Upper Motives of Algebraic Groups and Incompressibility of Severi-brauer Varieties

Let G be a semisimple affine algebraic group of inner type over a field F . We write XG 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 XG. More precisely, it is known that the motive of any variety in XG 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 PGLA of automorphisms of a given central simple F algebra A, for any variety in the class XG (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 XG are p-incompressible. If A is a division algebra of degree p for some n ≥ 0, then the list of p-incompressible varieties includes the generalized Severi-Brauer variety X(p;A) of ideals of reduced dimension p for m = 0, 1, . . . , n.