Quantization and reduction for torsion free CR manifolds

TL;DR

This paper studies the quantization process on torsion free CR manifolds with group actions, proving asymptotic expansion of Fourier-Szeg ext{"o} operators and demonstrating that quantization commutes with reduction under certain conditions.

Contribution

It establishes the asymptotic expansion of the Fourier-Szeg ext{"o} operator and shows quantization commutes with reduction for high tensor powers of line bundles.

Findings

Fourier-Szeg ext{"o} projector admits a full asymptotic expansion.

Quantization commutes with reduction for large tensor powers.

Results apply to compact torsion free CR manifolds with group actions.

Abstract

Consider a compact torsion free CR manifold $X$ and assume that $X$ admits a compact CR Lie group action $G$. Let $L$ be a $G$-equivariant rigid CR line bundle over $X$. It seems natural to consider the space of $G$-invariant CR sections in the high tensor powers as quantization space, on which a certain weighted $G$-invariant Fourier-Szeg\H{o} operator projects. Under certain natural assumptions, we show that the group invariant Fourier-Szeg\H{o} projector admits a full asymptotic expansion. As an application, if the tensor power of the line bundle is large enough, we prove that quantization commutes with reduction.