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.^[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 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/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 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 \mathrm{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 explicitly construct the metaplectic cover of SL₂(R).

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

g\in \mathrm{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(gz) 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.

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' = \mathrm{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 $[\psi]\in PU(\mathcal H)$ . 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 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.

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, 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]

Notes

↑ Weil, A. (1964). "Sur certains groupes d'opérateurs unitaires". Acta Math. 111: 143–211. doi:10.1007/BF02391012.
↑ http://arxiv.org/abs/0705.4556

References

Howe, Roger; Tan, Eng-Chye (1992), Nonabelian harmonic analysis. Applications of SL(2,R), Universitext, New York: Springer-Verlag, ISBN 0-387-97768-6
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 (http://arxiv.org/abs/math/0610818)
Gurevich, Shamgar; Hadani, Ronny (2005), Canonical quantization of symplectic vector spaces over finite fields, http://arxiv.org/abs/0705.4556