Berezin's quantization on flag manifolds and spherical modules

TL;DR

This paper demonstrates how spherical Harish-Chandra modules underpin various symbol algebras on flag manifolds and establishes a general correspondence principle connecting them.

Contribution

It provides a unified framework linking spherical modules to symbol algebras on flag manifolds and proves a general correspondence principle.

Findings

Spherical Harish-Chandra modules relate to covariant, contravariant, and mixed symbols.

A general proof of the correspondence principle for these symbol algebras.

Unified algebraic framework for quantization on flag manifolds.

Abstract

It is shown that the theory of spherical Harish-Chandra modules naturally provides the algebras of covariant, contravariant and mixed symbols on generalized flag manifolds. The general proof of the correspondence principle for all these symbol algebras is given.