Metaplectic group

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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).