The equivariant spectral function of an invariant elliptic operator. $L^p$-bounds, caustics, and concentration of eigenfunctions

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 $G=\{e\}$.