Characters of wreath products of compact groups with the infinite symmetric group and characters of their canonical subgroups

Abstract

Characters of wreath products $G=\mathfrak{S}_{\infty}(T)$ of any compact groups $T$ with the infinite symmetric group $\mathfrak{S}_{\infty}$ are studied. It is proved that the set $E(G)$ of all normalized characters is equal to the set $F(G)$ of all normalized factorizable continuous positive definite class functions. A general explicit formula of $f_{A} \in E(G)$ is given corresponding to a parameter $A=\left( \left( \alpha_{\zeta ,\epsilon} \right)_{({\zeta ,\epsilon})\in \hat{T}\times\{0,1\}} ; \mu \right)$. Similar results are obtained for certain canonical subgroups of $G$.