Microlocal lifts and quantum unique ergodicity on GL2(Qp)

Abstract

We prove that arithmetic quantum unique ergodicity holds on compact arithmetic quotients of GL2(p) for automorphic forms belonging to the principal series. We interpret this conclusion in terms of the equidistribution of eigen functions on covers of a fixed regular graph or along nested sequences of regular graphs. Our results are the first of their kind on any padic arithmetic quotient. They may be understood as analogues of Lindenstrauss’s theorem on the equidistribution of Maass forms on a compact arithmetic surface. The new ingredients here include the introduction of a representation-theoretic notion of “p-adic microlocal lifts” with favorable properties, such as diagonal invariance of limit measures; the proof of positive entropy of limit measures in a p-adic aspect, following the method of Bourgain–Lindenstrauss; and some analysis of local Rankin–Selberg integrals involving the microlocal lifts introduced here as well as classical newvectors. An important input is a measure-classification result of Einsiedler–Lindenstrauss.