Orthogonal multiple flag varieties of finite type I: Odd degree case

Let G be the split orthogonal group of degree 2n+1 over an arbitrary infinite field F of charF≠2. In this paper, we classify multiple flag varieties G/P1×⋯×G/Pk of finite type. Here a multiple flag variety is said to be of finite type if it has a finite number of G-orbits with respect to the diagonal action of G.