The Nil Hecke Ring and Singularity of Schubert Varieties

Let G be a semi-simple simply-connected complex algebraic group and T ⊂ B a maximal torus and a Borel subgroup respectively. Let h = Lie T be the Cartan subalgebra of the Lie algebra Lie G, and W := N(T )/T the Weyl group associated to the pair (G, T ), where N(T ) is the normalizer of T in G. We can view any element w = w mod T ∈ W as the element (denoted by the corresponding German character) w of G/B, defined as w = wB. For any w ∈ W , there is associated the Schubert variety Xw := BwB/B ⊂ G/B and the T−fixed points of Xw (under the canonical left action) are precisely Iw := {v : v ∈ W and v ≤ w}. We (together with B. Kostant) have defined a certain ring QW (T ) (which is the smash product of the group algebra Z[W ] with the W−field Q(T ) of rational functions on the torus T ) and certain elements yw ∈ QW (T ) (for any w ∈ W ). Expressing the elements yw in the {δv}v∈W basis: