Toeplitz operators and Hamiltonian torus actions
type: Item
instance of: scholarly article
title: Toeplitz operators and Hamiltonian torus actions
zbMATH Open document ID: 1099.53059
author: Laurent Charles
published in: Journal of Functional Analysis
publication date: 2006-06-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0405128
review text: This paper deals with semi-classical aspects of geometric quantization. In the case of compact Kähler phase space and a compact symplectic group action \textit{V. Guillemin} and \textit{S. Sternberg} [Invent. Math. 67, 491--513 (1982; Zbl 0503.58017); ibid. 67, 515--538 (1982; Zbl 0503.58018)] constructed an isomorphy between the \(G\)-invariant part of the quantizing Hilbert space and the quantizing space of the \(G\)-reduced phase space. The paper under review considers the semi-classical aspects in the case of the torus group acting either on a compact phase space or on \(Cn\) (\(n\)-dim. harmonic oscillator). The main results state that 1) the intertwining Guillemin/Sternberg isomorphism is a Fourier integral operator and 2) Toeplitz operators on the phase space correspond to Toeplitz operators on the reduce phase space. These results are extended to the case that the reduced phase space is only an orbifold.
Mathematics Subject Classification ID: 53D50
Mathematics Subject Classification ID: 58D30
Mathematics Subject Classification ID: 47B37
Mathematics Subject Classification ID: 58J40
zbMATH DE Number: 5037261
zbMATH Keywords: geometric quantization
zbMATH Keywords: microlocal
zbMATH Keywords: semi-classical
zbMATH Keywords: symplectic reduction Toeplitz operator
zbMATH Keywords: Fourier integral operator
reviewed by: Christian Günther
MaRDI profile type: Publication
OpenAlex ID: W1975796939
arXiv ID: math/0405128
cites work: Convexity and Commuting Hamiltonians
cites work: General concept of quantization
cites work: Semiclassical spectral estimates for Toeplitz operators
cites work: The Spectral Theory of Toeplitz Operators. (AM-99)
cites work: Q4111841
cites work: Berezin--Toeplitz operators, a semi-classical approach
cites work: Quasimodes and Bohr-Sommerfeld Conditions for the Toeplitz Operators
cites work: Q4409407
cites work: Q3776999
cites work: The heat kernel Lefschetz fixed point formula for the spin-c dirac operator
cites work: Q4855399
cites work: Convexity properties of the moment mapping
cites work: Geometric quantization and multiplicities of group representations
cites work: Homogeneous quantization and multiplicities of group representations
cites work: Q4783635
cites work: Q3999037
cites work: The Riemann-Roch theorem for complex V-manifolds
cites work: Reduction of symplectic manifolds with symmetry
cites work: Symplectic surgery and the Spin\(^c\)-Dirac operator
cites work: Q4409414
cites work: Q4409417
cites work: ON A GENERALIZATION OF THE NOTION OF MANIFOLD
cites work: Symplectic reduction and Riemann-Roch formulas for multiplicities
cites work: Q4337982
cites work: Q4192639
cites work: Fedosov *-products and quantum momentum maps
DOI: 10.1016/J.JFA.2005.10.011
description: scientific article