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Addendum to "Singular equivariant asymptotics and Weyl's law" (1507.05611v2)
Published 20 Jul 2015 in math.SP and math.DG
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 $L2(M)$. In forthcoming papers, the improved remainder will be used to prove an equivariant semiclassical Weyl law and a corresponding equivariant quantum ergodicity theorem.