Metaplectic group

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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.

Metaplectic group has a particularly important representation called the Weil representation that 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 theta correspondence.

1 Definition
2 Explicit construction for n=1
3 Construction of the Weil representation
4 Generalizations
5 See also
6 References

[edit] Definition

The fundamental group of the symplectic Lie group Sp_2n(R) is infinite cyclic, so SL₂(R) has a unique connected double cover, which is denoted Mp_2n(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 Sp_2n(F) admits a unique perfect central extension with the kernel Z_2, 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.

[edit] 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,

$g\cdot z=\frac{az+b}{cz+d}$ where $g = \begin{pmatrix}a&b\\c&d\end{pmatrix}\in SL_2(\mathbb{R})$

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 explicit construct the metaplactic cover of SL₂(R).

The elements of the metaplectic group Mp₂(R) are the pairs (g, ε), where

$g\in SL_2(\mathbb{R}), \epsilon^2=cz+d=j(g\cdot z),$

so that ε is a choice of one of the two branches of the complex square root function of j (g z) for z in the complex upper half-plane. The multiplication law is defined by:

$(g_1,\epsilon_1)\cdot (g_2,\epsilon_2)=(g_1 g_2, \epsilon),$ where $\epsilon(z)=\epsilon_1(g_2\cdot z)\epsilon_2(z).$

The associativity of this product follows from a certain cocycle condition satisfied by ε(z). The map

$(g,\epsilon)\mapsto g$

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.

[edit] Construction of the Weil representation

We first give a rather abstract reason why the Weil representation exists. The Heisenberg group has an essentially unique irreducible unitary representation on a Hilbert space H with the center acting as a given nonzero constant (the content of the Stone-von Neumann theorem). So any automorphism of the Heisenberg group acts on this representation H.

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 H. However if we look more closely we notice that the action of an automorphism of the Heisenberg group on H is not uniquely defined, but only defined up to multiplication by a non-zero constant.

So we only get a homomorphism from the symplectic group to the projective unitary group of 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 H. A little 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 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 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(ix²y) of x, for y real.

[edit] 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 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). 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. This case is used in the theory of theta functions of lattices, where typically G will be the discriminant group of an even lattice.

[edit] See also

[edit] References

Roger Howe, Eng-Chye Tan, Nonabelian harmonic analysis. Applications of SL(2,R). Universitext. Springer-Verlag, New York, 1992. ISBN 0-387-97768-6
Gerard Lion, Michele Vergne. The Weil representation, Maslov index and theta series, Progress in Mathematics, vol. 6, Birkhäuser Boston, 1980.
Weil, André. Sur certains groupes d'opérateurs unitaires. Acta Math. 111 1964 143--211

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Categories: Fourier analysis | Topology of Lie groups | Theta functions