  We study the ergodic properties of eigenfunctions of Schr\"odinger operators
on a closed connected Riemannian manifold $M$ in case that the underlying
Hamiltonian system possesses certain symmetries. More precisely, let $M$ carry
an isometric effective action of a compact connected Lie group $G$. We prove an
equivariant quantum ergodicity theorem assuming that the symmetry-reduced
Hamiltonian flow on the principal stratum of the singular symplectic reduction
of $M$ is ergodic. We deduce the theorem by proving an equivariant version of
the semiclassical Weyl law, relying on recent results on singular equivariant
asymptotics. It implies an equivariant version of the
Shnirelman-Zelditch-Colin-de-Verdi\`{e}re theorem, as well as a representation
theoretic equidistribution theorem. In case that $G$ is trivial, one recovers
the classical results.
