  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.
