Arithmetic quantum unique ergodicity for products of hyperbolic $2$- and $3$-spaces

TL;DR

This paper proves the arithmetic quantum unique ergodicity conjecture for sequences of Hecke--Maass forms on products of hyperbolic 2- and 3-spaces, showing that eigenfunctions become uniformly distributed.

Contribution

It establishes AQUE for mixed hyperbolic spaces by an inductive argument, extending previous results to more complex product spaces.

Findings

AQUE holds for Hecke--Maass forms on these product spaces.

Limit measures do not concentrate on closed orbits of proper subgroups.

The proof uses induction on the dimension of the orbit.

Abstract

We prove the arithemtic quantum unique ergodicity (AQUE) conjecture for sequences of Hecke--Maass forms on quotients $\Gamma\backslash (\mathbb{H}^{(2)})^r \times (\mathbb{H}^{(3)})^s$. An argument by induction on dimension of the orbit allows us to rule out the limit measure concentrating on closed orbits of proper subgroups despite many returns of the Hecke correspondence to neighborhoods of the orbit.