Fibering polarizations and Mabuchi rays on symmetric spaces of compact type

In this paper, we describe holomorphic quantizations of the cotangent bundle of a symmetric space of compact type T & lowast;(U/K)congruent to UC/KC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T<^>*(U/K)\cong U_\mathbb {C}/K_\mathbb {C}$$\end{document}, along Mabuchi rays of U-invariant K & auml;hler structures. At infinite geodesic time, the K & auml;hler polarizations converge to a mixed polarization P infinity\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {P}_\infty $$\end{document}. We show how a generalized coherent state transform (gCST) relates the quantizations along the Mabuchi geodesics such that holomorphic sections converge, as geodesic time goes to infinity, to distributional P infinity\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {P}_\infty $$\end{document}-polarized sections. Unlike in the case of T & lowast;(U)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T<^>*(U)$$\end{document}, the gCST mapping from the Hilbert space of vertically polarized sections are not asymptotically unitary due to the appearance of representation dependent factors associated to the isotypical decomposition for the U-action . In agreement with the general program outlined by Baier, Hilgert, Kaya, Mour & atilde;o and Nunes in Journal of Geometry and Physics, 2025, we also describe how the quantization in the limit polarization P infinity\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {P}_\infty $$\end{document} is given by the direct sum of the quantizations for all the symplectic reductions relative to the invariant torus action associated to the Hamiltonian action of U.