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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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

complex symplectic linear algebra linear symplectomorphisms Lagrangian submanifolds the metaplectic representation


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.

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