  Let G be a semisimple affine algebraic group over a field F. Assuming that G
becomes of inner type over some finite field extension of F of degree a power
of a prime p, we investigate the structure of the Chow motives with
coefficients in a finite field of characteristic p of the projective
G-homogeneous varieties. The complete motivic decomposition of any such variety
contains one specific summand, which is the most understandable among the
others and which we call the upper indecomposable summand of the variety. We
show that every indecomposable motivic summand of any projective G-homogeneous
variety is isomorphic to a shift of the upper summand of some (other)
projective G-homogeneous variety. This result is already known (and has
applications) in the case of G of inner type and is new for G of outer type
(over F).

Algebraic Geometry