Kyoto Journal of Mathematics

The moment map on symplectic vector space and oscillator representation

Takashi Hashimoto

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Abstract

Let G denote Sp(n,R), U(p,q), or 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,ω) naturally gives rise to the oscillator (or Segal–Shale–Weil) representation of g:=Lie(G)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 μ,X for each Xg, which we denote by μˆ,X. Then we show that the map Xiμˆ,X gives a representation of g. With a suitable choice of V in each case, the representation coincides with the oscillator representation of 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


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