Cohomological splitting of coadjoint orbits (Q1880182)

The author shows how to express the rational cohomology of a coadjoint orbit \({\mathcal O}\) of a compact Lie group \(G\) as the tensor product of the cohomology of other coadjoint orbits \({\mathcal O}_ k\), with \(\dim\,{\mathcal O}_ k<\dim\,{\mathcal O}\). Using the coadjoint orbit hierarchy and cohomological splittings of Hamiltonian bundles it is shown that if \(G_ 1\subset G_ 2\) are stabilizers of the coadjoint action of the compact connected Lie group \(G\) and \(G_ 2\) is connected, then there is an additive isomorphism \(H^ *(G/G_ 1,\mathbb Q) \cong H^ *(G/G_ 2,\mathbb Q)\otimes H^ *(G_ 2/G_ 1,\mathbb Q)\). As an illustration of this theorem, the author presents a cohomological splitting of flag manifolds.