Let G be a semisimple linear algebraic group of inner type over a field F, p be a prime and X be a projective homogeneous G-variety such that G splits over the function field of X. In the present paper we introduce a new discrete invariant of G called J-invariant which characterizes motivic decomposition of X modulo p, allows computing the p-relative canonical dimension of X and reflects the splitting behavior of G. In the case of orthogonal groups this invariant was constructed by A. Vishik.

When G is isotropic, the problem of computing the Chow motive of X was solved by B. Köck (in the split case), V. Chernousov, S. Gille and A. Merkurjev (in the case of an isotropic X) and P. Brosnan (in the general case). In all these proofs one constructs a relative cellular filtration on X which allows expressing the motive of the total space X in terms of motives of the base.

In the case of an anisotropic group the situation becomes much more complicated. When G is an orthogonal group the motive of X can be computed following the works of M. Rost (the case of Pfister quadrics), N. Karpenko, A. Merkurjev and A. Vishik (general case). The motive of a Severi-Brauer variety was computed by N. Karpenko. For some exceptional varieties motivic decompositions into indecomposables were found by J.-P. Bonnet (G_2-case) and by S. Nikolenko, N. Semenov and K. Zainoulline (F_4-case). To obtain all these results one essentially uses Rost Nilpotence Theorem which says that it suffices to provide a motivic decomposition of bar X=Xtimes_(Spec F)bar F with the property that all the respective idempotents are defined over the base field.

We uniformize all these proofs. Using a result of Edidin and Graham on the Chow groups of cellular fibrations we reduce to the case when X is the variety of Borel subgroups in G. Then we consider the pull-back map from the Chow ring of bar X to the Chow ring of the split group bar G modulo p. The latter has a nice description in terms of p-exceptional degrees introduced by V. Kac. An attempt to describe rational cycles on bar X naturally leads to the notion of J-invariant.

We show that the action of Steenrod operations on the Chow ring of bar G force some restrictions on possible values of J-invariant. Generalizing a result of K. Zainoulline we obtain the formula expressing the p-relative canonical dimension of X. We provide an upper bound on the smallest p-component of degrees of splitting fields of G. Finally, we give examples suggesting a strong connection between J-invariant and certain cohomological invariants of G.