Geometric quantization on CR manifolds

TL;DR

This paper investigates geometric quantization on CR manifolds with group actions, proving Fredholm properties of the CR Guillemin-Strernberg map and demonstrating that quantization commutes with reduction in Sasakian settings.

Contribution

It establishes the Fredholm property of the CR Guillemin-Strernberg map under pseudoconvexity and proves quantization commutes with reduction for Sasakian manifolds.

Findings

CR Guillemin-Strernberg map is Fredholm under pseudoconvexity

Quantization commutes with reduction in Sasakian manifolds

Application to holomorphic line bundles near momentum map inverse image

Abstract

Let $X$ be a compact connected orientable CR manifold with the action of a connected compact Lie group $G$. Under natural pseudoconvexity assumptions we show that the CR Guillemin-Strernberg map is Fredholm at the level of Sobolev spaces of CR functions. As application we study this map for holomorphic line bundles which are positive near the inverse image of $0$ by the momentum map. We also show that "quantization commutes with reduction" for Sasakian manifolds.