Metaplectic group

Definition

Over the reals

Let $m$ be a natural number. The metaplectic group $Mp(2m,\R)$, also denoted $Mp_{2m}(\R)$, is defined as the unique double cover of the symplectic group $Sp(2m,\R)$ over the field of real numbers. This makes sense because the fundamental group of any $Sp(2m,\R)$ is an infinite cyclic group and hence has a unique subgroup of index two.

Metaplectic group are examples of (finite-dimensional) real Lie groups that are not linear Lie groups, i.e., the metaplectic group does not have any faithful finite-dimensional representations over the field of real numbers.

Over the reals

Let $m$ be a natural number and $F$ be a local field other than the field of complex numbers. The group $Mp(2m,F)$ is the unique perfect central extension with the symplectic group $Sp(2m,F)$ as the quotient group and cyclic group:Z2 as the base normal subgroup (in this case, central subgroup).