Szeg\"{o} kernel equivariant asymptotics under Hamiltonian Lie group actions

TL;DR

This paper investigates the asymptotic behavior of Szeg"o kernels in the context of Hamiltonian Lie group actions on complex manifolds, revealing how these kernels concentrate near specific loci related to the moment map.

Contribution

It provides new asymptotic formulas for equivariant Szeg"o kernels under Hamiltonian Lie group actions, extending understanding of their concentration properties.

Findings

Szeg"o kernels concentrate near loci defined by the moment map.

Finite dimensionality of isotypical components under certain conditions.

Asymptotic formulas for equivariant kernels derived.

Abstract

Suppose that a compact and connected Lie group $G$ acts on a complex Hodge manifold $M$ in a holomorphic and Hamiltonian manner, and that the action linearizes to a positive holomorphic line bundle $A$ on $M$. Then there is an induced unitary representation on the associated Hardy space and, if the moment map of the action is nowhere vanishing, the corresponding isotypical components are all finite dimensional. We study the asymptotic concentration behavior of the corresponding equivariant Szeg\"{o} kernels near certain loci defined by the moment map.