Scaling asymptotics of Szego kernels under commuting Hamiltonian actions

TL;DR

This paper investigates the asymptotic behavior of Szego kernels under commuting Hamiltonian actions on complex projective manifolds, revealing how the structure of G-modules is reflected in local scaling asymptotics of equivariant Szego projectors.

Contribution

It provides a detailed analysis of the scaling asymptotics of Szego kernels in the presence of commuting Hamiltonian group actions, linking local geometric properties to representation-theoretic structures.

Findings

Finite-dimensionality of isotypical components under certain conditions

Asymptotic concentration of Szego projectors along loci defined by moment maps

Explicit local scaling asymptotics for equivariant Szego kernels

Abstract

Let M be a connected d-dimensional complex projective manifold, and let A be a holomorphic positive Hermitian line bundle on M, with normalized curvature. Let G be a compact and connected Lie group of dimension d(G), and let T be a compact torus T of dimension d(T). Suppose that both G and T act on M in a holomorphic and Hamiltonian manner, that the actions commute, and linearize to A. If X is the principal circle-bundle associated to A, then this set-up determines commuting unitary representations of G and T on the Hardy space H(X) of X, which may then be decomposed over the irreducible representations of the two groups. If the moment map for the T-action is nowhere zero, all isotypical components for the torus are finite-dimensional, and thus provide a collection of finite-dimensional G-modules. Given a non-zero integral weight n(T) for T, we consider the isotypical components associated to the multiples kn(T), k that goes to infinity, and focus on how their structure as G-modules is reflected by certain local scaling asymptotics on X (and M). More precisely, given a fixed irreducible character n(G) of G, we study the local scaling asymptotics of the equivariant Szeg\H{o} projectors associated to n(G) and kn(T), for k that goes to infinity, investigating their asymptotic concentration along certain loci defined by the moment maps.