Vol. 12, No. 9, 2018

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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 GL2(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.

Keywords
arithmetic quantum unique ergodicity, microlocal lifts, representation theory
Mathematical Subject Classification 2010
Primary: 58J51
Secondary: 22E50, 37A45
Milestones
Received: 26 January 2017
Revised: 9 April 2018
Accepted: 15 July 2018
Published: 21 December 2018
Authors
Paul D. Nelson
Departement Mathematik
ETH
Zurich
Switzerland