A remark on Berezin's quantization and cut locus

TL;DR

This paper investigates how Berezin's quantization interacts with the geometric structure of symmetric spaces, specifically focusing on the role of the cut locus in the properties of coherent states and related functions.

Contribution

It precisely characterizes the relationship between the cut locus and the domain of definition for various functions in Berezin's quantization on symmetric spaces.

Findings

Functions like coherent states and covariant symbols are well-defined outside the cut locus.

The set of orthogonal coherent vectors corresponds exactly to the cut locus.

The diastasis and two-point functions are defined when variables are outside the cut locus.

Abstract

The consequences for Berezin's quantization on symmetric spaces of the identity of the set of coherent vectors orthogonal to a fixed one with the cut locus are stated precisely. It is shown that functions expressing the coherent states, the covariant symbols of operators, the diastasis function, the characteristic and two-point functions are defined when one variable does not belong to the cut locus of the other one.