  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.
