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 o f or equal to B a maximal torus of G. The projective variety G/B is a gener alization of the classical ag variety. The structure sheaves of the Schuber t 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 prod uct of a dominant line bundle and a Schubert class in K(G/B). This result g eneralizes a formula of Chevalley which computes an analogous product in co homology. 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 thi s setting the new formula is a generalization of recent G = GL(n)(C) result s of Fulton and Lascoux.