Positive Entropy Using Hecke Operators at a Single Place

Source abstract

Abstract We prove the following statement: let $X=\textrm{SL}_n({{\mathbb{Z}}})\backslash \textrm{SL}_n({{\mathbb{R}}})$ and consider the standard action of the diagonal group $A<\textrm{SL}_n({{\mathbb{R}}})$ on it. Let $\mu$ be an $A$-invariant probability measure on $X$, which is a limit $$\begin{equation*} \mu=\lambda\lim_i|\phi_i|^2dx, \end{equation*}$$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 [2].