From motives of versal flag varieties to modular representations of Hecke-type algebras


Let G be a split semisimple linear algebraic group over a field k, let E be a G-torsor over k. Let h be an algebraic oriented cohomology theory in the sense of Levine-Morel (e.g.Chow ring or an algebraic cobordism). Consider a twisted form E/B of the variety of Borel subgroups G/B. Following the motivic Galois group approach and the Kostant-Kumar results on equivariant cohomology of flag varieties we establish an equivalence between the h-motivic subcategory generated by E/B and the category of projective modules of certain Hecke-type algebra H which depends on the root system of G, its isogeny class, on E, and on the formal group law of the theory h. In particular, taking h to be the Chow groups with finite coefficients F_p and E to be a generic torsor we obtain that all irreducible modules of the affine nil-Hecke algebra H of G with coefficients in F_p are isomorphic and correspond to the generalized Rost-Voevodsky motive for (G,p).