Metaplectic group
In mathematics, the metaplectic group Mp2n is a double cover of the symplectic group Sp2n. It can be defined over either real or p-adic numbers. The construction covers more generally the case of an arbitrary local or finite field, and even the ring of adeles.
The metaplectic group has a particularly significant infinite-dimensional linear representation, the Weil representation. It was used by André Weil to give a representation-theoretic interpretation of theta functions, and is important in the theory of modular forms of half-integral weight and the theta correspondence.
The fundamental group of the symplectic Lie group Sp2n(R) is infinite cyclic, so it has a unique connected double cover, which is denoted Mp2n(R) and called the metaplectic group.
The metaplectic group Mp2(R) is not a matrix group: it has no faithful finite-dimensional representations. Therefore, the question of its explicit realization is nontrivial. It has faithful irreducible infinite-dimensional representations, such as the Weil representation described below.
It can be proved that if F is any local field other than C, then the symplectic group Sp2n(F) admits a unique perfect central extension with the kernel Z/2Z, the cyclic group of order 2, which is called the metaplectic group over F. It serves as an algebraic replacement of the topological notion of a 2-fold cover used when F = R. The approach through the notion of central extension is useful even in the case of real metaplectic group, because it allows a description of the group operation via a certain cocycle.
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