Quantum ergodicity in higher rank for Benjamini-Schramm sequences of lattices

The quantum ergodicity theorem of Schnirelman, Colin de Verdière, and Zelditch states that on a compact Riemannian manifold with ergodic geodesic flow, most eigenfunctions of high frequency distribute their L2 mass according to the uniform measure. More recently, there has been increasing interest in the delocalization properties of eigenfunctions of bounded frequency on spaces of increasing volume. A particularly well-behaved setting is that of locally symmetric spaces, attached to cocompact lattices having uniform spectral gap in a fixed non-compact semisimple Lie group, which Benjamini-Schramm converge to their commun universal cover. In rank one, this problem has seen tremendous progress. A discrete version of this problem was solved in the breakthrough work of Anantharaman-Le Masson for large regular graphs. Shortly thereafter an alternative approach was given by Brooks-Le Masson-Lindenstrauss which relied critically on the construction of a wave propagator having good spectral and geometric properties. The latter approach was subsequently adapted to the setting of large hyperbolic surfaces by Le Masson-Sahlsten and then to higher dimensional hyperbolic manifolds by Abert-Bergeron-Le Masson. The kernels of these propagators are supported on spheres or annuli, and (among other things) the argument requires good estimates on the intersection volumes of their translates. In this talk, we shall present a higher rank construction of a wave propagator, which generalizes the one given for PGL(3,\Q_p) by Carsten Peterson in his thesis, and show how it can be used to prove quantum ergodicity for locally symmetric spaces associated with classical real Lie groups in the Benjamini-Schramm limit. In doing so, we repair an error in our previous work on this topic for quotients of SL(n,\R) with Jasmin Matz. Our approach to controlling intersection volumes is due to Simon Marshall, and reduces to establishing deep estimates on the Harish-Chandra spherical function. This is joint work with Simon Marshall, Jasmin Matz, and Carsten Peterson.