A Pieri-Chevalley formula in the K-theory of a G/B-bundle

Let G be a semisimple complex Lie group, B a Borel subgroup, and T subset of or equal to B a maximal torus of G. The projective variety G/B is a generalization of the classical ag variety. The structure sheaves of the Schubert subvarieties form a basis of the K-theory K(G/B) and every character of T gives rise to a line bundle on G/B. This note gives a formula for the product of a dominant line bundle and a Schubert class in K(G/B). This result generalizes a formula of Chevalley which computes an analogous product in cohomology. The new formula applies to the relative case, the K-theory of a G/B-bundle over a smooth base X, and is presented in this generality. In this setting the new formula is a generalization of recent G = GL(n)(C) results of Fulton and Lascoux.