Involve: A Journal of Mathematics

  • Involve
  • Volume 8, Number 4 (2015), 551-569.

Linear symplectomorphisms as $R$-Lagrangian subspaces

Chris Hellmann, Brennan Langenbach, and Michael VanValkenburgh

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Abstract

The graph of a real linear symplectomorphism is an R-Lagrangian subspace of a complex symplectic vector space. The restriction of the complex symplectic form is thus purely imaginary and may be expressed in terms of the generating function of the transformation. We provide explicit formulas; moreover, as an application, we give an explicit general formula for the metaplectic representation of the real symplectic group.

Article information

Source
Involve, Volume 8, Number 4 (2015), 551-569.

Dates
Received: 20 September 2013
Revised: 24 August 2014
Accepted: 31 October 2014
First available in Project Euclid: 22 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.involve/1511370910

Digital Object Identifier
doi:10.2140/involve.2015.8.551

Mathematical Reviews number (MathSciNet)
MR3366010

Zentralblatt MATH identifier
1380.37112

Subjects
Primary: 37J10: Symplectic mappings, fixed points 51A50: Polar geometry, symplectic spaces, orthogonal spaces 70H15: Canonical and symplectic transformations 81S10: Geometry and quantization, symplectic methods [See also 53D50]

Keywords
complex symplectic linear algebra linear symplectomorphisms Lagrangian submanifolds the metaplectic representation

Citation

Hellmann, Chris; Langenbach, Brennan; VanValkenburgh, Michael. Linear symplectomorphisms as $R$-Lagrangian subspaces. Involve 8 (2015), no. 4, 551--569. doi:10.2140/involve.2015.8.551. https://projecteuclid.org/euclid.involve/1511370910


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