B-sub-modules of Lie(G)/Lie(B) and Smooth Schubert Varieties in G/B

Abstract

Let G be a complex semi-simple linear algebraic group without G_2 factors, B a Borel subgroup of G and T a maximal torus in B. 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 Lie(G)/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 note we prove that X(w) is non-singular iff the following two conditions hold: 1) its Poincar\'e 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 Lie(G)/Lie(B). This gives two criteria in terms of the combinatorics of W which are necessary and sufficient for X(w) to be smooth: \sum_{x\le w} t^{\ell(x)} is palindromic, and every root of (G,T) in the convex hull of the set of negative roots whose reflection is less than w (in the Bruhat order on W) has the property that its T-weight space (in Lie(G)/Lie(B)) is contained in TE(X(w)). However, as we show by example, these conditions don't characterize the smooth Schubert varieties when G has type G_2.