  The classical Stone-von Neuman theorem relates the irreducible unitary
representations of the Heisenberg group $H_n$ to non-trivial unitary characters
of its center $Z$, and plays a crucial role in the construction of the
oscillator representation for the metaplectic group.
  In this paper we extend these ideas to non-unitary and non-irreducible
representations, thereby obtaining an equivalence of categories between certain
representations of $Z$ and those of $H_n$. Our main result is a smooth
equivalence, which involves the fundamental ideas of du Cloux on differentiable
representations and smooth imprimitivity systems for Nash groups. We show how
to extend the oscillator representation to the smooth setting and give an
application to degenerate Whittaker models for representations of reductive
groups.
  We also include an algebraic equivalence, which can be regarded as a
generalization of Kashiwara's lemma from the theory of $D$-modules.
