Poincar\'{e} series and rational cohomology rings of Kac-Moody groups and their flag manifolds

In this paper, we study the rational cohomology rings of indefinite Kac-Moody groups and their flag manifolds. By extracting the information of cohomology from the Poincar'{e} series, we are able to determine the rational cohomology rings of Kac-Moody groups and their flag manifolds. Since Kac-Moody groups and their flag manifolds are rational formal, we also determine their rational homotopy groups and rational homotopy types.