group theory

# Contents

## Definition

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

## Properties

### Relation to the metalinear group

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

$\mathrm{Gl}\left(n,ℝ\right)↪\mathrm{Sp}\left(2n,ℝ\right)$Gl(n,\mathbb{R}) \hookrightarrow Sp(2n,\mathbb{R})

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

$\begin{array}{ccc}\mathrm{Ml}\left(n,ℝ\right)& ↪& \mathrm{Mp}\left(2n,ℝ\right)\\ ↓& & ↓\\ \mathrm{Gl}\left(n,ℝ\right)& ↪& \mathrm{Sp}\left(2n,ℝ\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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)