  A Schubert variety in the complete flag manifold $GL_n/B$ is Levi-spherical
if the action of a Borel subgroup in a Levi subgroup of a standard parabolic
has a dense orbit. We give a combinatorial classification of these Schubert
varieties. This establishes a conjecture of the latter two authors, and a new
formulation in terms of standard Coxeter elements. Our proof uses the theory of
key polynomials (type A Demazure module characters).
