An automorphism $ T $ of a probability space hasasymptotic orthogonal powers if all ergodic joinings of the the product $ T ^ r \otimes T ^ s $ converge to the product joining, when the relatively prime integers $ r $ and $ s $ go to infinity. We show that the affine unipotent and ergodic diffeomorphisms of nilmanifolds have asymptotic orthogonal powers. Two consequences follow:

1. The conjecture of Sarnak on the orthogonality of the Möbius function to deterministic systems is true for any automorphism measurably isomorphic to an affine unipotent and ergodic diffeomorphism of a nilmanifold.
2. In addition, for these automorphims, the above mentioned Sarnak conjecture "holds on small intervals".