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 Poincaré 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.