Positive entropy using Hecke operators at a single place

Abstract

We prove the following statement: Let $X=\text{SL}_n(\mathbb{Z})\backslash \text{SL}_n(\mathbb{R})$, and consider the standard action of the diagonal group $A<\text{SL}_n(\mathbb{R})$ on it. Let $\mu$ be an $A$-invariant probability measure on $X$, which is a limit $$ \mu=\lambda\lim_i|\phi_i|^2dx, $$ where $\phi_i$ are normalized eigenfunctions of the Hecke algebra at some fixed place $p$, and $\lambda>0$ is some positive constant. Then any regular element $a\in A$ acts on $\mu$ with positive entropy on almost every ergodic component. We also prove a similar result for lattices coming from division algebras over $\mathbb{Q}$, and derive a quantum unique ergodicity result for the associated locally symmetric spaces. This generalizes a result of Brooks and Lindenstrauss.