Toeplitz operators and Hamiltonian torus actions

Abstract

This paper is devoted to semi-classical aspects of symplectic reduction. Consider a compact prequantizable Kähler manifold M with a Hamiltonian torus action. In the seminal paper [V. Guillemin, S. Sternberg, Geometric quantization and multiplicities of group representations, Invent. Math. 67 (3) (1982) 515–538], 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–Kawasaki theorem.