Harmonic analysis on the infinite symmetric group

Abstract

Let S be the group of finite permutations of the naturals 1,2,... The subject of the paper is harmonic analysis for the Gelfand pair (G,K), where G stands for the product of two copies of S while K is the diagonal subgroup in G. The spherical dual to (G,K) (that is, the set of irreducible spherical unitary representations) is an infinite-dimensional space. For such Gelfand pairs, the conventional scheme of harmonic analysis is not applicable and it has to be suitably modified. We construct a compactification of S called the space of virtual permutations. It is no longer a group but it is still a G-space. On this space, there exists a unique G-invariant probability measure which should be viewed as a true substitute of Haar measure. More generally, we define a 1-parameter family of probability measures on virtual permutations, which are quasi-invariant under the action of G. Using these measures we construct a family {T_z} of unitary representations of G depending on a complex parameter z. We prove that any T_z admits a unique decomposition into a multiplicity free integral of irreducible spherical representations of (G,K). Moreover, the spectral types of different representations (which are defined by measures on the spherical dual) are pairwise disjoint. Our main result concerns the case of integral values of parameter z: then we obtain an explicit decomposition of T_z into irreducibles. The case of nonintegral z is quite different. It was studied by Borodin and Olshanski, see e.g. the survey math.RT/0311369.