Möbius disjointness for homogeneous dynamics

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, Γ⊂G a lattice, and u∈G an Ad-unipotent element, then for every x∈Γ and every function f continuous on the 1-point compactification of Γ, the sequence f(xun) cannot correlate with the Möbius function on average.