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

Source abstract

We prove the arithmetic quantum unique ergodicity (AQUE) conjecture for non-degenerate sequences of Hecke eigenfunctions on quotients Γ ∖ G / K \Gamma \backslash G/K , where G ≃ PGL d ⁡ ( R ) G\simeq \operatorname {PGL}_{d}(\mathbb {R}) , K K is a maximal compact subgroup of G G and Γ > G \Gamma >G is a lattice associated to a division algebra over Q \mathbb {Q} of prime degree d 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.