If the reflection group W corresponds to the Weyl group of a compact semisimple group K with maximal torus T, then the Kostant polynomials describe the structure of the de Rham cohomology of the generalized flag manifold K/T, also isomorphic to G/B where G is the complexification of K and B is the corresponding Borel subgroup.
Armand Borel showed that its cohomology ring is isomorphic to the quotient of the ring of polynomials by the ideal generated by the invariant homogeneous polynomials of positive degree.
This ring had already been considered by Claude Chevalley in establishing the foundations of the cohomology of compact Lie groups and their homogeneous spaces with André Weil, Jean-Louis Koszul and Henri Cartan; the existence of such a basis was used by Chevalley to prove that the ring of invariants was itself a polynomial ring.
A detailed account of Kostant polynomials was given by Bernstein, Gelfand & Gelfand (1973) and independently Demazure (1973) as a tool to understand the Schubert calculus of the flag manifold.
Their structure is governed by difference operators associated to the corresponding root system.
If α is a root, then sα denotes the corresponding reflection operator.
The choice of Δ gives rise to a Bruhat order on the Weyl group determined by the ways of writing elements minimally as products of simple root reflection.
It can be checked directly that as is invariant under W. In fact δi satisfies the derivation property Hence Since or 0, it follows that so that by the invertibility of N for all i, i.e. at is invariant under W. As above let Φ be a root system in a real inner product space V, and Φ+ a subset of positive roots.
For each element s in W, let Δs be the subset of Δ consisting of the simple roots satisfying s−1α < 0, and put where the sum is calculated in the weight lattice P. The set of linear combinations of the exponentials eμ with integer coefficients for μ in P becomes a ring over Z isomorphic to the group algebra of P, or equivalently to the representation ring R(T) of T, where T is a maximal torus in K, the simply connected, connected compact semisimple Lie group with root system Φ.
If W is the Weyl group of Φ, then the representation ring R(K) of K can be identified with R(T)W. Steinberg's theorem.
The corresponding Weyl group equals the stabilizer of λs in W. It is generated by the simple reflections sj for which sαj is a positive root.