Vol. 8, No. 4, 2015

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Linear symplectomorphisms as $R$-Lagrangian subspaces

Chris Hellmann, Brennan Langenbach and Michael VanValkenburgh

Vol. 8 (2015), No. 4, 551–569

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.

complex symplectic linear algebra, linear symplectomorphisms, Lagrangian submanifolds, the metaplectic representation
Mathematical Subject Classification 2010
Primary: 37J10, 51A50, 70H15, 81S10
Received: 20 September 2013
Revised: 24 August 2014
Accepted: 31 October 2014
Published: 23 June 2015

Communicated by Ravi Vakil
Chris Hellmann
Department of Mathematics and Statistics
Grinnell College
Grinnell, IA 50112
United States
Brennan Langenbach
Department of Mathematics and Statistics
Grinnell College
Grinnell, IA 50112
United States
Michael VanValkenburgh
Department of Mathematics and Statistics
California State University
Sacramento, CA 95819
United States