  Let G be a semisimple affine algebraic group of inner type over a field F. We
write C for the class of all finite direct products of projective G-homogeneous
F-varieties. We determine the structure of the Chow motives with coefficients
in a finite field of the varieties in C. More precisely, it is known that the
motive of any variety in C decomposes (in a unique way) into a sum of
indecomposable motives, and we describe the indecomposable summands which
appear in the decompositions.
  In the case where G is the group of automorphisms of a given central simple
F-algebra A, for any variety in the class C (which includes the generalized
Severi-Brauer varieties of the algebra A) we determine its canonical dimension
at any prime p. In particular, we find out which varieties in C are
p-incompressible. If A is a division algebra of degree p^n for some n, then the
list of p-incompressible varieties includes the generalized Severi-Brauer
variety X(p^m; A) of ideals of reduced dimension p^m for m=0,1,...,n.
