## Kyoto Journal of Mathematics

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

### 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 \stackrel{\u02c6}{\mu},X\rangle $. Then we show that the map $X\mapsto \mathrm{i}\langle \stackrel{\u02c6}{\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**

Received: 11 May 2015

Revised: 10 November 2015

Accepted: 21 April 2016

First available in Project Euclid: 3 May 2017

**Permanent link to this document**

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

**Subjects**

Primary: 22E46: Semisimple Lie groups and their representations

Secondary: 17B20: Simple, semisimple, reductive (super)algebras 81S10: Geometry and quantization, symplectic methods [See also 53D50]

**Keywords**

symplectic vector space moment map canonical quantization oscillator representation Howe duality

#### 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