A quantization of the harmonic analysis on the infinite-dimensional unitary group
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Author(s)
Gorin, Vadim
Olshanski, Grigori
Date Issued
June 2015
Journal
Journal of Functional Analysis
Publisher
Elsevier
Citation
Gorin, Vadim and Grigori Olshanski. “A Quantization of the Harmonic Analysis on the Infinite-Dimensional Unitary Group.” Journal of Functional Analysis 270, 1 (January 2016): 375–418 © 2015 Elsevier Inc
Version
Author's final manuscript
Abstract
The present work stemmed from the study of the problem of harmonic analysis on the infinite-dimensional unitary group 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(∞). 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 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 [17]). Keywords:
Noncommutative harmonic analysis; Gelfand–Tsetlin graph; Determinantal measures
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-NonCommercial-NoDerivs License
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DOI of Published Version
https://doi.org/10.1016/J.JFA.2015.06.006
