Mass equidistribution for lifts on hyperbolic $4$-manifolds

TL;DR

This paper proves the quantum unique ergodicity conjecture for a specific sequence of lifts on hyperbolic 4-manifolds, using a novel amplification method to handle non-tempered representations.

Contribution

It introduces a new amplification technique tailored for non-tempered lifts, achieving unconditional mass equidistribution results on hyperbolic 4-manifolds.

Findings

Proves QUE for Pitale lifts on hyperbolic 4-manifolds

Develops a geometric amplifier construction for non-tempered forms

First application of amplification to escape non-tempered subgroups

Abstract

We prove the quantum unique ergodicity (QUE) conjecture of Rudnick and Sarnak for the sequence of Pitale lifts, which are Hecke-Maass forms on a congruence quotient of $\mathbb{H}^4$ constructed as lifts from half-integral weight forms (i.e. non-holomorphic analogues of the Saito-Kurokawa lifts). The result is unconditional, unlike other mass equidistribution results for similar lifts. Our main innovation is the delicate construction of an amplifier with favorable geometric properties (while we do use the non-temperedness of the lifts, it alone is not enough). To the best of our knowledge, this is the first successful use of the amplification method for escaping a non-tempered subgroup.