Pushforward of Siegel flag varieties in the Chow ring

Given a reductive group G over an algebraically closed field and subsets I , J ⊂ Δ of the simple roots Δ determined by a choice of maximal torus and Borel subgroup, there is a closed embedding of flag varieties L J / L J ∩ P I ↪ G / P I . In this paper we compute the class of the sub flag variety [ L J / L J ∩ P I ] ∈ A • ( G / P I ) in the Chow ring for the ‘Siegel’ case where G is a general symplectic group of semisimple rank g and P I is the parabolic stabilising a maximal isotropic subspace. This corresponds, under the isomorphism with the tautological ring of the moduli space of principally polarised abelian varieties T • ( A g tor ) ≅ A • ( G / P I ) , to the generator of the classes in the tautological ring which are supported on the toroidal boundary. This provides basic evidence for a conjecture describing the tautological ring of a Hodge-type Shimura variety.