters of $\mathfrak{S}_{\infty}(T)$. This paper takes a dual approach to these problems. We study harmonic functions on $\mathbb{Y}(\widehat{T})$, the branching graph of the inductive system of $\mathfrak{S}_{n}(T)$’s, and give a classification of the minimal nonnegative harmonic functions on it. This immediately implies a classification of the characters of $\mathfrak{S}_{\infty}(T)$, which is a logically independent proof of the one obtained in earlier works. We obtain explicit formulas for minimal nonnegative harmonic functions on $\mathbb{Y}(\widehat{T})$ and Martin integral expressions for harmonic functions.

#### Article information

Source
Kyoto J. Math., Volume 54, Number 4 (2014), 775-817.

Dates
First available in Project Euclid: 5 November 2014

https://projecteuclid.org/euclid.kjm/1415196156

Digital Object Identifier
doi:10.1215/21562261-2801822

Mathematical Reviews number (MathSciNet)
MR3276417

Zentralblatt MATH identifier
1306.22002

#### Citation

Hora, Akihito; Hirai, Takeshi. Harmonic functions on the branching graph associated with the infinite wreath product of a compact group. Kyoto J. Math. 54 (2014), no. 4, 775--817. doi:10.1215/21562261-2801822. https://projecteuclid.org/euclid.kjm/1415196156

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