  The aim of this paper is to show that the canonical quantization of the
moment maps on symplectic vector spaces naturally gives rise to the oscillator
representations. More precisely, let $(W,\omega)$ denote a real symplectic
vector space, on which a Lie group $G$ acts symplectically on the left, where
$G$ denotes a real reductive Lie group $\mathrm{Sp}(n,\mathbb R), \mathrm
U(p,q)$ or $\mathrm O^*(2n)$ in this paper. Then we quantize the moment map
$\mu: W \to \mathfrak g_0^*$, where $\mathfrak g_0^*$ denotes the dual space of
the Lie algebra $\mathfrak g_0$ of $G$. Namely, after taking a complex
Lagrangian subspace $V$ of the complexification of $W$, we assign an element of
the Weyl algebra for $V$ to $< \mu, X >$, which we denote by $< \hat{\mu}, X
>$, for each $X \in \mathfrak g_0$. It is shown that the map $X \mapsto \mathrm
i <\hat{\mu}, X >$ gives a representation of $\mathfrak g_0$ which extends to
the one of $\mathfrak g$, the complexification of $\mathfrak g_0$, by
linearity. With a suitable choice of the complex Lagrangian subspace $V$ in
each case, the representation coincides with the oscillator representation of
$\mathfrak g$.
