Schubert polynomials and Arakelov theory of orthogonal flag varieties

Abstract. Let X = Sp 2n/B the flag variety of the symplectic group. We propose a theory of combinatorially explicit Schubert polynomials which rep-resent the Schubert classes in the Borel presentation of the cohomology ring of X. We use these polynomials to describe the arithmetic Schubert calculus on X. Moreover, we give a method to compute the natural arithmetic Chern numbers on X, and show that they are all rational numbers. Let X = Sp2n/B be the flag variety for the symplectic group Sp2n of rank n. The cohomology (or Chow) ring of X has a standard presentation, due to Borel [Bo], as a quotient ring Z[x1,..., xn]/In, where the variables xi come from the characters of B and In is the ideal generated by the invariant polynomials under the actio