Linear Symplectomorphisms as R-Lagrangian Subspaces

TL;DR

This paper explores the structure of symplectomorphisms as R-Lagrangian subspaces, providing explicit formulas and a general expression for the metaplectic representation of the real symplectic group.

Contribution

It introduces explicit formulas for R-Lagrangian subspaces associated with symplectomorphisms and derives a general formula for the metaplectic representation.

Findings

Explicit formulas for R-Lagrangian subspaces of symplectomorphisms

A general formula for the metaplectic representation of the real symplectic group

Clarification of the imaginary restriction of the symplectic form

Abstract

The graph of a real symplectic linear transformation 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.