Harmonic analysis on the infinite symmetric group

• 成果类型: Article
• 署名作者: Kerov, S; Olshanski, G; Vershik, A
• 署名单位: Kharkevich Institute for Information Transmission Problems of the RAS; Russian Academy of Sciences; Steklov Mathematical Institute of the Russian Academy of Sciences; St. Petersburg Department of the Steklov Mathematical Institute of the Russian Academy of Sciences
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-004-0381-4
• 发表日期: 2004
• 页码: 551-642
• 关键词: point-processes; young-diagrams; REPRESENTATIONS; deformation

摘要

The infinite symmetric group S(infinity), 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(infinity) in the Hilbert space l(2)(S(infinity)) generates a type II1 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(infinity): 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 G superset of S(infinity), which we call the space of virtual permutations. Although G is no longer a group, it still admits a natural two-sided action of S(infinity). Thus, G is a G-space, where G stands for the product of two copies of S(infinity). On G, there exists a unique G-invariant probability measure mu(1), which has to be viewed as a true Haar measure for S(infinity). More generally, we include mu(1) into a family {mu(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 is an element of C} of unitary representations of G, called generalized regular representations (each representation T-z with z not equal ) can be realized in the Hilbert space L-2(G, mu(t)) where t = \z\(2)). As \z\ --> infinity, 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(infinity)). 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-z1 and T-z2 are shown to be disjoint. In the case z is an element of Z, a complete description of the spectral type is obtained. Further work on the case z is an element of C\Z reveals a remarkable link with stochastic point processes and random matrix theory.