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We study the spectral properties of Schr\"odinger operators on a compactconnected Riemannian manifold $M$ without boundary in case that the underlyingHamiltonian system possesses certain symmetries. More precisely, if $M$ carriesan isometric and effective action of a compact connected Lie group $G$, weprove a generalized equivariant version of the semiclassical Weyl law with anestimate for the remainder, using a semiclassical functional calculus for$h$-dependent functions and relying on recent results on singular equivariantasymptotics. These results will be used to derive an equivariant quantumergodicity theorem in Part II of this work. When $G$ is trivial, one recoversthe classical results.

Semiclassical analysis and symmetry reduction I. Equivariant Weyl law for invariant Schr\"odinger operators on compact manifolds