Metaplectic group

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In mathematics, the metaplectic group Mp2n is a double cover of the symplectic group Sp2n. It has a particularly important representation called the Weil representation. It is often defined over the real numbers, but there are also analogues over local fields and for finite groups. It was used by Weil to give a representation theoretic interpretation of theta functions, and is important in the theory of modular forms of half-integral weight.

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[edit] Definition

The group Sp2(R) = SL2(R) has infinite cyclic fundamental group, so it has a unique connected double cover, which is called Mp2(R). Its elements can be written explicitly as pairs (A, f) where

A = \begin{pmatrix}a&b\\c&d\end{pmatrix}

is in SL2(R) and f is a function on the upper half-plane whose square is cτ + d.

The metaplectic group Mp2(R) is not algebraic: it has no faithful finite-dimensional representation. It has faithful infinite-dimensional representations, the most important of which is the Weil representation described below.

If n>1 then the symplectic group Sp2n(R) also has a unique connected double cover Mp2n(R). Most of the basic theory of this group is a straightforward generalization of the case n = 1.

[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 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.

[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 Sp2n(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 Sp2n(F).
  • 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

  • Weil, André. Sur certains groupes d'opérateurs unitaires. Acta Math. 111 1964 143--211
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