nLab
metaplectic group

Contents

Definition

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

Properties

Relation to the metalinear group

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

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

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

Ml(n,) Mp(2n,) Gl(n,) Sp(2n,). \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)