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Let $G$ be a split semisimple algebraic group over a field and let $A^*$ be an oriented cohomology theory in the Levine–Morel sense. We provide a uniform approach to the $A^*$ -motives of geometrically cellular smooth projective $G$ -varieties based on the Hopf algebra structure of $A^*(G)$. Using this approach, we provide various applications to the structure of motives of twisted flag varieties. [ABSTRACT FROM AUTHOR]

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]

We compute the algebraic Morava K-theory ring of split special orthogonal and spin groups. In particular, we establish certain stabilization results for the Morava K-theory of special orthogonal and spin groups. Besides, we apply these results to study Morava motivic decompositions of orthogonal Grassmannians. For instance, we determine all indecomposable summands of the Morava motives of a generic quadric. [ABSTRACT FROM AUTHOR]