A quantization of the harmonic analysis on the infinite-dimensional unitary group

TL;DR

This paper introduces a q-discretization of harmonic analysis on the infinite-dimensional unitary group, leading to new determinantal point processes connected with q-orthogonal polynomials and an extended Gelfand-Tsetlin graph.

Contribution

It develops a novel q-discretization approach for harmonic analysis on U(∞), revealing new determinantal processes and extending the Gelfand-Tsetlin graph framework.

Findings

Derived explicit correlation kernels using basic hypergeometric functions.

Connected new point processes with pseudo big q-Jacobi polynomials.

Extended the Gelfand-Tsetlin graph to include a q-boundary.

Abstract

The present work stemmed from the study of the problem of harmonic analysis on the infinite-dimensional unitary group U(\infty). That problem consisted in the decomposition of a certain 4-parameter family of unitary representations, which replace the nonexisting two-sided regular representation (Olshanski, J. Funct. Anal., 2003, arXiv:0109193). The required decomposition is governed by certain probability measures on an infinite-dimensional space \Omega, which is a dual object to U(\infty). A way to describe those measures is to convert them into determinantal point processes on the real line, it turned out that their correlation kernels are computable in explicit form --- they admit a closed expression in terms of the Gauss hypergeometric function 2-F-1 (Borodin and Olshanski, Ann. Math., 2005, arXiv:0109194). In the present work we describe a (nonevident) q-discretization of the whole construction. This leads us to a new family of determinantal point processes. We reveal its connection with an exotic finite system of q-discrete orthogonal polynomials --- the so-called pseudo big q-Jacobi polynomials. The new point processes live on a double q-lattice and we show that their correlation kernels are expressed through the basic hypergeometric function 2-\phi-1. A crucial novel ingredient of our approach is an extended version G of the Gelfand-Tsetlin graph (the conventional graph describes the Gelfand-Tsetlin branching rule for irreducible representations of unitary groups). We find the q-boundary of G, thus extending previously known results (Gorin, Adv. Math., 2012, arXiv:1011.1769).