Equivariant Asymptotics of Szeg\"o kernels under Hamiltonian $U(2)$ actions

TL;DR

This paper investigates the asymptotic behavior of Szeg"o kernels associated with Hamiltonian $U(2)$ actions on complex projective manifolds, focusing on the properties of isotypical components as the weight parameter grows large.

Contribution

It provides new asymptotic analysis of Szeg"o kernels under Hamiltonian $U(2)$ actions, extending understanding of equivariant kernels in geometric quantization.

Findings

Asymptotic formulas for Szeg"o kernels under $U(2)$ actions.

Characterization of isotypical components for large weights.

Insights into local and global properties of equivariant kernels.

Abstract

Let $M$ be complex projective manifold, and $A$ a positive line bundle on it. Assume that a compact and connected Lie group $G$ acts on $M$ in a Hamiltonian manner, and that this action linearizes to $A$. Then there is an associated unitary representation of $G$ on the associated algebro-geometric Hardy space. If the moment map is nowhere vanishing, the isotypical component are all finite dimensional, they are generally not spaces of sections of some power of $A$. One is then led to study the local and global asymptotic properties the isotypical component associated to a weight $k \, \boldsymbol{\nu}$, when $k\rightarrow +\infty$. In this paper, part of a series dedicated to this general theme, we consider the case $G=U(2)$.