%0 Journal Article %1 keyhere %A Kerov, Sergei %A Olshanski, Grigori %A Vershik, Anatoly %D 2004 %J Inventiones Mathematicae %K CitesCSP Kerov Olshanski RepresentationTheory Vershik %N 3 %P 551--642 %T Harmonic analysis on the infinite symmetric group %U http://dx.doi.org/10.1007/s00222-004-0381-4 %V 158 %X 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 $SS(ınfty)$ , 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 G-invariant 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 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 $L^2(S, \mu_t)$ , 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 ℓ 2( 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 z 1, z 2, 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. ER -

@article{keyhere, 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 $\mathfrak{S}\supset{S(\infty)}$ , 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 μ 1, which has to be viewed as a “true” Haar measure for S(∞). More generally, we include μ 1 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 ℓ 2( 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 z 1, z 2, 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. ER -}, added-at = {2008-08-28T17:56:58.000+0200}, author = {Kerov, Sergei and Olshanski, Grigori and Vershik, Anatoly}, biburl = {https://www.bibsonomy.org/bibtex/2aba39504aa1b3de49c964fc15363e51a/haulk}, interhash = {9672f4ceb27810cff075d9067bf32a1f}, intrahash = {aba39504aa1b3de49c964fc15363e51a}, journal = {Inventiones Mathematicae}, keywords = {CitesCSP Kerov Olshanski RepresentationTheory Vershik}, month = {#dec#}, number = 3, pages = {551--642}, timestamp = {2008-08-28T17:56:58.000+0200}, title = {Harmonic analysis on the infinite symmetric group}, url = {http://dx.doi.org/10.1007/s00222-004-0381-4}, volume = 158, year = 2004 }