Realizations of factor representations of finite type with emphasis on their characters for wreath products of compact groups with the infinite symmetric group

Abstract

Characters of factor representations of finite type of the wreath products $G = \mathfrak{S}_{\infty}(T)$ of any compact groups $T$ with the infinite symmetric group $\mathfrak{S}_{\infty}$ were explicitly given in [HH4]-[HH6], as the extremal continuous positive definite class functions $f_{A}$ on $G$ determined by a parameter $A$. In this paper, we give a special kind of realization of a factor representation $\pi ^{A}$ associated to $f_{A}$. This realization is better than the Gelfand-Raikov realization $\pi _{f}$, $f = f_{A}$, in [GR] at least at the point where a matrix element $\langle \pi ^{A}(g)v_{0}, v_{0}\rangle$ of $\pi ^{A}$ for a cyclic vector $v_{0}$ can be calculated explicitly, which is exactly equal to the character $f_{A}$ (and so $\pi ^{A}$ has a trace-element $v_{0}$). So the positive-definiteness of class functions $f_{A}$ given in [HH4]-[HH6] is automatically guaranteed, a proof of which occupies the first half of [HH6] in the case of $T$ infinite. The case where $T$ is abelian contains the cases of infinite Weyl groups and the limits $\mathfrak{S}_{\infty}(\mathbf{Z}_{r}) = \lim _{n\to\infty}G(r,1,n)$ of complex reflexion groups.