Metaplectic group

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.

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,

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

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 SL₂(R).

The elements of the metaplectic group Mp₂(R) are the pairs (g, ε), where ${\displaystyle g\in \operatorname {SL} _{2}(\mathbf {R} )}$ and ε is a holomorphic function on the upper half-plane such that ${\displaystyle \epsilon (z)^{2}=cz+d=j(g,z)}$. The multiplication law is defined by:

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

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

${\displaystyle (g,\epsilon )\mapsto g}$

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.

We first give a rather abstract reason why the Weil representation exists. The Heisenberg group has an irreducible unitary representation on a Hilbert space ${\displaystyle {\mathcal {H}}}$, that is,

${\displaystyle \rho$ :\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 ${\displaystyle \rho '}$ is another such representation, there exists an automorphism

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

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

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 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]

• Heisenberg group
• Metaplectic structure
• Reductive dual pair
• Spin group, another double cover
• Symplectic group
• Theta function

• Howe, Roger; Tan, Eng-Chye (1992), Nonabelian harmonic analysis. Applications of SL(2,R), Universitext, New York: Springer-Verlag, ISBN 978-0-387-97768-3
• 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, arXiv:math/0610818
• Gurevich, Shamgar; Hadani, Ronny (2005), Canonical quantization of symplectic vector spaces over finite fields, https://arxiv.org/abs/0705.4556CS1 maint: location (link)