Killing Forms on 2-Step Nilmanifolds

We study left-invariant Killing k -forms on simply connected 2-step nilpotent Lie groups endowed with a left-invariant Riemannian metric. For k = 2 , 3 , we show that every left-invariant Killing k -form is a sum of Killing forms on the factors of the de Rham decomposition. Moreover, on each irreducible factor, non-zero Killing 2-forms define (after some modification) a bi-invariant orthogonal complex structure and non-zero Killing 3-forms arise only if the Riemannian Lie group is naturally reductive when viewed as a homogeneous space under the action of its isometry group. In both cases, k = 2 or k = 3 , we show that the space of left-invariant Killing k -forms of an irreducible Riemannian 2-step nilpotent Lie group is at most one-dimensional.