For a finite Coxeter system and a subset of its diagram nodes, we define spherical elements (a generalization of Coxeter elements). Conjecturally, for Weyl groups, spherical elements index Schubert varieties in a flag manifold G/B that are spherical for the action of a Levi subgroup. We evidence the conjecture, employing the combinatorics of Demazure modules, and work of R. Avdeev-A.

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Abstract. Let Lw be the Levi part of the stabilizer Qw in GLN (for left multiplication) of a Schubert variety X(w) in the Grassmannian G d,N . For the natural action of Lw on C[X(w)], the homogeneous coordinate ring of X(w) (for the Plücker embedding), we give a combinatorial description of the decomposition of C[X(w)] into irreducible Lw-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 Qw. 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 Lw-varieties.

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