Toeplitz operators and Hamiltonian torus actions
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.
scientific article