In mathematics, the **metaplectic group** Mp2*n* is a double cover of the symplectic group Sp2*n*. 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**.^{[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.

# Definition

The fundamental group of the symplectic Lie group Sp2n(**R**) is infinite cyclic, so it has a unique connected double cover, which is denoted Mp2*n*(**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 Sp2*n*(*F*) admits a unique perfect central extension with the kernel **Z**/2**Z**, 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.

# Explicit construction for *n* = 1

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,

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**).

is a surjection from Mp2(**R**) to SL2(**R**) which does not admit a continuous section. Hence, we have constructed a non-trivial 2-fold cover of the latter group.

# Construction of the Weil representation

Now we give a more concrete construction in the simplest case of
Mp2(**R**). The Hilbert space *H* is then the space of all *L*2 functions on the reals. The Heisenberg group is generated by translations and by multiplication by the functions *e**ixy* of *x*, for *y* real. Then the action of the metaplectic group on *H* is generated by the Fourier transform and multiplication by the functions exp(*ix*2*y*) of *x*, for *y* real.

# Generalizations

Weil showed how to extend the theory above by replacing ℝ by any locally compact group *G* that is isomorphic to its Pontryagin dual (the group of characters). The Hilbert space *H* is then the space of all *L*2 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). There is an analogue of the symplectic group acting on the Heisenberg group, and this action lifts to a projective representation on *H*. The corresponding central extension of the symplectic group is called the metaplectic group.

Some important examples of this construction are given by:

*G*is a vector space over the reals of dimension*n*. This gives a metaplectic group that is a double cover of the symplectic group Sp2*n*(**R**).- More generally
*G*can be a vector space over any local field*F*of dimension*n*. This gives a metaplectic group that is a double cover of the symplectic group Sp2*n*(*F*). *G*is a vector space over the adeles of a number field (or global field). This case is used in the representation-theoretic approach to automorphic forms.*G*is a finite group. The corresponding metaplectic group is then also finite, and the central cover is trivial. This case is used in the theory of theta functions of lattices, where typically*G*will be the discriminant group of an even lattice.- A modern point of view on the existence of the
*linear*(not projective) Weil representation over a finite field, namely, that it admits a canonical Hilbert space realization, was proposed by David Kazhdan. Using the notion of canonical intertwining operators suggested by Joseph Bernstein, such a realization was constructed by Gurevich-Hadani.^{[2]}

# See also

- Heisenberg group
- Metaplectic structure
- Reductive dual pair
- Spin group, another double cover
- Symplectic group
- Theta function