Pushforward of Siegel flag varieties in the Chow ring

Given a reductive group, choice of maximal torus and Borel subgroup, and two subsets of the simple roots, one obtains a closed embedding of sub flag varieties. In this paper we compute the class of the sub flag variety in the Chow ring for the Siegel case where the group is the general symplectic group and the parabolic stabilises a maximal isotropic subspace. This corresponds, under the isomorphism with the tautological ring of the compactified moduli space of abelian varieties, to the generator of the classes in the tautological ring which are supported on the toroidal boundary. We conjecture that this is not a coincidence.