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

Lior Silberman, Akshay Venkatesh

Abstract:

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.
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