such that α(v) > 0 for every positive root. If αi is a simple root with reflection operator si then the corresponding divided difference operator is defined by If and s has reduced expression then is independent of the reduced expression. Moreover if and 0 otherwise. If w0 is the longest element of W, the element of greatest length or equivalently the element sending Φ+ to −Φ+, then More generally for some constants as,t. Set and Then Ps is a homogeneous polynomial of degree . These polynomials are the Kostant polynomials. Properties Theorem. The Kostant polynomials form a free basis of the ring of polynomials over the W-invariant polynomials. In fact the matrix is unitriangular for any total order such that s ≥ t implies . Hence Thus if with as invariant under W, then Thus where another unitriangular matrix with polynomial entries. 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. Steinberg basis As above let Φ be a root system in a real inner product space V, and Φ+ a subset of positive roots. From these data we obtain the subset Δ = {α1, α2, …, αn} of the simple roots, the coroots and the fundamental weights λ1, λ2, ..., λn as the dual basis of the coroots. 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 exponentials λs (s in W) form a free basis for the ring of exponentials over the subring of W-invariant exponentials. Let ρ denote the half sum of the positive roots, and A denote the antisymmetrisation operator The positive roots β with sβ positive can