Asymptotic Problems in Representation Theory

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Abstract Okounkov The aim of this project is to investigate a number of interrelated asymptotic problems arising in representation theory and mathematical physics. The two main parts of the project are the study of admissible representations of Gelfand pairs associated to infinite-dimensional symmetric spaces and the investigation of the thermodynamic limit of certain quantum many-body systems. The methods to be used are the various asymptotic methods developed recently by G. Olshanski and the PI. They include, in particular, the use of certain interpolation analogs of the multivariate Jacobi polynomials and binomial-type expansions of the multivariate Jacobi polynomials. In mathematics, many problems of interest are just too complicated to attack them directly. There are two fundamental principles which help gain some insight into complicated phenomena: the symmetry, which helps understand the structure of the problem, and the law of large numbers, which forces systems of a very large size behave, in a sense, in a simplified and predictable fashion. The interaction of these principles plays the key role in the project. For example, we study a certain quantum mechanical system of particles and use its rich symmetries to obtain information about the behavior of this system as the number of particles becomes very, very large. This sort of information is of great interest because most physical objects consist of a huge number of very small components.