Harmonic analysis on the infinite symmetric group

Abstract

The infinite symmetric group S(∞), whose elements are finite permutations of {1,2,3,...}, is a model example of a “big” group. By virtue of an old result of Murray–von Neumann, the one–sided regular representation of S(∞) in the Hilbert space ℓ²(S(∞)) generates a type II₁ von Neumann factor while the two–sided regular representation is irreducible. This shows that the conventional scheme of harmonic analysis is not applicable to S(∞): for the former representation, decomposition into irreducibles is highly non–unique, and for the latter representation, there is no need of any decomposition at all. We start with constructing a compactification $\mathfrak{S}\supset{S(\infty)}$, which we call the space of virtual permutations. Although $\mathfrak{S}$ is no longer a group, it still admits a natural two–sided action of S(∞). Thus, $\mathfrak{S}$ is a G–space, where G stands for the product of two copies of S(∞). On $\mathfrak{S}$, there exists a unique G-invariant probability measure μ₁, which has to be viewed as a “true” Haar measure for S(∞). More generally, we include μ₁ into a family {μ _t : t>0} of distinguished G-quasiinvariant probability measures on virtual permutations. By making use of these measures, we construct a family {T _z : z∈ℂ} of unitary representations of G, called generalized regular representations (each representation T _z with z≠=0 can be realized in the Hilbert space $L^2(\mathfrak{S}, \mu_t)$, where t=|z|²). As |z|→∞, the generalized regular representations T _z approach, in a suitable sense, the “naive” two–sided regular representation of the group G in the space ℓ²(S(∞)). In contrast with the latter representation, the generalized regular representations T _z are highly reducible and have a rich structure. We prove that any T _z admits a (unique) decomposition into a multiplicity free continuous integral of irreducible representations of G. For any two distinct (and not conjugate) complex numbers z₁, z₂, the spectral types of the representations $T_{z_1}$ and $T_{z_2}$ are shown to be disjoint. In the case z∈ℤ, a complete description of the spectral type is obtained. Further work on the case z∈ℂ∖ℤ reveals a remarkable link with stochastic point processes and random matrix theory.