Levi Subgroup Actions on Schubert Varieties, and Some Geometric Consequences

In this chapter, we consider certain Levi subgroup actions on Schubert varieties in the Grassmannian, and derive some geometric consequences on the singularities of Schubert varieties as well as sphericity consequences. To be more precise, let L ω be the Levi part of the stabilizer Q ω in GL N (ℂ) (for left multiplication) of a Schubert variety X(ω) in the Grassmannian G d,N .