Consider the Laplace-Beltrami operator on a Riemannian manifold. Through functional calculus, one defines spectral projectors at a given frequency, with a given bandwidth. What is the L^2 --> L^p operator norm of these projectors? Sogge proved 30 years ago universal (valid for any manifold) and optimal bounds if the bandwidth is sufficiently broad. For narrower bandwidth, the global geometry of the manifold comes into play, and the question is mostly open. I will present progress on these questions for some locally symmetric spaces: the Euclidean torus, and quotients of the hyperbolic plane.

This is joint work with J.-P. Anker, T. Leger and S. Myerson.