Affine Hecke algebras and the Schubert calculus

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]∣w∈W} (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]∈KT(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–Chevalley formulas” for the products [Xλ][OXw], [X−λ][OXw], [Xw0λ][OXw], and [OXw0si][OXw], where λ is dominant. By using the Chern character and comparing lowest degree terms the products which are computed in this paper also give results for the Grothendieck polynomials, double Schubert polynomials, and ordinary Schubert polynomials in, respectively K(G/B), HT∗(G/B) and H∗(G/B).