J-invariant of linear algebraic groups
  Let G be a linear algebraic group over a field F and X be a projective homogeneous G-variety such that G splits over the function field of X. In the present paper we introduce an invariant of G called J-invariant which characterizes the motivic behaviour of X. This generalizes the respective notion invented by A. Vishik in the context of quadratic forms. As a main application we obtain a uniform proof of all known motivic decompositions of generically split projective homogeneous varieties (Severi-Brauer varieties, Pfister quadrics, maximal orthogonal Grassmannians, G2- and F4-varieties) as well as provide new examples (exceptional varieties of types E6, E7 and E8). We also discuss relations with torsion indices, canonical dimensions and cohomological invariants of the group G. 
Victor Petrov, Nikita Semenov, Kirill Zainoulline