| Résume | Let $G$ be a linear algebraic group over a field $F$ and $X$ be aprojective homogeneous $G$-variety such that $G$ splitsover the function field of $X$.We introduce an invariant of $G$ called $J$-invariantwhich characterizes the splitting properties of the Chow motive of $X$.As a main application we obtain a uniform proof ofall known motivic decompositions of generically splitprojective homogeneous varieties(Severi-Brauer varieties, Pfister quadrics, maximal orthogonalGrassmannians,$G_2$- and $F_4$-varieties)as well as provide new ones (exceptional varieties of types$E_6$, $E_7$ and $E_8$).We also discuss applications totorsion indices, canonical dimensionsand splitting properies of the group $G$. |
