We prove that arithmetic quantum unique ergodicity holds on compact arithmetic quotients
of
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
-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
“-adic
microlocal lifts” with favorable properties, such as diagonal invariance
of limit measures; the proof of positive entropy of limit measures in a
-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.
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