Generalized regular representations of big wreath products

Let G be a finite group with k conjugacy classes, and S(∞) be the infinite symmetric group, i.e., the group of finite permutations of {1, 2, 3, …}. Then the wreath product G_∞ = G ∼ S (∞) of G with S (∞) (called the big wreath product) can be defined. The group G_∞ 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_∞. In particular, we construct an analogue SG of the space of virtual permutations. We then formulate and prove a theorem characterizing all central probability measures on SG. Next, we introduce generalized regular representations {Tz1,⋯,zk:z1∈C,⋯,zk∈C} of the big wreath product G_∞, which are analogues of the Kerov–Olshanski–Vershik generalized regular representations of the infinite symmetric group. We derive an explicit formula for the characters of Tz1,⋯,zk. The spectral measures of these representations are characterized in different ways. In particular, these spectral measures are associated with point processes whose correlation functions are explicitly computed. Thus, in representation-theoretic terms, the paper solves a natural problem of harmonic analysis for the big wreath products: our results describe the decomposition of Tz1,⋯,zk into irreducible components.