On some invariants of orbits in the flag variety under a symmetric subgroup

TL;DR

This paper investigates invariants of orbits in the flag variety under a symmetric subgroup, linking algebraic and geometric properties to Weyl group subsets and admissible paths, with applications to real form orbit intersections.

Contribution

It introduces new invariants of K-orbits in the flag variety using Weyl group subsets and admissible paths, enhancing understanding of orbit intersections for real forms of G.

Findings

Invariants are characterized by subsets of the Weyl group.

Criteria for non-empty intersections of real form and Borel subgroup orbits.

Invariants are described via admissible paths in K-orbit sets.

Abstract

Let $G$ be a connected reductive algebraic group over an algebraically closed field ${\bf k}$ of characteristic not equal to 2, let $\B$ be the variety of all Borel subgroups of $G$, and let $K$ be a symmetric subgroup of $G$. Fixing a closed $K$-orbit in $\B$, we associate to every $K$-orbit on $\B$ some subsets of the Weyl group of $G$, and we study them as invariants of the $K$-orbits. When ${\bf k} = {\mathbb C}$, these invariants are used to determine when an orbit of a real form of $G$ and an orbit of a Borel subgroup of $G$ have non-empty intersection in $\B$. We also characterize the invariants in terms of admissible paths in the set of $K$-orbits in $\B$.