Non-escape of Mass for Arithmetic Quantum Limits on Hyperbolic 4-Manifolds

The Arithmetic Quantum Unique Ergodicity (AQUE) conjecture predicts that the L2 mass of Hecke-Maass cusp forms on an arithmetic hyperbolic manifold becomes equidistributed as the Laplace eigenvalue grows. If the underlying manifold is non-compact, mass could “escape to infinity”. This possibility was ruled out by Soundararajan for arithmetic surfaces, which when combined with celebrated work of Lindenstrauss completed the proof of AQUE for surfaces.

We establish non-escape of mass for Hecke-Maass cusp forms on a congruence quotient of hyperbolic 4-space. Unlike in the setting of hyperbolic 2- or 3-manifolds (for which AQUE has been proved), the number of terms in the Hecke relations is unbounded, which prevents us from naively applying Cauchy-Schwarz. We instead view the isometry group as a group of quaternionic matrices, and rely on non-commutative unique factorization along with certain structural features of the Hecke action. Joint work with Zvi Shem-Tov.