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^*(U/K)≅ U_ℂ/K_ℂ, along Mabuchi rays of U-invariant Kähler structures. At infinite geodesic time, the Kähler polarizations converge to a mixed polarization 𝒫_∞. We show how a generalized coherent state transform relates the quantizations along the Mabuchi geodesics such that holomorphic sections converge, as geodesic time goes to infinity, to distributional 𝒫_∞-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 in [Bai+23], we also describe how the quantization in the limit polarization 𝒫_∞ 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.