Toeplitz operators and Hamiltonian torus action

Details Toeplitz operators and Hamiltonian torus action Toeplitz operators and Hamiltonian torus action

2004-05-07

arxiv.org/abs/math/0405128v2

Mathematics

Symplectic Geometry

corrected typos, accepted for publication in J. Funct. Anal

Scientific paper

This paper is devoted to semi-classical aspects of symplectic reduction. Consider a compact prequantizable Kahler manifold M with a Hamiltonian torus action. Guillemin and Sternberg introduced an isomorphism between the invariant part of the quantum space associated to M and the quantum space associated to the symplectic quotient of M, provided this quotient is non-singular. We prove that this isomorphism is a Fourier integral operator and that the Toeplitz operators of M descend to Toeplitz operators of the reduced phase space. We also extend these results to the case where the symplectic quotient is an orbifold and estimate the spectral density of a reduced Toeplitz operator, a result related to the Riemann-Roch-Kawazaki theorem.

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