  Let $L_w$ be the Levi part of the stabilizer $Q_w$ in $GL_N$ (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 Pl\"ucker 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 $G_{2,N}$ are
spherical $L_w$-varieties.
