Metaplectic group
# Metaplectic group

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. 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/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.

## 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,

    $g\cdot z={\frac {az+b}{cz+d}}$ where $g={\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in \operatorname {SL} _{2}(\mathbf {R} )$

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 $g\in \operatorname {SL} _{2}(\mathbf {R} )$ and ε is a holomorphic function on the upper half-plane such that $\epsilon (z)^{2}=cz+d=j(g,z)$. The multiplication law is defined by:

    $(g_{1},\epsilon _{1})\cdot (g_{2},\epsilon _{2})=(g_{1}g_{2},\epsilon ),$    where $\epsilon (z)=\epsilon _{1}(g_{2}\cdot z)\epsilon _{2}(z).$

That this product is well-defined follows from the cocycle relation $j(g_{1}g_{2},z)=j(g_{1},g_{2}\cdot z)j(g_{2},z)$. The map

    $(g,\epsilon )\mapsto g$

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 ${\mathcal {H}}$, that is,

    $\rho :\mathbb {H} (V)\longrightarrow U({\mathcal {H}})$

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

    $\psi \in U({\mathcal {H}})$ such that $\rho '=\operatorname {Ad} _{\psi }(\rho )$.

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 ${\mathcal {H}}$—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 ${\mathcal {H}}$. 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 $[\psi ]\in \operatorname {PU} ({\mathcal {H}})$. So we only get a homomorphism from the symplectic group to the projective unitary group of ${\mathcal {H}}$; 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 ${\mathcal {H}}$. 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 $exp(ix^2y)$ 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.
