Mass equidistribution for lifts on hyperbolic $4$-manifolds

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.