Homogeneity of arithmetic quantum limits for hyperbolic $4$-manifolds

Abstract

We work toward the arithmetic quantum unique ergodicity (AQUE) conjecture for sequences of Hecke--Maass forms on hyperbolic $4$-manifolds. We show that limits of such forms can only scar on totally geodesic $3$-submanifolds, and in fact that all ergodic components of the microlocal lift other than the uniform measure arise from the uniform measures on these submanifolds.