Positive Entropy Using Hecke Operators at a Single Place

We prove the following statement: let X= SL n(Z)backslash SL n(R) and consider the standard action of the diagonal group A< SL n(R) on it. Let μ be an A-invariant probability measure on X, which is a limit ∗ μ=λ lim i|φ i|2dx, ∗where φ i are normalized eigenfunctions of the Hecke algebra at some fixed place p and λ0 is some positive constant. Then any regular element a\in A acts on μ with positive entropy on almost every ergodic component. We also prove a similar result for lattices coming from division algebras over Q and derive a quantum unique ergodicity result for the associated locally symmetric spaces. This generalizes a result of Brooks and Lindenstrauss [2].