In mathematics, the **metaplectic group** Mp_{2n} is a double cover of the symplectic group Sp_{2n}. 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 Sp_{2n}(**R**) is infinite cyclic, so it has a unique connected double cover, which is denoted Mp_{2n}(**R**) and called the **metaplectic group**.

The metaplectic group Mp_{2}(**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 Sp_{2n}(*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 SL_{2}(**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 SL_{2}(**R**).

The elements of the metaplectic group Mp_{2}(**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:

- where

That this product is well-defined follows from the cocycle relation . The map

is a surjection from Mp_{2}(**R**) to SL_{2}(**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

We first give a rather abstract reason why the Weil representation exists. The Heisenberg group has an irreducible unitary representation on a Hilbert space , that is,

with the center acting as a given nonzero constant. The Stone–von Neumann theorem states that this representation is essentially unique: if is another such representation, there exists an automorphism

- such that .

and the conjugating automorphism is projectively unique, i.e., up to a multiplicative modulus 1 constant. So any automorphism of the Heisenberg group, inducing the identity on the center, acts on this representation —to be precise, the action is only well-defined up to multiplication by a non-zero constant.

The automorphisms of the Heisenberg group (fixing its center) form the symplectic group, so at first sight this seems to give an action of the symplectic group on . However, the action is only defined up to multiplication by a nonzero constant, in other words, one can only map the automorphism of the group to the class .
So we only get a homomorphism from the symplectic group to the **projective** unitary group of ; in other words a projective representation. The general theory of projective representations then applies, to give an action of some central extension of the symplectic group on . A calculation shows that this central extension can be taken to be a double cover, and this double cover is the metaplectic group.

Now we give a more concrete construction in the simplest case of
Mp_{2}(**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 abelian group *G*, which by Pontryagin duality is isomorphic to its 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 Sp_{2n}(**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 Sp_{2n}(*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

## Notes

**^**Weil, A. (1964). "Sur certains groupes d'opérateurs unitaires".*Acta Math*.**111**: 143–211. doi:10.1007/BF02391012.**^**Gurevich, Shamgar; Hadani, Ronny (31 May 2007). "Quantization of symplectic vector spaces over finite fields". arXiv:0705.4556 [math.RT].

## References

- Howe, Roger; Tan, Eng-Chye (1992),
*Nonabelian harmonic analysis. Applications of SL(2,*, Universitext, New York: Springer-Verlag, ISBN 978-0-387-97768-3**R**) - Lion, Gerard; Vergne, Michele (1980),
*The Weil representation, Maslov index and theta series*, Progress in Mathematics,**6**, Boston: Birkhäuser - Weil, André (1964), "Sur certains groupes d'opérateurs unitaires",
*Acta Math.*,**111**: 143–211, doi:10.1007/BF02391012 - Gurevich, Shamgar; Hadani, Ronny (2006), "The geometric Weil representation",
*Selecta Mathematica. New Series*, arXiv:math/0610818 - Gurevich, Shamgar; Hadani, Ronny (2005),
*Canonical quantization of symplectic vector spaces over finite fields*, https://arxiv.org/abs/0705.4556CS1 maint: location (link)