Upper motives of outer algebraic groups

Abstract

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).