Abstract: Given a connected simply-connected s-step real nilpotent Lie group G with a left invariant integrable almost complex structure J and a lattice T of maximal rank of the Lie group G, the existence of a nilpotent complex structure on the compact homogeneous spaceG/T is considered. It is shown in this paper, through the use of inductive methods, that the complex structure J can induce an integrable complex structure $\widetilde{J}$ on the compact homogeneous spaceG/T, and that $\widetilde{J}$ is nilpotent.

Key words: connected simply-connected nilpotent Lie group, left invariant integrable almost complex stru-cture, nilpotent complex structure, compact homogenous space