## The problem of harmonic analysis on the infinite-dimensional unitary group

Venue: | J. Funct. Anal. 205, no |

Citations: | 32 - 10 self |

### BibTeX

@ARTICLE{Borodin_theproblem,

author = {Alexei Borodin and Grigori Olshanski},

title = {The problem of harmonic analysis on the infinite-dimensional unitary group},

journal = {J. Funct. Anal. 205, no},

year = {},

pages = {464--524}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. The infinite–dimensional unitary group U(∞) is the inductive limit of growing compact unitary groups U(N). In this paper we solve a problem of harmonic analysis on U(∞) stated in [Ol3]. The problem consists in computing spectral decomposition for a remarkable 4–parameter family of characters of U(∞). These characters generate representations which should be viewed as analogs of nonexisting regular representation of U(∞). The spectral decomposition of a character of U(∞) is described by the spectral measure which lives on an infinite–dimensional space Ω of indecomposable characters. The key idea which allows us to solve the problem is to embed Ω 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