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# Mathematics > Representation Theory

# Title: Positive entropy using Hecke operators at a single place

(Submitted on 19 Feb 2020 (v1), last revised 25 Sep 2020 (this version, v3))

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.

## Submission history

From: Zvi Shem-Tov [view email]**[v1]**Wed, 19 Feb 2020 08:49:11 GMT (22kb)

**[v2]**Wed, 18 Mar 2020 13:57:33 GMT (22kb)

**[v3]**Fri, 25 Sep 2020 20:55:56 GMT (25kb)

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