Addendum to "Singular equivariant asymptotics and Weyl's law"

TL;DR

This paper refines the analysis of oscillatory integrals on Riemannian manifolds with symmetry, leading to improved eigenvalue distribution estimates and representation multiplicity formulas, with applications to Weyl's law and quantum ergodicity.

Contribution

It provides a refined remainder estimate in stationary phase approximation for singular critical sets, advancing the understanding of eigenvalue asymptotics under symmetry.

Findings

Derived a refined remainder estimate for oscillatory integrals with singular critical sets.

Established an asymptotic multiplicity formula for irreducible representations in $L^2(M)$.

Set the stage for future proofs of equivariant Weyl law and quantum ergodicity.

Abstract

Let $M$ be a closed Riemannian manifold carrying an effective and isometric action of a compact connected Lie group $G$. We derive a refined remainder estimate in the stationary phase approximation of certain oscillatory integrals on $T^\ast M \times G$ with singular critical sets that were examined previously in order to determine the asymptotic distribution of eigenvalues of an invariant elliptic operator on $M$. As an immediate consequence, we deduce from this an asymptotic multiplicity formula for families of irreducible representations in $L^2(M)$. In forthcoming papers, the improved remainder will be used to prove an equivariant semiclassical Weyl law and a corresponding equivariant quantum ergodicity theorem.