Motivic decompositions of twisted flag varieties and representations of Hecke-type algebras

Let G be a split semisimple linear algebraic group over a field k0. Let E be a G-torsor over a field extension k of k0. Let h be an algebraic oriented cohomology theory in the sense of Levine-Morel. Consider a twisted form E/B of the variety of Borel subgroups G/B over k. Following the Kostant-Kumar results on equivariant cohomology of flag varieties we establish an isomorphism between the Grothendieck groups of the h-motivic subcategory generated by E/B and the category of finitely generated projective modules of certain Hecke-type algebra H which depends on the root datum of G, on the torsor E and on the formal group law of the theory h. In particular, taking h to be the Chow groups with finite coefficients Fp and E to be a generic G-torsor we prove that all indecomposable submodules of an affine nil-Hecke algebra H of G with coefficients in Fp are isomorphic to each other and correspond to the (non-graded) generalized Rost-Voevodsky motive for (G,p).