Entropy bounds and quantum unique ergodicity for Hecke eigenfunctions on division algebras

Abstract

We prove the arithmetic quantum unique ergodicity (AQUE) conjecture for non-degenerate sequences of Hecke eigenfunctions on quotients $\Gamma \backslash G/K$, where $G\simeq {\mathrm{PGL}}_d(\mathbb{R})$, $K$ is a maximal compact subgroup of $G$ and $\Gamma <G$ is a lattice associated to a division algebra over $\mathbb{Q}$ of prime degree $d$. More generally, we introduce a new method of proving positive entropy of quantum limits, which applies to higher-rank groups. The result on AQUE is obtained by combining this with a measure-rigidity theorem due to Einsiedler-Katok, following a strategy first pioneered by Lindenstrauss

26 pages; This paper dates from 2006 but was not published. We make it available here because the results and techniques may still be of interest