Summary: We prove that arithmetic quantum unique ergodicity holds on compact arithmetic quotients of \(\mathrm{GL}_2(\mathbb Q_p)\) 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.