Levi Subgroup Actions on Schubert Varieties, Induced Decompositions of their Coordinate Rings, and Sphericity Consequences

Abstract

Let L w be the Levi part of the stabilizer Q w in G L N (for left multiplication) of a Schubert variety X(w) in the Grassmannian G d,N . For the natural action of L w on $\mathbb {C}[X(w)]$ , the homogeneous coordinate ring of X(w) (for the Plucker embedding), we give a combinatorial description of the decomposition of $\mathbb {C}[X(w)]$ into irreducible L w -modules; in fact, our description holds more generally for the action of the Levi part L of any parabolic subgroup Q that is contained in Q w . This decomposition is then used to show that all smooth Schubert varieties, all determinantal Schubert varieties, and all Schubert varieties in G2,N are spherical L w -varieties.