Asymptotics of Szeg\"{o} kernels under Hamiltonian torus actions

TL;DR

This paper investigates the asymptotic behavior of Szeg"o kernels on circle bundles over symplectic manifolds, especially under Hamiltonian torus actions, revealing local decompositions related to group representations.

Contribution

It extends the Tian-Zelditch expansion to the setting of Hamiltonian torus actions, providing local asymptotic expansions reflecting equivariant decompositions.

Findings

Derived asymptotic expansions for Szeg"o kernels under torus actions

Connected local kernel behavior to group representation theory

Enhanced understanding of Hardy space decomposition in symplectic geometry

Abstract

Let $X$ be the circle bundle associated to a positive line bundle on a complex projective (or, more generally, compact symplectic) manifold. The Tian-Zelditch expansion on $X$ may be seen as a local manifestation of the decomposition of the (generalized) Hardy space $H(X)$ into isotypes for the $S^1$-action. More generally, given a compatible action of a compact Lie group, and under general assumptions guaranteeing finite dimensionality of isotypes, we may look for asymptotic expansions locally reflecting the equivariant decomposition of $H(X)$ over the irreducible representations of the group. We focus here on the case of compact tori.