The moment map on symplectic vector space and oscillator representation

Abstract

Let $G$ denote $Sp(n,\mathbb{R})$, $\mathrm{U}(p,q)$, or ${\mathrm{O}}^{\ast }\left(2n\right)$. The main aim of this article is to show that the canonical quantization of the moment map on a symplectic $G$-vector space $(W,\omega )$ naturally gives rise to the oscillator (or Segal–Shale–Weil) representation of $\mathfrak{g}:=Lie\left(G\right)\otimes \mathbb{C}$. More precisely, after taking a complex Lagrangian subspace $V$ of the complexification of $W$, we assign an element of the Weyl algebra for $V$ to $\langle \mu ,X\rangle$ for each $X\in \mathfrak{g}$, which we denote by $\langle \hat{\mu },X\rangle$. Then we show that the map $X\mapsto \mathrm{i}\langle \hat{\mu },X\rangle$ gives a representation of $\mathfrak{g}$. With a suitable choice of $V$ in each case, the representation coincides with the oscillator representation of $\mathfrak{g}$.