M\"obius disjointness for homogeneous dynamics

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