Quantization and reduction for torsion free CR manifolds

2024/11/01 by Andrea Galasso, Galasso, Andrea, Chin-Yu Hsiao +1
Mathematics · #Advanced Topics in Algebra #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Holomorphic and Operator Theory #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.2411.00478

openalex publication_date 2024/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• G-invariant Szegö kernel asymptotics and CR reduction
• Geometric quantization on CR manifolds
• Szegő kernels and equivariant embedding theorems for CR manifolds
• Functional calculus and quantization commutes with reduction for Toeplitz operators on CR manifolds
• Asymptotics of G-equivariant Szegő kernels