Combinatorial geometry of flag domains in G/B

Abstract

A real form $G_0$ of a complex semisimple Lie group $G$ has only finitely many orbits in any given compact $G$-homogeneous projective algebraic manifold $Z=G/Q$. A maximal compact subgroup $K_0$ of $G_0$ has special orbits $C$ which are complex sub-manifolds in the open orbits of $G_0$. These are referred to as cycles. The cycles intersect Shubert varieties $S$ transversely in finitely many points. In particular, determining these points of intersection yields a description of the topological class of the given cycle. This was carried out for all real forms of $SL(n,\mathbb{C})$ in the work of A. Brecan. Our work here is devoted to the real forms of the other classical groups, $Sp(2n,\mathbb{C})$ and $SO(n,\mathbb{C})$. For the manifold $Z=G/B$ of complete flags the points of intersection in $S\cap C$ are described, in particular the number of such is computed. For certain real forms, e.g., $Sp(2n,\mathbb{R})$ and $SO^*(2n)$, remarkably simple formulas are proved. In other cases the results are algorithmic in nature.