Schubert polynomials, theta and eta polynomials, and Weyl group invariants

We examine the relationship between the (double) Schubert polynomials of Billey–Haiman and Ikeda–Mihalcea–Naruse and the (double) theta and eta polynomials of Buch–Kresch–Tamvakis and Wilson from the perspective of Weyl group invariants. We obtain generators for the kernel of the natural map from the corresponding ring of Schubert polynomials to the (equivariant) cohomology ring of symplectic and orthogonal flag manifolds.