Chow motives of twisted flag varieties

Compositio Mathematica, 142, issue 4, 2006, p. 1063-1080
(1^st online appearance of the preprint on the 12-5-2005)

Chow motives of quadrics and Severi-Brauer varieties are quite well understood, after the works of M. Rost, N. Karpenko, O. Izhboldin, A. Vishik and others, but the motives of other projective homogeneous varieties (quotients of a semi-simple linear algebraic group by a parabolic subgroup) are not that well-known.

In this article, under mild assumptions, we decompose the motives of flag varieties G/P corresponding to non maximal parabolic subgroups P as direct sums of motives of flag varieties corresponding to maximal parabolics, even for anisotropic varieties. As a consequence, we obtain a new counter example to the unicity of the decomposition of an integral Chow motive as a direct sum of indecomposable ones.