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**Harmonic analysis on the infinite-dimensional unitary group and determinantal point processes.**
*(English)*
Zbl 1082.43003

The article is devoted to the problem of decomposition of the natural representations of a given group on irreducible representations. The object of the authors’ investigation is the infinite-dimensional unitary group \(U(\infty)\) that is the inductive limit of growing compact unitary groups \(U(N)\), i.e. \(U(\infty)\) is defined as the union of the ascending chains of compact groups: \(U(1)\subset U(2)\subset U(3)\subset\dots\). The problem considered here consists in computing the spectral decomposition for a remarkable \(4\)-parameter family of characters of \(U(\infty)\). These characters generate representations which can be viewed as analogs of nonexisting regular representations of \(U(\infty)\).

The spectral decomposition of a character of \(U(\infty)\) is described by the spectral measure that lives on an infinite-dimensional space \(\Omega\) of indecomposable characters.

The key idea that allows the authors to solve the problem is to embed \(\Omega\) into a space of point configurations on the real line without two points. This allows them to transform the spectral measure into a stochastic point process on the real line.

The main result of the article (Theorem 10.1) is a complete description of the processes corresponding to the considered family of characters. Unfortunately, we cannot cite it here because the formulation of this statement occupies almost one and a half pages. It is proven that each of the processes is a determinantal point process. Thus, its correlation functions have a determinantal form with a certain kernel. These kernels are expressed by the Gauss hypergeometric function.

The authors reduce the problem of computing the correlation kernels to a problem of evaluating uniform asymptotics of the Askey-Lesky orthogonal polynomials. To solve the last problem, the authors express the polynomials in terms of a solution of a discrete Riemann-Hilbert problem.

As the authors note, from the point of view of statistical physics, they study thermodynamic limits of a discrete log-gas system. An interesting feature of this log-gas is that its density function is asymptotically equal to the characteristic function of an interval. The point processes investigated in the article describe how different the random particle configuration is from the typical “densely packed” configuration.

The article contains an extended Introduction that presents a detailed description of the problems and ideas of the article. The purpose of this topic is to make the article accessible and interesting for a wide category of readers. The Introduction is followed by 11 topics that contain detailed investigation of the problems, and an Appendix that is devoted to some manipulations with hypergeometric functions that are widely exploited in the article. The references contain a wide range of sources that reflect several results and methods concerning the development area.

Let us note that the present article is a serious investigation that is based on several authors’ earlier works. Detailed links can be found in the article. Let us also mention one additional work that should be of interest to the reader. This is an extended version of Olshanski’s lecture at the 4th European congress of mathematics in Stockholm [A. Borodin and G. Olshanski, Representation theory and random point processes. Laptev, Ari (ed.), Proceedings of the 4th European congress of mathematics (ECM), Stockholm, Sweden, June 27 – July 2, 2004. Zürich: European Mathematical Society (EMS). 73–94 (2005; Zbl 1087.15025)].

The present article should be interesting for specialists in abstract harmonic analysis, group representations, statistics, and statistical physics.

The spectral decomposition of a character of \(U(\infty)\) is described by the spectral measure that lives on an infinite-dimensional space \(\Omega\) of indecomposable characters.

The key idea that allows the authors to solve the problem is to embed \(\Omega\) into a space of point configurations on the real line without two points. This allows them to transform the spectral measure into a stochastic point process on the real line.

The main result of the article (Theorem 10.1) is a complete description of the processes corresponding to the considered family of characters. Unfortunately, we cannot cite it here because the formulation of this statement occupies almost one and a half pages. It is proven that each of the processes is a determinantal point process. Thus, its correlation functions have a determinantal form with a certain kernel. These kernels are expressed by the Gauss hypergeometric function.

The authors reduce the problem of computing the correlation kernels to a problem of evaluating uniform asymptotics of the Askey-Lesky orthogonal polynomials. To solve the last problem, the authors express the polynomials in terms of a solution of a discrete Riemann-Hilbert problem.

As the authors note, from the point of view of statistical physics, they study thermodynamic limits of a discrete log-gas system. An interesting feature of this log-gas is that its density function is asymptotically equal to the characteristic function of an interval. The point processes investigated in the article describe how different the random particle configuration is from the typical “densely packed” configuration.

The article contains an extended Introduction that presents a detailed description of the problems and ideas of the article. The purpose of this topic is to make the article accessible and interesting for a wide category of readers. The Introduction is followed by 11 topics that contain detailed investigation of the problems, and an Appendix that is devoted to some manipulations with hypergeometric functions that are widely exploited in the article. The references contain a wide range of sources that reflect several results and methods concerning the development area.

Let us note that the present article is a serious investigation that is based on several authors’ earlier works. Detailed links can be found in the article. Let us also mention one additional work that should be of interest to the reader. This is an extended version of Olshanski’s lecture at the 4th European congress of mathematics in Stockholm [A. Borodin and G. Olshanski, Representation theory and random point processes. Laptev, Ari (ed.), Proceedings of the 4th European congress of mathematics (ECM), Stockholm, Sweden, June 27 – July 2, 2004. Zürich: European Mathematical Society (EMS). 73–94 (2005; Zbl 1087.15025)].

The present article should be interesting for specialists in abstract harmonic analysis, group representations, statistics, and statistical physics.

Reviewer: Alexander Tovstolis (Donetsk)

### MSC:

43A65 | Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis) |

22E65 | Infinite-dimensional Lie groups and their Lie algebras: general properties |

43A05 | Measures on groups and semigroups, etc. |

43A80 | Analysis on other specific Lie groups |

60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |