Cycle Intersection for - Flag Domains

A real form of a complex semisimple Lie group G has only finitely many orbits in any given compact G -homogeneous projective algebraic manifold . A maximal compact subgroup of has special orbits C which are complex submanifolds in the open orbits of . These special orbits C are characterized as the closed orbits in Z of the complexification K of . These are referred to as cycles . The cycles intersect Schubert varieties S transversely at finitely many points. Describing these points and their multiplicities was carried out for all real forms of by Brecan (Brecan, 2014) and (Brecan, 2017) and for the other real forms by Abu-Shoga (Abu-Shoga, 2017) and Huckleberry (Abu-Shoga and Huckleberry). In the present paper, we deal with the real form acting on the SO (2 n , C)-manifold of maximal isotropic full flags. We give a precise description of the relevant Schubert varieties in terms of certain subsets of the Weyl group and compute their total number. Furthermore, we give an explicit description of the points of intersection in terms of flags and their number. The results in the case of for all real forms will be given by Abu-Shoga and Huckleberry.