A generalization of Kostant theorem to integral cohomology

In this paper, we find weight decomposition and rank of a weight in the integral (co)homology of the positive system of a semi-simple Lie algebra over $\mathbb{C}$ and prove that the (co)homology of the weight subcomplex over a field of characteristic p is 0 if the rank of the weight is not divisible by p. This generalizes Kostant theorem to the integral cohomology of the positive system.