The moment map on symplectic vector space and oscillator representation

Abstract

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 $\mathfrak g$.