Quantum Unique Ergodicity for Locally Symmetric Spaces Ii

We prove the arithmetic quantum unique ergodicity (AQUE) conjecture for non-degenerate sequences of Hecke eigenfunctions on quotients Γ\G/K, where G ' PGLd(R), K is a maximal compact subgroup of G and Γ < G is a lattice associated to a division algebra over Q of prime degree d. The primary novelty of the present paper is a new method of proving positive entropy of quantum limits, which avoids sieves and yields better bounds than previous techniques. 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.