Relative equivariant motives and modules

Canadian Journal of Mathematics, vol. 73, no. 1, 2021, p. 131-159
(1^st online appearance of the preprint on the 22-9-2016)

This paper is based on the techniques developped in Invariants, torsion indices and oriented cohomology of flag varieties, A coproduct structure on the formal affine Demazure algebra and Formal Hecke algebras and algebraic oriented cohomology theories. We establish a link between the category of modules over the Demazure algebra and some categories of correspondences for oriented cohomology theories. It allows us to describe decompositions of motives of projective homogeneous varieties under a semi-simple group using algebraic and combinatorial methods, since the Demazure algebra is precisely defined using such combinatorial data as root data, Weyl groups, etc. Until now, this type of approach only produced results in the split case, but a link between motives that are equivariant under a split group and motives of generic varieties is used here to control in a systematic and combinatorial way the direct factors of motives of quotients of generic torsors (thus non split).