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Quantization and reduction for torsion free CR manifolds

Andrea Galasso, Chin-Yu Hsiao

TL;DR

The work develops a CR-analytic framework to study geometric quantization with symmetry on compact torsion-free CR manifolds. It constructs a $G$-invariant weighted Fourier-Szegő projector on $L^k$-valued CR sections and proves a full semiclassical expansion for its kernel via Melin–Sjöstrand stationary phase, with the leading term controlled by the Levi form and curvature data. A CR reduction theorem is established through curvature-data reduction, showing that the reduced space $X_G=Y/G$ inherits a CR structure compatible with the curvature data. Using the asymptotics, the authors prove that for large $k$ and appropriate spectral windows, quantization commutes with reduction in this CR context, yielding exact dimension counts and an isomorphism between invariant CR sections on $X$ and reduced CR sections on $X_G$. This extends the Guillemin–Sternberg paradigm to torsion-free CR geometry and provides technical tools for CR quantization with symmetry.

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-Szegő operator projects. Under certain natural assumptions, we show that the group invariant Fourier-Szegő 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.

Quantization and reduction for torsion free CR manifolds

TL;DR

The work develops a CR-analytic framework to study geometric quantization with symmetry on compact torsion-free CR manifolds. It constructs a -invariant weighted Fourier-Szegő projector on -valued CR sections and proves a full semiclassical expansion for its kernel via Melin–Sjöstrand stationary phase, with the leading term controlled by the Levi form and curvature data. A CR reduction theorem is established through curvature-data reduction, showing that the reduced space inherits a CR structure compatible with the curvature data. Using the asymptotics, the authors prove that for large and appropriate spectral windows, quantization commutes with reduction in this CR context, yielding exact dimension counts and an isomorphism between invariant CR sections on and reduced CR sections on . This extends the Guillemin–Sternberg paradigm to torsion-free CR geometry and provides technical tools for CR quantization with symmetry.

Abstract

Consider a compact torsion free CR manifold and assume that admits a compact CR Lie group action . Let be a -equivariant rigid CR line bundle over . It seems natural to consider the space of -invariant CR sections in the high tensor powers as quantization space, on which a certain weighted -invariant Fourier-Szegő operator projects. Under certain natural assumptions, we show that the group invariant Fourier-Szegő 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.

Paper Structure

This paper contains 15 sections, 29 theorems, 262 equations.

Table of Contents

  1. Introduction
  2. Examples
  3. Preliminaries
  4. Standard notations
  5. Some standard notations in semi-classical analysis
  6. CR geometry and CR line bundles
  7. The weighted Fourier-Szegő operator
  8. CR manifolds with group action
  9. CR reduction with respect to curvature data
  10. Proofs of Theorem \ref{['thm:Gszego']} and Theorem \ref{['thm:2']}
  11. Local coordinates
  12. $G$-invariant Szegő kernels asymptoics near $Y$
  13. Proof of Theorem \ref{['thm:quantandred']}
  14. Preparation
  15. Theorem \ref{['thm:quantandred']}

Key Result

Theorem 1.1

With the notations and assumptions above, let $\chi\in\mathcal{C}^\infty(X)$ with ${\rm supp\,}\chi\cap Y=\emptyset$. Then, Let $p\in Y$ and let $s$ be a local $G\times\mathbb R$-invariant CR trivializing section defined on an open set $D\subset X$, $p\in D$, $\left\vert s\right\vert^2_{h^L}=e^{-2\Phi}$. Then on $D\times D$, where is a symbol with expansion Furthermore $A\in\mathcal{C}^{\infty

Theorems & Definitions (58)

  • Remark 1.1
  • Theorem 1.1: Semi-classical $G$-invariant Fourier Szegő kernel
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4: Quantization commutes with reduction
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 48 more