The Schubert calculus, braid relations, and generalized cohomology

Abstract

Let X X be the flag variety of a compact Lie group and let h ∗ {h^{\ast }} be a complex-oriented generalized cohomology theory. We introduce operators on h ∗ ( X ) {h^{\ast }}(X) which generalize operators introduced by Bernstein, Gel’fand, and Gel’fand for rational cohomology and by Demazure for K K -theory. Using the Becker-Gottlieb transfer, we give a formula for these operators, which enables us to prove that they satisfy braid relations only for the two classical cases, thereby giving a topological interpretation of a theorem proved by the authors and extended by Gutkin.