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

The present work stemmed from the study of the problem of harmonic analysis on the infinite-dimensional unitary group U(∞)U(∞). That problem consisted in the decomposition of a certain 4-parameter family of unitary representations, which replace the nonexisting two-sided regular representation (Olshanski [31]). The required decomposition is governed by certain probability measures on an infinite-dimensional space Ω, which is a dual object to U(∞)U(∞). 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 F12 (Borodin and Olshanski [8]).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 ϕ12.A crucial novel ingredient of our approach is an extended version GG 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 GG, thus extending previously known results (Gorin [17]).