A Representation Theoretic Approach to Toeplitz Quantization on Flag Manifolds

TL;DR

This paper introduces a representation-theoretic framework for analyzing Toeplitz operators on flag manifolds, deriving properties, commuting families, and a Szeg"o Limit Theorem through harmonic analysis techniques.

Contribution

It provides a novel representation-theoretic approach to Toeplitz quantization on flag manifolds, including explicit formulas, invariant families, and the realization of the Berezin transform as a convolution.

Findings

Derived an abstract formula for matrix coefficients of Toeplitz operators.

Identified large commuting families of Toeplitz operators via subgroup invariance.

Proved a Szeg"o Limit Theorem using convolution approximations of the Berezin transform.

Abstract

In this paper, we study Toeplitz operators on generalized flag manifolds of compact Lie groups using a representation-theoretic point of view. We prove several basic properties of these Toeplitz operators, including an abstract formula for their matrix coefficients in terms of the decomposition of certain tensor product representations. We also show how to identify large commuting families of Toeplitz operators based on invariance of their symbols under certain subgroups. Finally, we realize the Berezin transform as a convolution with certain functions that form an approximate identity on the generalized flag manifold, which allows us to prove a Szeg\"o Limit Theorem using certain results due to Hirschman, Liang, and Wilson.