INVARIANTS, TORSION INDICES AND ORIENTED COHOMOLOGY OF COMPLETE FLAGS

Let G be a split semisimple linear algebraic group over a field and let T be a split maximal torus of G. Let h be an oriented cohomology (algebraic cobordism, connective K-theory, Chow groups, Grothendieck's K-0, etc.) with formal group law F. We construct a ring from F and the characters of T, that we call a formal group ring, and we define a characteristic ring morphism c from this formal group ring to h (G/B) where G/B is the variety of Borel subgroups of G. Our main result says that when the torsion index of G is inverted, c is surjective and its kernel is generated by elements invariant under the Weyl group of G. As an application, we provide an algorithm to compute the ring structure of h (G/B) and to describe the classes of desingularized Schubert varieties and their products.