#### Vol. 8, No. 4, 2015

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editors’ Addresses Editors’ Interests Scientific Advantages Submission Guidelines Submission Form Ethics Statement Editorial Login Author Index Coming Soon Contacts ISSN: 1944-4184 (e-only) ISSN: 1944-4176 (print)
Linear symplectomorphisms as $R$-Lagrangian subspaces

### Chris Hellmann, Brennan Langenbach and Michael VanValkenburgh

Vol. 8 (2015), No. 4, 551–569
##### 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.

##### Keywords
complex symplectic linear algebra, linear symplectomorphisms, Lagrangian submanifolds, the metaplectic representation
##### Mathematical Subject Classification 2010
Primary: 37J10, 51A50, 70H15, 81S10