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Abstract

In the paper "The second cohomology of nilpotent orbits in classical Lie algebras, Kyoto J. Math. 60 (2020), no. 2, 717-799" by I. Biswas, P. Chatterjee, and C. Maity, explicit descriptions of the second and first real de Rham cohomology groups of a general homogeneous space of a Lie group are given, extending an earlier result in "On the exactness of Kostant-Kirillov form and the second cohomology of nilpotent orbits, Internat. J. Math. 23 (2012), no. 8, 1250086" by I. Biswas and P. Chatterjee. From the computational viewpoint, they turned out to be new and very useful, and in fact played a crucial role in determining the second cohomology of nilpotent orbits as done in the above two papers. In this paper, we give computable and explicit descriptions of the third and fourth real de Rham cohomologies of a general homogeneous space, in terms of the associated Lie-theoretic data, along the lines mentioned above. We also draw numerous corollaries of our main results in important special settings. Moreover, as a consequence, we obtain a new and interesting invariant by showing that for a large class of homogeneous spaces, the difference between the third and fourth Betti numbers coincides with the difference between the numbers of simple factors of the ambient group and the associated closed subgroup.