group theory

# Contents

## Definition

For $n \in \mathbb{N}$, the metaplectic group $Mp(2n, \mathbb{R})$ is the Lie group which is the unique double cover of the symplectic group $Sp(2n, \mathbb{R})$.

## Properties

### Relation to the metalinear group

Inside the symplectic group $Sp(2n, \mathbb{R})$ sits the general linear group

$Gl(n,\mathbb{R}) \hookrightarrow Sp(2n,\mathbb{R})$

as the subgroup that preserves the standard Lagrangian submanifold $\mathbb{R}^n \hookrightarrow \mathbb{R}^{2n}$. Restriction of the metaplectic group extension along this inclusion defines the metalinear group $Ml(n)$

$\array{ Ml(n, \mathbb{R}) &\hookrightarrow& Mp(2n, \mathbb{R}) \\ \downarrow && \downarrow \\ Gl(n, \mathbb{R}) &\hookrightarrow& Sp(2n, \mathbb{R}) } \,.$

Hence a metaplectic structure on a symplectic manifold induces a metalinear structure on its Lagrangian submanifolds.

## References

An original reference is

• Andre Weil, Sur certains groupes d’opérateurs unitaires, Acta Math. 111: 143–211. (1964).

Revised on July 11, 2012 19:22:00 by Urs Schreiber (134.76.83.9)