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

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