  Let $M$ be a compact boundaryless Riemannian manifold, carrying an effective
and isometric action of a compact Lie group $G$, and $P_0$ an invariant
elliptic classical pseudodifferential operator on $M$. Using Fourier integral
operator techniques, we prove a local Weyl law with remainder estimate for the
equivariant (or reduced) spectral function of $P_0$ for each isotpyic component
in the Peter-Weyl decomposition of $L^2(M)$, generalizing work of
Avacumovi\v{c}, Levitan, and H\"ormander. From this we deduce a generalized
Kuznecov sum formula for periods of G-orbits, and recover the local Weyl law
for orbifolds shown by Stanhope and Uribe. Relying on recent results on
singular equivariant asymptotics of oscillatory integrals, we further
characterize the caustic behaviour of the reduced spectral function near
singular orbits, which allows us to give corresponding point-wise bounds for
clusters of eigenfunctions in specific isotypic components. In case that $G$
acts on $M$ without singular orbits, we are able to deduce hybrid $L^p$-bounds
for $2 \leq p \leq \infty$ in the eigenvalue and isotypic aspect that improve
on the classical estimates of Seeger and Sogge for generic eigenfunctions. Our
results are sharp in the eigenvalue aspect, but not in the isotypic aspect, and
reduce to the classical ones in the case $G=\{e\}$.
