There is a classical theorem of Sogge which provides bounds for the $L^p$ norms of a Laplace eigenfunction on a compact Riemannian manifold, which are sharp on the sphere and for spectral clusters. I will present a generalization of this theorem to eigenfunctions of the full ring of invariant differential operators on a locally symmetric space, as well as a theorem on the restriction of eigenfunctions to maximal flat subspaces. Time permitting, I will discuss ways in which these bounds can be improved using inputs from number theory.