Chow motives of twisted flag varieties

Abstract

Let $G$ be an adjoint simple algebraic group of inner type. We express the Chow motive (with integral coefficients) of an anisotropic projective $G$-homogeneous variety in terms of motives of simpler $G$-homogeneous varieties, namely, those that correspond to maximal parabolic subgroups of $G$. We decompose the motive of a generalized Severi–Brauer variety $\mathrm{SB}_2(A)$ of a division algebra $A$ of degree 5 into a direct sum of twisted motives of the Severi–Brauer variety $\mathrm{SB}(B)$ of a division algebra $B$ Brauer-equivalent to the tensor square $A^{\otimes 2}$. As an application we provide another counter-example to the uniqueness of a direct sum decomposition in the category of motives with integral coefficients.