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

Abstract Let G be a split semisimple linear algebraic group over a field k 0. Let E be a G -torsor over a field extension k of k 0. 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 F p and E to be a generic G -torsor we prove that all finitely generated projective indecomposable submodules of an affine nil-Hecke algebra H of G with coefficients in F p are isomorphic to each other and correspond to the (non-graded) generalized Rost–Voevodsky motive for (G , p). [ABSTRACT FROM AUTHOR]