An example of orthogonal triple flag variety of finite type

Let G be the split special orthogonal group of degree 2n+1 over a field F of . Then we describe G-orbits on the triple flag varieties G/P×G/P×G/P and G/P×G/P×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 GLn-orbits on G/B, Q2n-orbits on the full flag variety of GL2n where Q2n is the fixed-point subgroup in Sp2n of a nonzero vector in F2n and 1×Sp2n-orbits on the full flag variety of GL2n+1. In the same way, we can also solve the same problem for SO2n where the maximal parabolic subgroup P is of the shape (n,n).