The nil Hecke ring and cohomology of G/P for a Kac-Moody group G

Let G be the group with Borel subgroup B, associated to a Kac-Moody Lie algebra [unk] (with Weyl group W and Cartan subalgebra [unk]). Then H(*)(G/B) has, among others, four distinguished structures (i) an algebra structure, (ii) a distinguished basis, given by the Schubert cells, (iii) a module for W, and (iv) a module for Hecke-type operators A(w), for w [unk] W. We construct a ring R, which we refer to as the nil Hecke ring, which is very simply and explicitly defined as a functor of W together with the W-module [unk] alone and such that all these four structures on H(*)(G/B) arise naturally from the ring R.