Invariants, exponents and formal group laws

Let W be the Weyl group of a crystallographic root system acting on the associated weight lattice by means of reflections. In the present note we extend the notion of exponent of the W-action to the context of an arbitrary algebraic oriented cohomology theory of Levine-Morel and Panin-Smirnov and the associated formal group law. From this point of view the classical Dynkin index of the associated Lie algebra will be the second exponent of the deformation map from the multiplicative to the additive formal group law. We apply this generalized exponent to study the torsion part of an arbitrary oriented cohomology theory of a twisted flag variety.