Metaplectic group

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).