Asymptotics of G-equivariant Szeg\H{o} kernels

TL;DR

This paper investigates the asymptotic behavior of G-equivariant Szeg ext{"o} kernels on CR manifolds with group actions, extending classical theorems and analyzing Fourier components under additional circle actions.

Contribution

It introduces G-equivariant Szeg ext{"o} kernels on CR manifolds and establishes asymptotic theorems, including Fourier analysis under circle actions, expanding the understanding of geometric analysis in this context.

Findings

Established G-equivariant Szeg ext{"o} kernel asymptotics

Derived Boutet de Monvel-Sj"ostrand type theorems for these kernels

Analyzed Fourier components under transversal CR S^1 actions

Abstract

Let $(X, T^{1,0}X)$ be a compact connected orientable CR manifold of dimension $2n+1$ with non-degenerate Levi curvature. Assume that $X$ admits a connected compact Lie group $G$ action. Under certain natural assumptions about the group $G$ action, we define $G$-equivariant Szeg\H{o} kernels and establish the associated Boutet de Monvel-Sj\"ostrand type theorems. When $X$ admits also a transversal CR $S^1$ action, we study the asymptotics of Fourier components of $G$-equivariant Szeg\H{o} kernels with respect to the $S^1$ action.