0 Orthogonal Divided Differences and Schubert Polynomials , P̃ - Functions , and Vertex Operators

Using divided differences associated with the orthogonal groups, we investigate the structure of the polynomial rings over the rings of invariants of the corresponding Weyl groups. We study in more detail the action of orthogonal divided differences on some distinguished symmetric polynomials (P̃ polynomials) and relate it to vertex operators. Relevant families of orthogonal Schubert polynomials, generalizing P̃ -polynomials, and well-suited to intersection theory computations, are also studied.