infinite-dimensional酉群的调和分析与行列式点过程
Harmonic analysis on the infinite-dimensional unitary group and determinantal point processes
摘要：
无限维酉群U（无穷）是增长紧酉群U（N）的归纳极限。在本文中，我们解决了一个问题的调和分析上U（无穷大）在以前的文件数学/0109193。问题在于计算U（无穷大）的一个显著的4-参数特征标族的谱分解。这些字符生成的表示应被视为U（无穷大）不存在的正则表示的类似物。U（无穷大）的特征标的谱分解是由不可分解特征标的无穷维空间Ω上的谱测度描述的。解决这一问题的关键思想是将Omega嵌入到不含两个点的实直线上的点配置空间中。这将谱测度转化为实数线上的随机点过程。本文的主要结果是一个完整的描述过程对应于我们的具体家庭的字符。我们证明了每个过程都是一个决定点过程。也就是说，它的相关函数具有确定核的行列式形式。我们的内核有一个特殊的“可积”的形式，并表示通过高斯超几何函数。在无限对称群上的调和分析和无限埃尔米特矩阵上酉不变测度的调和分析的较简单的情况下，我们的论文math/9810015，math/9904010，math-ph/0010015中得到了类似的结果。
英文摘要：
The infinite-dimensional unitary group U(infinity) is the inductive limit of growing compact unitary groups U(N). In this paper we solve a problem of harmonic analysis on U(infinity) stated in the previous paper math/0109193. The problem consists in computing spectral decomposition for a remarkable 4-parameter family of characters of U(infinity). These characters generate representations which should be viewed as analogs of nonexisting regular representation of U(infinity). The spectral decomposition of a character of U(infinity) is described by the spectral measure which lives on an infinite-dimensional space Omega of indecomposable characters. The key idea which allows us to solve the problem is to embed Omega into the space of point configurations on the real line without 2 points. This turns the spectral measure into a stochastic point process on the real line. The main result of the paper is a complete description of the processes corresponding to our concrete family of characters. We prove that each of the processes is a determinantal point process. That is, its correlation functions have determinantal form with a certain kernel. Our kernels have a special `integrable' form and are expressed through the Gauss hypergeometric function. In simpler situations of harmonic analysis on infinite symmetric group and harmonic analysis of unitarily invariant measures on infinite hermitian matrices similar results were obtained in our papers math/9810015, math/9904010, math-ph/0010015.
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