Microlocal lifts and quantum unique ergodicity on $GL(2,\mathbb{Q}_p)$

Abstract

We prove that arithmetic quantum unique ergodicity holds on compact arithmetic quotients of $GL(2,\mathbb{Q}_p)$ for automorphic forms belonging to the principal series. We interpret this conclusion in terms of the equidistribution of eigenfunctions on covers of a fixed regular graph or along nested sequences of regular graphs. Our results are the first of their kind on any p-adic 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.