Motives and oriented cohomology of a linear algebraic group

Abstract

For a cellular variety $X$ over a field $k$ of characteristic 0 and an algebraic oriented cohomology theory $\hh$ of Levine-Morel we construct a filtration on the cohomology ring $\hh(X)$ such that the associated graded ring is isomorphic to the Chow ring of $X$. Taking $X$ to be the variety of Borel subgroups of a split semisimple linear algebraic group $G$ over $k$ we apply this filtration to relate the oriented cohomology of $G$ to its Chow ring. As an immediate application we compute the algebraic cobordism ring of a group of type $G_2$, of groups $SO_n$ and $Spin_m$ for $n=3,4$ and $m=3,4,5,6$ and $PGL_k$ for $k\geqslant 2$. Using this filtration we also establish the following comparison result between Chow motives and $\hh$-motives of generically cellular varieties: any irreducible Chow-motivic decomposition of a generically split variety $Y$ gives rise to a $\hh$-motivic decomposition of $Y$ with the same generating function. Moreover, under some conditions on the coefficient ring of $\hh$ the obtained $\hh$-motivic decomposition will be irreducible. We also prove that if Chow motives of two twisted forms of $Y$ coincide, then their $\hh$-motives coincide as well.