#### Vol. 12, No. 9, 2018

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print) Author Index To Appear Other MSP Journals
Microlocal lifts and quantum unique ergodicity on $GL_2(\mathbb{Q}_p)$

### Paul D. Nelson

Vol. 12 (2018), No. 9, 2033–2064
##### Abstract

We prove that arithmetic quantum unique ergodicity holds on compact arithmetic quotients of ${GL}_{2}\left({ℚ}_{p}\right)$ 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.

##### Keywords
arithmetic quantum unique ergodicity, microlocal lifts, representation theory
##### Mathematical Subject Classification 2010
Primary: 58J51
Secondary: 22E50, 37A45