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

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