  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.
