A Representation Theoretic Approach to Toeplitz Quantization on Flag Manifolds

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.