Möbius disjointness for homogeneous dynamics

Abstract

We prove Sarnak’s Möbius disjointness conjecture for all unipotent translations on homogeneous spaces of real connected Lie groups. Namely, we show that if $G$ is any such group, $\Gamma \subset G$ a lattice, and $u\in G$ an Ad-unipotent element, then for every $x\in \Gamma \backslash G$ and every function $f$ continuous on the $1$-point compactification of $\Gamma \backslash G$, the sequence $f\left(x{u}^n\right)$ cannot correlate with the Möbius function on average.