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

TL;DR

This paper proves the arithmetic quantum unique ergodicity conjecture for Hecke eigenfunctions on certain quotients related to division algebras, introducing a new method for establishing positive entropy of quantum limits in higher-rank groups.

Contribution

It establishes AQUE for non-degenerate Hecke eigenfunctions on division algebra quotients and introduces a novel approach to proving positive entropy in higher-rank groups.

Findings

Proves AQUE for Hecke eigenfunctions on division algebra quotients.

Develops a new method for positive entropy of quantum limits.

Combines measure rigidity with entropy methods for higher-rank groups.

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