Sphericity of Smooth Schubert and Richardson Varieties

We investigate Schubert varieties that are spherical under a Levi subgroup action in a partial flag variety. We show that if the Schubert variety is nonsingular, then it is a spherical $L$-variety, where $L$ is the largest Levi subgroup that acts the Schubert variety. Moreover, we classify all partial flag varieties that are spherical with respect to a Levi subgroup action; we show that the closures of $L$-orbits in a spherical Schubert $L$-variety are Richardson varieties.