Affine Hecke algebras and the Schubert calculus

Abstract

Using a combinatorial approach that avoids geometry, this paper studies the structure of KT(G/B), the T-equivariant K-theory of the generalized flag variety G/B. This ring has a natural basis {[OXw] wεW} (the double Grothendieck polynomials), where OXw is the structure sheaf of the Schubert variety Xw. For rank two cases we compute the corresponding structure constants of the ring KT(G/B) and, based on this data, make a positivity conjecture for general G which generalizes the theorems of M. Brion (for K(G/B)) and W. Graham (for HT*(G/B)). Let [Xλ]εKT(G/B) be the class of the homogeneous line bundle on G/B corresponding to the character of T indexed by λ. For general G we prove "Pieri-Cheval..