Singular equivariant asymptotics and Weyl's law

TL;DR

This paper investigates the spectral distribution of invariant elliptic operators on manifolds with group actions, establishing Weyl's law for eigenvalues within isotypic components using singularity resolution techniques.

Contribution

It introduces a method to analyze eigenvalue asymptotics for invariant operators on G-manifolds, linking spectral properties with Hamiltonian flow reduction.

Findings

Weyl's law holds for equivariant spectral counting functions.

Asymptotic eigenvalue distribution is characterized along isotypic components.

Provides estimates for the spectral counting function remainder.

Abstract

We study the spectrum of an invariant, elliptic, classical pseudodifferential operator on a closed G-manifold M, where G is a compact, connected Lie group acting effectively and isometrically on M. Using resolution of singularities, we determine the asymptotic distribution of eigenvalues along the isotypic components, and relate it with the reduction of the corresponding Hamiltonian flow, proving that the equivariant spectral counting function satisfies Weyl's law, together with an estimate for the remainder.