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 $$\mu =\lambda \underset{i}{\lim }|{\varphi }_i{|}^{2}dx,$$ where ${\varphi }_i$ 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.