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

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• 作者：Vadim Gorin^a ; ^b ; ^{vadicgor@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Grigori Olshanski^b ; ^c ; ^{olsh2007@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Noncommutative harmonic analysis ; Gelfand&ndash ; Tsetlin graph ; Determinantal measures
• 刊名：Journal of Functional Analysis
• 出版年：2016
• 出版时间：1 January 2016
• 年：2016
• 卷：270
• 期：1
• 页码：375-418
• 全文大小：697 K

文摘

The present work stemmed from the study of the problem of harmonic analysis on the infinite-dimensional unitary group U(∞)$U(\infty )$. That problem consisted in the decomposition of a certain 4-parameter family of unitary representations, which replace the nonexisting two-sided regular representation (Olshanski [31]). The required decomposition is governed by certain probability measures on an infinite-dimensional space Ω, which is a dual object to U(∞)$U(\infty )$. 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 [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 a055ea87f004b4058f0f7e4e9d2">${}_2\varphi _1$.

A crucial novel ingredient of our approach is an extended version G$\mathbb{G}$ 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$\mathbb{G}$, thus extending previously known results (Gorin [17]).