Quantum ergodicity in the Benjamini--Schramm limit for locally symmetric spaces

Abstract

We prove that for almost all symmetric spaces $X$ and for any sequence of compact locally symmetric spaces $Y_n$ which is uniformly discrete, has a uniform spectral gap, and converges in the sense of Benjamini--Schramm to $X$, the joint eigenfunctions of all invariant differential operators on $Y_n$ delocalize on average when their spectral parameters are taken to lie in a fixed spectral window.