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Concept
Metaplectic group
Summary
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.
In the case n = 1, the symplectic group coincides with the special linear group SL2(R). This group biholomorphically acts on the complex upper half-plane by fractional-linear transformations,
where
is a real 2-by-2 matrix with the unit determinant and z is in the upper half-plane, and this action can be used to explicitly construct the metaplectic cover of SL2(R).
The elements of the metaplectic group Mp2(R) are the pairs (g, ε), where and ε is a holomorphic function on the upper half-plane such that .
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