Singular equivariant asymptotics and Weyl’s law. On the distribution of eigenvalues of an invariant elliptic operator

We study the spectrum of an invariant, elliptic, classical pseudodifferential operator on a closed Riemannian manifold carrying an effective and isometric action of a compact, connected Lie group . Using resolution of singularities, we determine the asymptotic distribution of eigenvalues along the isotypic components, and relate it to the reduction of the corresponding Hamiltonian flow, proving that the reduced spectral counting function satisfies Weyl’s law, together with an estimate for the remainder.