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 ℓ 2 (S(∞)) generates a type II 1 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 S ⊃ S(∞), which we call the space of virtual permutations. Although S is no longer a group, it still admits a natural two-sided action of S(∞). Thus, S is a G-space, where G stands for the product of two copies of S(∞). On S, there exists a unique Ginvariant probability measure µ 1 , which has to be viewed as a "true" Haar measure for S(∞). More generally, we include µ 1 into a family {µt : t > 0} of distinguished G-quasiinvariant probability measures on virtual permutations. By making use of these measures, we construct a family {Tz : z ∈ C} of unitary representations of G, called generalized regular representations (each representation Tz with z = 0 can be realized in the Hilbert space L 2 (S, µt), where t = |z| 2 ). As |z| → ∞, the generalized regular representations Tz approach, in a suitable sense, the "naive" two-sided regular representation of the group G in the space ℓ 2 (S(∞)). In contrast with the latter representation, the generalized regular representations Tz are highly reducible and have a rich structure. We prove that any Tz admits a (unique) decomposition into a multiplicity free continuous integral of irreducible representations of G. For any two distinct (and not conjugate) complex numbers z 1 , z 2 , the spectral types of the representations Tz 1 and Tz 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.
