Smooth Schubert varieties in G/B and B-submodules of $ {{\mathfrak{g}} \left/ {\mathfrak{b}} \right.} $

Abstract

Let G be a semisimple linear algebraic group over $ \mathbb{C} $ without G 2-factors, B a Borel subgroup of G and T ⊂ B a maximal torus. The flag variety G/B is a projective G-homogeneous variety whose tangent space at the identity coset is isomorphic, as a B-module, to $ {{\mathfrak{g}} \left/ {\mathfrak{b}} \right.} $ , where $ \mathfrak{g} $ = Lie(G) and $ \mathfrak{b} $ = Lie(B). Recall that if w is an element of the Weyl group W of the pair (G, T), the Schubert variety X(w) in G/B is by definition the closure of the Bruhat cell BwB. In this paper we prove that X(w) is nonsingular if and only if: (1) its Poincare polynomial is palindromic; and (2) the tangent space TE(X(w)) to the set T-stable curves in X(w) through the identity is a B-submodule of $ {{\mathfrak{g}} \left/ {\mathfrak{b}} \right.} $ . The second condition can be interpreted as saying that the roots of (G, T) in the convex hull of a certain set of roots canonically associated to w arise as tangent weights to T-stable curves in X(w) at the identity. A corollary is that X(w) is smooth if and only if X(w -1) is smooth. Condition (2) also gives a pattern avoidance criterion for TE(X(w)) to be B-stable.