## Kyoto Journal of Mathematics

### The moment map on symplectic vector space and oscillator representation

Takashi Hashimoto

#### Abstract

Let $G$ denote $\operatorname{Sp}(n,\mathbb{R})$, $\mathrm{U}(p,q)$, or $\mathrm{O}^{*}(2n)$. 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 $\mathfrak{g}:=\operatorname{Lie}(G)\otimes\mathbb{C}$. 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 $\langle \widehat{\mu},X\rangle$. Then we show that the map $X\mapsto\mathrm{i}\langle \widehat{\mu},X\rangle$ gives a representation of $\mathfrak{g}$. With a suitable choice of $V$ in each case, the representation coincides with the oscillator representation of $\mathfrak{g}$.

#### Article information

Source
Kyoto J. Math., Volume 57, Number 3 (2017), 553-583.

Dates
Revised: 10 November 2015
Accepted: 21 April 2016
First available in Project Euclid: 3 May 2017

https://projecteuclid.org/euclid.kjm/1493798414

Digital Object Identifier
doi:10.1215/21562261-2017-0006

Mathematical Reviews number (MathSciNet)
MR3685055

Zentralblatt MATH identifier
1373.22023

#### Citation

Hashimoto, Takashi. The moment map on symplectic vector space and oscillator representation. Kyoto J. Math. 57 (2017), no. 3, 553--583. doi:10.1215/21562261-2017-0006. https://projecteuclid.org/euclid.kjm/1493798414

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