Classification of Levi-spherical Schubert varieties

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 an open 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 and contributes to the theory of key polynomials (type A Demazure module characters).