  Let $G$ be a finite subgroup of the linear group of a finite-dimensional
complex vector $V$, $B={\operatorname S}(V)$ be the symmetric algebra,
${\mathcal D}=\mathcal D^G_B$ the ring of $G$-invariant differential operators,
and ${\mathcal D}^-$ its subring of negative degree operators. We prove that
$M\mapsto M^{ann}= {\operatorname Ann}_{\mathcal D^-}(M)$ defines an
isomorphism between the category of ${\mathcal D}$-submodules of $B$ and a
category of modules formed as lowest weight spaces. This is applied to a
construction of simple ${\mathcal D}$-submodules of $B$ when $G$ is a
generalized symmetric group, to show that $B^{ann}$ is a so-called Gelfand
model. Using differential algebra and lowest weight methods we also prove
branching rules, entailing the main results in the representation theory of the
symmetric group, such as a differential construction of the Young basis.
