Kyoto Journal of Mathematics
- Kyoto J. Math.
- Volume 57, Number 3 (2017), 553-583.
The moment map on symplectic vector space and oscillator representation
Let denote , , or . The main aim of this article is to show that the canonical quantization of the moment map on a symplectic -vector space naturally gives rise to the oscillator (or Segal–Shale–Weil) representation of . More precisely, after taking a complex Lagrangian subspace of the complexification of , we assign an element of the Weyl algebra for to for each , which we denote by . Then we show that the map gives a representation of . With a suitable choice of in each case, the representation coincides with the oscillator representation of .
Kyoto J. Math., Volume 57, Number 3 (2017), 553-583.
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
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 22E46: Semisimple Lie groups and their representations
Secondary: 17B20: Simple, semisimple, reductive (super)algebras 81S10: Geometry and quantization, symplectic methods [See also 53D50]
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