The Nil Hecke Ring and Singularity of Schubert Varieties

Abstract

Let G be a semi-simple simply-connected complex algebraic group and T ⊂ B a maximal torus and a Borel subgroup respectively. Let ŋ= 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 $\omega = \overline \omega$ mod T Î W as the element (denoted by the corresponding German character) xo of G/B, denned as $m = \overline \omega B$. For any w Î W, there is associated the Schubert variety ${{\rm X}_{\omega }}: = \overline {B\omega B/B} \subset G/B$ and the T-fixed points of X _w (under the canonical left action) are precisely ${\rm I}\omega : = \left\{ {o:\upsilon \in Wand\upsilon \leqslant \omega } \right\}$.

Dedicated to Professor Bertram Kostant