Equivariant spectral asymptotics for h-pseudodifferential operators

We prove equivariant spectral asymptotics for h-pseudodifferential operators for compact orthogonal group actions generalizing results of El Houakmi and Helffer ["Comportement semi-classique en présence de symétries: Action d'un groupe de Lie compact," Asymp. Anal. 5(2), 91-113 (1991)] and Cassanas ["Reduced Gutzwiller formula with symmetry: Case of a Lie group," J. Math. Pures Appl. 85(6), 719-742 (2006)]. Using recent results for certain oscillatory integrals with singular critical sets [P. Ramacher, "Singular equivariant asymptotics and Weyl's law: On the distribution of eigenvalues of an invariant elliptic operator," J. Reine Angew. Math. (Crelles J.) (to be published)], we can deduce a weak equivariant Weyl law. Furthermore, we can prove a complete asymptotic expansion for the Gutzwiller trace formula without any additional condition on the group action by a suitable generalization of the dynamical assumptions on the Hamilton flow.