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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 Plücker 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.

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