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

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.