Affine Hecke algebras and the Schubert calculus

Using a combinatorial approach which avoids geometry, this paper studies the ring structure of KT (G/B), the T -equivariant K-theory of the (generalized) flag variety G/B. Here, the data G ⊇ B ⊇ T is a complex reductive algebraic group (or symmetrizable Kac-Moody group)G, a Borel subgroup B, and a maximal torus T , and KT (G/B) is the Grothendieck group of T -equivariant coherent sheaves on G/B. Because of the T -equivariance the ring KT (G/B) is an R-algebra, where R is the representation ring of T . As explained by Grothendieck [Gd] (in the non Kac-Moody case) and Kostant and Kumar [KK] (in the general Kac-Moody case), the ring KT (G/B) has a natural R-basis {[OXw ] | w ∈ W}, where W is the Weyl group and OXw is the structure sheaf of the Schubert variety Xw ⊆ G/B. One of the main problems in the field is to understand the structure constants of the ring KT (G/B) with this basis, that is, the coeffients c z wv in the equations