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Harmonic analysis on the infinite symmetric group. (English) Zbl 1057.43005

The best studied groups in representation theory are tame. Irreducible unitary representations of tame groups can be classified, and the basic problem of harmonic analysis consists in decomposing interesting reducible representations into irreducible components.
The infinite symmetric group \(S(\infty)\), whose elements are finite permutations of \(\{1,2,3,\dots\}\), is wild, not tame. The wild groups do not admit any sensible classification of irreducible representations, and reducible representations can be decomposed into irreducible ones in essentially different ways. Unlike tame groups, wild groups have factor representations of von Neumann types II and III. All those facts imply that the representation theory of wild groups should be based on principles distinct from those used for tame groups.
In the paper under review the authors develop a new approach to harmonic analysis on the group \(S(\infty)\). They construct a compactification \({\mathfrak S}\supset S(\infty)\), which they call the space of virtual permutations. Although \(\mathfrak S\) is no longer a group, it still admits a natural two-sided action of \(S(\infty)\). Thus, \(\mathfrak S\) is a \(G\)-space, where \(G\) stands for the product of two copies of \(S(\infty)\). On \(\mathfrak S\) there is a family \(\{\mu_t: t>0\}\) of distinguished \(G\)-quasiinvariant probability measures. By making use of these measures, the authors introduce and study a family \(\{ T_z: z\in{\mathbb C}\}\) of unitary representations of \(G\), called generalized regular representations (each representation \(T_z\) with \(z\neq 0\) can be realized in the Hilbert space \(L^2({\mathfrak S}, \mu_t)\), where \(t=| z| ^2\)). As \(| z| \to\infty\), the generalized regular representations \(T_z\) approach, in a suitable sense, the “naive” two-sided regular representation of the group \(G\) in the space \(l^2(S(\infty))\). In contrast with the latter representation, the generalized regular representations \(T_z\) are highly reducible and have a rich structure. The authors 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 \(z\in{\mathbb Z}\), a complete description of the spectral type is obtained. Further work on the case \(z\in{\mathbb C}\setminus{\mathbb Z}\) reveals a remarkable link with stochastic point processes and random matrix theory.

MSC:

43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
20C32 Representations of infinite symmetric groups
05E10 Combinatorial aspects of representation theory
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