Topology of the Octonionic Flag Manifold Augustin-liviu Mare and Matthieu Willems

The octonionic flag manifold Fl(O) is the space of all pairs in OP 2 × OP 2 (where OP 2 denotes the octonionic projective plane) which satisfy a certain “incidence” relation. It comes equipped with the projections π1, π2 : Fl(O) → OP , which are OP 1 bundles, as well as with an action of the group Spin(8). The first two results of this paper give Borel type descriptions of the usual, respectively Spin(8)-equivariant cohomology of Fl(O) in terms of π1 and π2 (actually the Euler classes of the tangent spaces to the fibers of π1, respectively π2, which are rank 8 vector bundles on Fl(O)). Then we obtain a GoreskyKottwitz-MacPherson type description of the ring H∗ Spin(8)(Fl(O)). Finally, we consider the Spin(8)-equivariant K-theory ring of Fl(O) and obtain a Goresky-Kottwitz-MacPherson type description of this ring.