An example of orthogonal triple flag variety of finite type

TL;DR

This paper classifies the orbits of certain algebraic groups on triple flag varieties over fields with characteristic not equal to 2, extending understanding of group actions on geometric structures in algebraic geometry.

Contribution

It provides a detailed description of G-orbits on specific triple flag varieties for split special orthogonal groups, including new orbit classifications for related subgroup actions.

Findings

Classification of G-orbits on G/P×G/P×G/P and G/P×G/P×G/B

Descriptions of GL_n-orbits on G/B and Q_{2n}-orbits on the full flag variety

Orbit classifications for SO_{2n} with different parabolic subgroups

Abstract

Let G be the split special orthogonal group of degree 2n+1 over a field F of char F \ne 2. Then we describe G-orbits on the triple flag varieties G/P\times G/P\times G/P and G/P\times G/P\times G/B with respect to the diagonal action of G where P is a maximal parabolic subgroup of G of the shape (n,1,n) and B is a Borel subgroup. As by-products, we also describe GL_n-orbits on G/B, Q_{2n}-orbits on the full flag variety of GL_{2n} where Q_{2n} is the fixed-point subgroup in Sp_{2n} of a nonzero vector in F^{2n} and 1\times Sp_{2n}-orbits on the full flag variety of GL_{2n+1}. In the same way, we can also solve the same problem for SO_{2n} where the maximal parabolic subgroup P is of the shape (n,n).