Kyoto Journal of Mathematics

The moment map on symplectic vector space and oscillator representation

Takashi Hashimoto

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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

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

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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]

symplectic vector space moment map canonical quantization oscillator representation Howe duality


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.

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