Combinatorial geometry of flag domains in G/B

TL;DR

This paper explores the combinatorial geometry of flag domains in complex Lie groups, focusing on intersection points of cycles with Schubert varieties for classical groups, providing explicit formulas and algorithms.

Contribution

It extends the understanding of cycle intersections in flag domains from $SL(n,\,\mathbb{C})$ to other classical groups like $Sp(2n,\,\mathbb{C})$ and $SO(n,\,\mathbb{C})$, with explicit formulas and algorithms.

Findings

Explicit intersection point counts for certain real forms.

Simple formulas for $Sp(2n,\mathbb{R})$ and $SO^*(2n)$.

Algorithmic methods for other cases.

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.