Probability measures on families of partitions related to harmonic analysis on big wreath products

TL;DR

This paper develops explicit formulas for probability measures on partitions related to harmonic analysis on big wreath products, extending the understanding of spectral measures for generalized regular representations involving arbitrary compact groups.

Contribution

It introduces a general formula for $z$-measures on partitions associated with wreath products of any compact group, broadening previous specific cases.

Findings

Explicit formula for $z$-measures on partitions for arbitrary compact groups.

Description of spectral measures of generalized regular representations.

Extension of harmonic analysis techniques to big wreath products.

Abstract

We construct generalized regular representations of the wreath product of a compact group with the infinite symmetric group. The characters of these representations are determined by probability measures on families of partitions called the $z$-measures for the wreath product of a compact group with the symmetric group in the present paper. Our main result is an explicit formula for these $z$-measures which holds true for an arbitrary compact group. The result enables us to describe the spectral measures of the generalized regular representations of big wreath products.