Functional calculus and quantization commutes with reduction for Toeplitz operators on CR manifolds

TL;DR

This paper demonstrates that functional calculus for Toeplitz operators on CR manifolds involves complex Fourier integral operators and establishes that quantization commutes with reduction under group actions, providing spectral dimension results.

Contribution

It proves that functional calculus operators are Fourier integral operators and shows quantization commutes with reduction for Toeplitz operators on CR manifolds.

Findings

Functional calculus operators are Szegő type Fourier integral operators.

Quantization commutes with reduction for Toeplitz operators with group actions.

Semi-classical spectral dimensions are computed for Toeplitz operators.

Abstract

Given a CR manifold with non-degenerate Levi form, we show that the operators of the functional calculus for Toeplitz operators are complex Fourier integral operators of Szeg\H{o} type. As an application, we establish semi-classical spectral dimensions for Toeplitz operators. We then consider a CR manifold with a compact Lie group action $G$ and we establish quantization commutes with reduction for Toeplitz operators. Moreover, we also compute semi-classical spectral dimensions for $G$-invariant Toeplitz operators.