Neshitov, Alexander; Petrov, Victor; Semenov, Nikita; Zainoulline, Kirill
(2018):
Motivic decompositions of twisted flag varieties and representations of Hecketype algebras.
In: Advances in Mathematics, Vol. 340: pp. 791818

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Abstract
Let G be a split semisimple linear algebraic group over a field Ice. Let E be a Gtorsor over a field extension k of k(0). Let h be an algebraic oriented cohomology theory in the sense of LevineMorel. Consider a twisted form E/B of the variety of Borel subgroups G/B over k. Following the KostantKumar results on equivariant cohomology of flag varieties we establish an isomorphism between the Grothendieck groups of the hmotivic subcategory generated by E/B and the category of finitely generated projective modules of certain Hecketype 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 Gtorsor we prove that all finitely generated projective indecomposable submodules of an affine nilHecke algebra H of G with coefficients in Fp, are isomorphic to each other and correspond to the (nongraded) generalized RostVoevodsky motive for (G, p).