Quantization in fibering polarizations, Mabuchi rays and geometric Peter--Weyl theorem

TL;DR

This paper uses geometric quantization to interpret the Peter--Weyl theorem, revealing connections between non-Kähler polarizations, Mabuchi geometry, and representation theory of compact groups.

Contribution

It introduces a novel geometric quantization approach for non-Kähler polarizations and links it to the classical Peter--Weyl theorem via Mabuchi geodesics and coherent state transforms.

Findings

Polarization occurs at the geodesic boundary of Kähler polarizations.

Half-form corrected quantization matches the Kähler case.

Unitary parallel transport relates Fourier transform and Borel--Weil representations.

Abstract

In this paper we use techniques of geometric quantization to give a geometric interpretation of the Peter--Weyl theorem. We present a novel approach to half-form corrected geometric quantization in a specific type of non-K\"ahler polarizations and study one important class of examples, namely cotangent bundles of compact semi-simple groups $K$. Our main results state that this canonically defined polarization occurs in the geodesic boundary of the space of $K\times K$-invariant K\"ahler polarizations equipped with Mabuchi's metric, and that its half-form corrected quantization is isomorphic to the K\"ahler case. An important role is played by invariance of the limit polarization under a torus action. Unitary parallel transport on the bundle of quantum states along a specific Mabuchi geodesic, given by the coherent state transform of Hall, relates the non-commutative Fourier transform for $K$ with the Borel--Weil description of irreducible representations of $K$.