Generalized regular representations of big wreath products

Let $G$ be a finite group with $k$ conjugacy classes, and $S(\infty)$ be the infinite symmetric group, i.e. the group of finite permutations of $\left\{1,2,3,\ldots\right\}$. Then the wreath product $G_{\infty}=G\sim S(\infty)$ of $G$ with $S(\infty)$ (called the big wreath product) can be defined. The group $G_{\infty}$ is a generalization of the infinite symmetric group, and it is an example of a ``big'' group, in Vershik's terminology. For such groups the two-sided regular representations are irreducible, the conventional scheme of harmonic analysis is not applicable, and the problem of harmonic analysis is a nontrivial problem with connections to different areas of mathematics and mathematical physics. Harmonic analysis on the infinite symmetric group was developed in the works by Kerov, Olshanski, and Vershik, and Borodin and Olshanski. The goal of this paper is to extend this theory to the case of $G_{\infty}$. In particular, we construct an analogue $\mathfrak{S}_{G}$ of the space of virtual permutations. We then formulate and prove a theorem characterizing all central probability measures on $\mathfrak{S}_{G}$, and introduce generalized regular representations $T_{z_1,\ldots,z_k}$ of the big wreath product $G_{\infty}$. The paper solves a natural problem of harmonic analysis for the big wreath products: our results describe the decomposition of $T_{z_1,\ldots,z_k}$ into irreducible components.