Fibering polarizations and Mabuchi rays on symmetric spaces of compact type

Abstract

In this paper, we describe holomorphic quantizations of the cotangent bundle of a symmetric space of compact type \(T^*(U/K)\cong U_\mathbb {C}/K_\mathbb {C}\), along Mabuchi rays of U-invariant Kähler structures. At infinite geodesic time, the Kähler polarizations converge to a mixed polarization \(\mathcal {P}_\infty \). 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 \(\mathcal {P}_\infty \)-polarized sections. Unlike in the case of \(T^*(U)\), 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, Mourão and Nunes in Journal of Geometry and Physics, 2025, we also describe how the quantization in the limit polarization \(\mathcal {P}_\infty \) 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.