The construction covers more generally the case of an arbitrary local or finite field, and even the ring of adeles.
[1] 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.
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.
This group biholomorphically acts on the complex upper half-plane by fractional-linear transformations, such as the Möbius transformation, 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
The multiplication law is defined by: That this product is well-defined follows from the cocycle relation
The Heisenberg group has an irreducible unitary representation on a Hilbert space
The Stone–von Neumann theorem states that this representation is essentially unique: if
So any automorphism of the Heisenberg group that induces the identity on the center acts on this representation
—more precisely, the action is only well-defined up to multiplication by a nonzero constant.
But the action above is only defined up to multiplication by a nonzero constant, so an automorphism of the group is mapped to an equivalence class
The general theory of projective representations gives an action of some central extension of the symplectic group on
This central extension can be taken to be a double cover, which is the metaplectic group.
The Heisenberg group is generated by translations and by multiplication by the functions eixy of x, for y real.
—the Weil representation—is generated by the Fourier transform and multiplication by the functions exp(ix2y) of x, for y real.
The Hilbert space H is then the space of all L2 functions on G. The (analogue of) the Heisenberg group is generated by translations by elements of G, and multiplication by elements of the dual group (considered as functions from G to the unit circle).