Fibering polarizations and Mabuchi rays on symmetric spaces of compact type

TL;DR

This paper explores the behavior of holomorphic quantizations on symmetric spaces of compact type along Mabuchi rays, revealing convergence properties and the structure of limit polarizations, with implications for geometric quantization.

Contribution

It introduces a detailed analysis of quantizations along Mabuchi geodesics on symmetric spaces, including the convergence to a mixed polarization and the role of generalized coherent state transforms.

Findings

Holomorphic sections converge to distributional polarized sections at infinity.

The generalized coherent state transform is not asymptotically unitary in this setting.

Limit quantization decomposes into a sum over symplectic reductions.

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 K\"ahler structures. At infinite geodesic time, the K\"ahler 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 $T^*U$, 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 $U$.