Submission: 2015, May 26
Motivated by the motivic Galois group and the Kostant-Kumar results on equivariant cohomology of flag varieties, we provide a uniform description of motivic (direct sum) decompositions with integer coefficients of versal flag varieties in terms of integer representations of the associated affine nil-Hecke algebra $H$. More generally, we establish an equivalence between the $h$-motivic subcategory generated by the motive of $E/B$ and the category of projective modules of the associated rational algebra $D$ of push-pull operators, where $E$ is a torsor for a split semisimple linear algebraic group $G$ over a field $k$, $B$ is a Borel subgroup of $G$, $h$ is an algebraic oriented cohomology theory in the sense of Levine-Morel (e.g. Chow ring $CH$ or an algebraic cobordism $\Omega$). The algebra $D$ can be think of as an integer-analogue of the 'Hopf-algebra of the $h$-motivic Galois group of $E/B$. As an application, taking $h=CH$ and specializing the coefficients to the finite field $\F_p$ we obtain that $p$-modular projective representations of $D=H$ are generated by an irreducible $H$-module corresponding to the generalized Rost-Voevodsky motive for $(G,p)$.
2010 Mathematics Subject Classification: 14F43, 14M15, 20C08, 14C15
Keywords and Phrases: linear algebraic group, torsor, flag variety, equivariant oriented cohomology, motivic decomposition, Hecke algebra
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