Weighted Bergman Spaces Associated with the Hyperbolic Group

A new way of describing weighted poly-Bergman spaces and true weighted poly-Bergman spaces on the upper half-plane Π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi $$\end{document} emerges using a system of coordinates induced by the R+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}_+$$\end{document}-action on Π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi $$\end{document}, the moment map associated with this action, and the standard symplectic structure. We give an isomorphic description of such spaces using Romanovski polynomials. We also provide a unitary description of Toeplitz operators on these spaces whose symbols are associated with the moment map.