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.[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.
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 . 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, 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, such as the Möbius transformation,
g ⋅ z=
| az+b | |
| cz+d |
where
g=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\operatorname{SL}2(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 SL2(R).
The elements of the metaplectic group Mp2(R) are the pairs (g, ε), where
g\in\operatorname{SL}2(R)
\epsilon(z)2=cz+d=j(g,z)
(g1,\epsilon1) ⋅ (g2,\epsilon2)=(g1g2,\epsilon),
\epsilon(z)=\epsilon1(g2 ⋅ z)\epsilon2(z).
That this product is well-defined follows from the cocycle relation
j(g1g2,z)=j(g1,g2 ⋅ z)j(g2,z)
(g,\epsilon)\mapstog
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.
We first give a rather abstract reason why the Weil representation exists. The Heisenberg group has an irreducible unitary representation on a Hilbert space
lH
\rho:H(V)\longrightarrowU(lH)
\rho'
\psi\inU(lH)
\rho'=\operatorname{Ad}\psi(\rho)
lH
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
lH
[\psi]\in\operatorname{PU}(lH)
lH
lH
Now we give a more concrete construction in the simplest case of Mp2(R). The Hilbert space H is then the space of all L2 functions on the reals. The Heisenberg group is generated by translations and by multiplication by the functions eixy 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(ix2y) of x, for y real.
Weil showed how to extend the theory above by replacing
R
Some important examples of this construction are given by: