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A Representation Theoretic Approach to Toeplitz Quantization on Flag Manifolds

Matthew Dawson, Yessica Hernández-Eliseo

TL;DR

The paper develops a representation-theoretic framework for Toeplitz operators on generalized flag manifolds $G/L$, deriving a matrix-coefficient formula from the decomposition of the tensor product $oldsymbol{\sigma^oldsymbol{\lambda} ensorar{oldsymbol{\sigma^oldsymbol{\lambda}}}$ and showing how invariant symbols yield large commuting families when the restriction to a subgroup is multiplicity-free. It realizes the Berezin transform as a convolution with the kernel $|oldsymbol{ riangle^oldsymbol{\lambda}}|^2$, proves surjectivity of the Toeplitz quantization map and characterizes its kernel via tensor-product multiplicities, and establishes a Szegő-type limit theorem by constructing approximate identities from the Berezin kernels. The work unites Harish-Chandra/Borel–Weil–conical-function machinery with Toeplitz quantization on flag manifolds, yielding semiclassical limits, asymptotic multiplicities, and a pathway to Szegő theorems in a broad representation-theoretic setting. This advances understanding of quantization on non-symmetric flag manifolds and offers concrete tools for analyzing commuting Toeplitz algebras and Berezin kernels in a broad class of compact Lie groups.

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 Szegö Limit Theorem using certain results due to Hirschman, Liang, and Wilson.

A Representation Theoretic Approach to Toeplitz Quantization on Flag Manifolds

TL;DR

The paper develops a representation-theoretic framework for Toeplitz operators on generalized flag manifolds , deriving a matrix-coefficient formula from the decomposition of the tensor product and showing how invariant symbols yield large commuting families when the restriction to a subgroup is multiplicity-free. It realizes the Berezin transform as a convolution with the kernel , proves surjectivity of the Toeplitz quantization map and characterizes its kernel via tensor-product multiplicities, and establishes a Szegő-type limit theorem by constructing approximate identities from the Berezin kernels. The work unites Harish-Chandra/Borel–Weil–conical-function machinery with Toeplitz quantization on flag manifolds, yielding semiclassical limits, asymptotic multiplicities, and a pathway to Szegő theorems in a broad representation-theoretic setting. This advances understanding of quantization on non-symmetric flag manifolds and offers concrete tools for analyzing commuting Toeplitz algebras and Berezin kernels in a broad class of compact Lie groups.

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 Szegö Limit Theorem using certain results due to Hirschman, Liang, and Wilson.
Paper Structure (12 sections, 26 theorems, 138 equations)

This paper contains 12 sections, 26 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Notation and Background
  3. Parabolic Subalgebras and Flag Manifolds
  4. The Borel-Weil Theorem and Matrix-Coefficient Functions
  5. Conical Functions
  6. Holomorphic Sections of Line Bundles on Partial Flag Manifolds
  7. Definition and Basic Properties of Toeplitz Operators
  8. Invariant Symbols and Commutative Toeplitz Algebras
  9. A Tensor Product and the Kernel of the Toeplitz Quantization Map
  10. The Berezin Transform
  11. The Berezin Transform and Approximate Identities
  12. Concluding Remarks and Further Research

Key Result

Lemma 2.1

If $\lambda = \lambda_1 \omega_1 + \cdots +\lambda_r \omega_r + \lambda_Z\in \Lambda^+$, where $\lambda_i \in\mathbb{Z}^{\geq 0}$ for $1\leq i \leq r$ and $\lambda_{Z}\in iZ(\mathfrak{g})^*$, then the corresponding conical function is given by: for all $g\in G^\mathbb{C}$, where $\chi_{\lambda_Z}$ is the central character of $G$ corresponding to the weight $\lambda_Z\in iZ(\mathfrak{g})^*$.

Theorems & Definitions (49)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • Definition 3.1
  • Lemma 3.3
  • proof
  • Corollary 3.4
  • ...and 39 more