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 $T^{\ast }(U/K)\cong U_{\mathbb{C}}/K_{\mathbb{C}}$, 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 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 $T^{\ast }U$, 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 ${\mathcal{P}}_{\infty }$ 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 $U$.

36 pages

References in corpus (2)

• Automorphisms of multiplicity free Hamiltonian manifolds
• Limit of geometric quantizations on Kähler manifolds with T-symmetry