nLab
metaplectic group

Definition

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

Properties

Relation to the metalinear group

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

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

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

$$\begin{array}{ccc}\mathrm{Ml}(n,ℝ)& \hookrightarrow & \mathrm{Mp}(2n,ℝ)\\ \downarrow & & \downarrow \\ \mathrm{Gl}(n,ℝ)& \hookrightarrow & \mathrm{Sp}(2n,ℝ)\end{array}.$$\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.

Related concepts

• metaplectic structure

• metaplectic correction (in geometric quantization)

• metaplectic representation

References

An original reference is

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