  We consider the action of the Levi subgroup of a parabolic subgroup that
stabilizes a Schubert variety. We show that a smooth Schubert variety is a
homogeneous space for a parabolic subgroup, or it has a smooth Schubert
divisor. Further, we show that all smooth Schubert varieties in a (partial)
flag variety of a rank two simple algebraic group are spherical.
