a Metaplectic Group?

Metaplectic groups “cover” symplectic groups, fitting into a short exact sequence like

$$1 \rightarrow \{ \pm 1 \} \rightarrow \operatorname {Mp}_{2n}(\mathbb{R}) \rightarrow \operatorname {Sp}_{2n}(\mathbb{R}) \rightarrow 1.$$

They arise when one realizes that a group of operators almost satisfies relations among symplectic matrices. In this formulation, they are creatures of geometry and harmonic analysis. More profoundly, modular forms of half-integer weight like those studied by Shimura Shi73 come from automorphic representations of metaplectic groups Gel76. In this setting, and via the theta correspondence, metaplectic groups belong to number theory and the Langlands program. Here we introduce metaplectic groups from multiple perspectives.

1. Topological

The two-by-two special linear group is the Lie group

$$\operatorname {SL}_2(\mathbb{R}) = \left\{ \left( \begin{array}{cc} a & b \\c & d \\\end{array} \right) : ad - bc = 1, \ a,b,c,d \in \mathbb{R}\right\}.$$

The group structure here is by matrix multiplication, and geometric structure arises from its embedding as a smooth submanifold of $\mathbb{R}^4$. Its subgroup of rotations is diffeomorphic to the circle,

$$\operatorname {SO}_2(\mathbb{R}) = \left\{ \left( \begin{array}{cc} \phantom {-} \cos \theta & \sin \theta \\- \sin \theta & \cos \theta \\\end{array} \right) : 0 \leq \theta < 2 \pi \right\}.$$

Write ${\mathfrak{p}}$ for the real vector space of symmetric trace-zero matrices,

$${\mathfrak{p}} = \left\{ \left( \begin{array}{cc} u & \phantom {-} v \\v & -u \\\end{array} \right) : u,v \in \mathbb{R}\right\}.$$

Then multiplication and the matrix exponential give a diffeomorphism

$$\phi \colon \operatorname {SO}_2(\mathbb{R}) \times {\mathfrak{p}} \rightarrow \operatorname {SL}_2(\mathbb{R}), \quad \phi (r, X) = r \cdot \exp (X).$$

E. Cartan proved such a statement for connected semisimple Lie groups in 1927, though we refer to Mostow Mos49 for a tidier proof. In such a general context, one finds a diffeomorphism,

$$\begin{equation} \phi \colon K \times {\mathfrak{p}} \xrightarrow {\text{diffeo}} G, {\tag{1.1}} \end{equation}$$

where $K$ is a maximal compact subgroup of $G$, and ${\mathfrak{p}}$ is a real vector space. From this, it follows that the fundamental groups of $K$ and $G$ (with base point $1$) can be identified. In particular, we have

$$\pi _1 \left( \operatorname {SL}_2(\mathbb{R}) \right) \equiv \pi _1 \left( \operatorname {SO}_2(\mathbb{R}) \right) = \pi _1( \bigcirc ) = \mathbb{Z}.$$

Associated to the quotient $\mathbb{Z}/ 2 \mathbb{Z}$ of $\pi _1(\operatorname {SL}_2(\mathbb{R}))$, one finds a two-fold covering space of $\operatorname {SL}_2(\mathbb{R})$. This covering space is naturally a Lie group, and the covering map a Lie group homomorphism, with kernel of order two. This “covering group” $\operatorname {Mp}_2(\mathbb{R})$ is the simplest example of a metaplectic group, and it fits into a short exact sequence,

$$1 \rightarrow \mu _2 \rightarrow \operatorname {Mp}_2(\mathbb{R}) \rightarrow \operatorname {SL}_2(\mathbb{R}) \rightarrow 1.$$

Here $\mu _2 \coloneq \{ \pm 1 \} \cong \mathbb{Z}/ 2 \mathbb{Z}$ is a group of order two, and it lies in the center of $\operatorname {Mp}_2(\mathbb{R})$.

The term “metaplectic” (“groupe métaplectique”) is due to André Weil Wei64, §34, in a more general context, where metaplectic groups cover symplectic groups. Indeed, Cartan’s diffeomorphism 1.1 holds for the symplectic groups $\operatorname {Sp}_{2n}(\mathbb{R})$, where they provide an identification,

$$\pi _1 \left( \operatorname {Sp}_{2n}(\mathbb{R}) \right) = \pi _1 \left( \operatorname {U}(n) \right) = \mathbb{Z}, \text{ when } n \geq 1.$$

In this way, the previous construction gives a central extension of Lie groups,

$$1 \rightarrow \mu _2 \rightarrow \operatorname {Mp}_{2n}(\mathbb{R}) \rightarrow \operatorname {Sp}_{2n}(\mathbb{R}) \rightarrow 1.$$

Note $n=1$ gives $\operatorname {Mp}_2(\mathbb{R})$, since $\operatorname {SL}_2(\mathbb{R}) = \operatorname {Sp}_2(\mathbb{R})$. These 2-fold coverings $\operatorname {Mp}_{2n}(\mathbb{R})$ of symplectic groups $\operatorname {Sp}_{2n}(\mathbb{R})$ are called metaplectic groups, since Weil’s landmark paper.

On the other hand, this topological construction also provides a $d$-fold covering of $\operatorname {Sp}_{2n}(\mathbb{R})$ for every positive integer $d$, corresponding to the quotient $\mathbb{Z}/ d \mathbb{Z}$ of the fundamental group $\mathbb{Z}$. These are sometimes called “higher metaplectic groups” though naming is not so consistent.

If one attempts to work with $\operatorname {SL}_n(\mathbb{R})$, things change more dramatically. While $\operatorname {SO}_2(\mathbb{R})$ is the circle, with fundamental group $\mathbb{Z}$, the space-rotation group $\operatorname {SO}_3(\mathbb{R})$ has fundamental group of order 2. This corresponds to the famous double cover,

$$1 \rightarrow \mu _2 \rightarrow \operatorname {SU}(2) \rightarrow \operatorname {SO}_3(\mathbb{R}) \rightarrow 1.$$

In fact, the fundamental groups $\pi _1(\operatorname {SO}_n(\mathbb{R}))$ are all just $\mathbb{Z}/ 2 \mathbb{Z}$, for $n \geq 3$. Since $\operatorname {SO}_n(\mathbb{R})$ is a maximal compact subgroup of $\operatorname {SL}_n(\mathbb{R})$, one finds only 2-fold covers of $\operatorname {SL}_n(\mathbb{R})$ for $n \geq 3$. Some describe these 2-fold covers of $SL_n(\mathbb{R})$ as “metaplectic covers of $SL_n(\mathbb{R})$” even though the symplectic group is no longer present.

2. Weil’s Groups of Operators

We have given a topological characterization of $\operatorname {Mp}_2(\mathbb{R})$, but not really a construction. One might compare to the case of spin groups, where the fundamental group of $\operatorname {SO}_n(\mathbb{R})$ has order two (assuming $n > 2$). As a result, there is a central extension,

$$1 \rightarrow \mu _2 \rightarrow \operatorname {Spin}_n(\mathbb{R}) \rightarrow \operatorname {SO}_n(\mathbb{R}) \rightarrow 1.$$

This “covering group” of $\operatorname {SO}_n(\mathbb{R})$ is the compact group known as the spin group. But a more direct construction of $\operatorname {Spin}_n(\mathbb{R})$ is possible using the Clifford algebra of dimension $2^n$. With it, one may embed $\operatorname {Spin}_n(\mathbb{R})$ in the group of invertible $2^m \times 2^m$ matrices with $m = \lfloor (n-1)/2 \rfloor$ – a great cost, to be sure, but at least the spin groups are groups of matrices. (Should we call $\operatorname {Spin}_n(\mathbb{R})$ a metaplectic group? I think not, but others may disagree.)

The traditional metaplectic groups $\operatorname {Mp}_{2n}(\mathbb{R})$ admit no such matrix representation. In fact, this is a theorem.

Theorem 2.1.

Let $N$ be a positive integer and let $\rho \colon \operatorname {Mp}_{2n}(\mathbb{R}) \rightarrow \operatorname {GL}_N(\mathbb{C})$ be a continuous homomorphism. Then $\operatorname {Ker}(\rho )$ contains $\mu _2$, i.e., $\rho$ factors through $\operatorname {Sp}_{2n}(\mathbb{R})$. In particular, $\rho$ is not injective.

A proof is difficult to locate in the literature; Bourbaki suggests it as an exercise in Bou98, Ch.III, Exercises for §6. It suffices to assume $n=1$, since $\operatorname {Mp}_2(\mathbb{R}) \subset \operatorname {Mp}_{2n}(\mathbb{R})$ for all $n \geq 1$. Finite-dimensional representations of $\operatorname {Mp}_2(\mathbb{R})$ are determined by associated representations of its Lie algebra, ${\mathfrak{mp}} _2(\mathbb{R}) = {\mathfrak{sl}} _2(\mathbb{R})$. Those can all be written down, and the corresponding representations of $\operatorname {Mp}_2(\mathbb{R})$ factor through $\operatorname {Mp}_2(\mathbb{R}) / \mu _2 = \operatorname {SL}_2(\mathbb{R})$.

Since $\operatorname {Mp}_{2n}(\mathbb{R})$ does not sit inside a group of finite matrices, one must go further and work with groups of unitary operators on a Hilbert space. Such representations of $\operatorname {Mp}_2(\mathbb{R})$—in fact, of the universal covering of $\operatorname {SL}_2(\mathbb{R})$—were classified by Pukánzsky Puk64 in 1964. In the same year, Weil Wei64 published “Sur certains groupes d’opérateurs unitaires,” where he constructed metaplectic groups via such operators.

Here we describe the metaplectic group $\operatorname {Mp}_2(\mathbb{R})$ in terms of unitary operators $\operatorname {U}({\mathcal{H}})$ on the Hilbert space ${\mathcal{H}}= {\mathcal{L}}^2(\mathbb{R})$ of square-integrable functions. One learns about various unitary operators in a first course in harmonic analysis. One example is the Fourier transform,

$$[{\mathbf{F}}f](x) = \int _{-\infty }^\infty f(y) e^{- 2 \pi i x y} dy.$$

It will be convenient to scale the Fourier transform by an eighth root of unity, to define

$${\mathbf{W}}f = \zeta \cdot {\mathbf{F}}f, \text{ and } \zeta = e^{i \pi / 4}.$$

Fourier inversion gives the formulae,

$$\begin{equation} [{\mathbf{F}}{\mathbf{F}}f](x) = f(-x), \text{ so } [{\mathbf{W}}{\mathbf{W}}f](x) = i f(-x). {\tag{2.2}} \end{equation}$$

Note that ${\mathbf{F}}^4 = \operatorname {Id}$, but ${\mathbf{W}}^4 = -1$, in the group $\operatorname {U}({\mathcal{H}})$.

Another operator is obtained by multiplying by a function of absolute value one. For $b \in \mathbb{R}$, define $\sigma _b(x) = e^{\pi i b x^2}$ (a “quadratic sinusoid”), and define unitary operators ${\mathbf{E}}_+(b)$, ${\mathbf{E}}_-(b)$ by

$$\begin{align*} [{\mathbf{E}}_+(b) f](x) &= \sigma _b(x) \cdot f(x),\text{ and } \\ {\mathbf{E}}_-(b) &= {\mathbf{W}}{\mathbf{E}}_+(b) {\mathbf{W}}^{-1}. \end{align*}$$

A computation in Fourier analysis (admittedly not in ${\mathcal{L}}^2$) shows that ${\mathbf{F}}\sigma _b = \vert b \vert ^{-1/2} \zeta ^{\operatorname {sgn}(b)} \sigma _{-1/b}$. This allows us to compute

$$[{\mathbf{E}}_-(b) f](x) = \vert b \vert ^{-1/2} \zeta ^{\operatorname {sgn}(b)} \cdot [\sigma _{-1/b} \ast f](x).$$

Here the star $\ast$ denotes convolution.

For $0 \neq u \in \mathbb{R}$, define the operator ${\mathbf{H}}(u)$ by

$$[{\mathbf{H}}(u) f](x) = \vert u \vert ^{1/2} f(u x) \cdot \begin{cases} 1 & \text{ if } u > 0; \\ -i & \text{ if } u < 0. \end{cases}$$

This is a familiar “time-scaling” operator in Fourier analysis, scaled by $\vert u \vert ^{1/2}$ to make it unitary, and scaled by $1$ or $-i$ to make the following equation true.

$$\begin{equation*} {\mathbf{E}}_-(u^{-1}) {\mathbf{E}}_+(u) = {\mathbf{E}}_+(-u) {\mathbf{H}}(u) {\mathbf{W}}\text{ for all } u \in \mathbb{R}^\times . \end{equation*}$$

This implies

$${\mathbf{H}}(u) = {\mathbf{E}}_+(u) {\mathbf{E}}_-(u^{-1}) {\mathbf{E}}_+(u) {\mathbf{W}}^{-1}.$$

The proofs of these formulae are left as a computational exercise. And if the reader has made it through the zoo of unitary operators, ${\mathbf{W}}$ and ${\mathbf{E}}_{\pm }(b)$, and ${\mathbf{H}}(u)$, they can find the following relations.

Proposition 2.3.

The operators above satisfy the following relations (for all $a,b \in \mathbb{R}$, $u,v \in \mathbb{R}^\times$).

i.

${\mathbf{W}}^2 = {\mathbf{H}}(-1)$;

ii.

${\mathbf{E}}_+(a + b) = {\mathbf{E}}_+(a) {\mathbf{E}}_+(b)$;

iii.

${\mathbf{H}}(u) {\mathbf{H}}(v) = (u,v)_2 {\mathbf{H}}(uv)$;

iv.

${\mathbf{H}}(u) {\mathbf{E}}_+(a) {\mathbf{H}}(u^{-1}) = {\mathbf{E}}_+(a u^2)$.

In (iii), the expression $(u,v)_2 \in \{ 1, -1 \}$ is the Hilbert symbol, defined by

$$(u,v)_2 = \begin{cases} \phantom {-} 1 & \text{ if } u>0 \text{ or } v>0; \\ -1 & \text{ if } u<0 \text{ and } v<0. \end{cases}$$

We have focused on the operators and relations above, because they are analogous to matrices and relations in $SL_2$. For any field $F$, define matrices in $\operatorname {SL}_2(F)$,

$$w \coloneq \left( \begin{array}{cc} 0 & 1 \\-1 & 0 \\\end{array} \right) , \ e_+(b) \coloneq \left( \begin{array}{cc} 1 & b \\0 & 1 \\\end{array} \right) \text{ for all } b \in F.$$

Define also, for all $b \in F$, $u \in F^\times$,

$$\begin{align*} e_-(b) &\coloneq w e_+(b) w^{-1} = \left( \begin{array}{cc} 1 & 0 \\-b & 1 \\\end{array} \right) , \\ h(u) &\coloneq e_+(u) e_-(u^{-1}) e_+(u) w^{-1} = \left( \begin{array}{cc} u & 0 \\0 & u^{-1} \\\end{array} \right) . \end{align*}$$

In Shalika’s thesis Sha04, Thm. 1.2.1 he proves:

Proposition 2.4.

Let $F$ be a field. Then $\operatorname {SL}_2(F)$ can be presented by generators $\{ w, e_+(b) : b \in F \}$, subject to the following relations.

i.

$w^2 = h(-1)$.

ii.

$e_+(a + b) = e_+(a) e_+(b)$ for all $a,b \in F$.

iii.

$h(u) h(v) = h(uv)$ for all $u,v \in F^\times$.

iv.

$h(u) e_+(a) h(u^{-1}) = e(a u^2)$ for $a \in F, u \in F^\times$.

The only difference in the relations, between Propositions 2.3 and 2.4, is the sign in relation (iii). Taken together, we find the following.

Theorem 2.5.

Let $M$ be the subgroup of $\operatorname {U}({\mathcal{H}})$ generated by ${\mathbf{W}}$ and $\{ {\mathbf{E}}_+(b) : b \in \mathbb{R}\}$. Then there is a unique isomorphism $\iota \colon SL_2(\mathbb{R}) \rightarrow M / \{ \pm 1 \}$ satisfying

$$\iota \left( \begin{array}{cc} 0 & 1 \\-1 & 0 \\\end{array} \right) = {\mathbf{W}}\text{ and } \iota \left( \begin{array}{cc} 1 & b \\0 & 1 \\\end{array} \right) = {\mathbf{E}}_+(b).$$

Indeed, the relations (i-iv) of $SL_2(\mathbb{R})$ are satisfied by ${\mathbf{W}}$ and ${\mathbf{E}}_+(b)$, once we quotient out by $\pm 1$. This provides the unique homomorphism from $SL_2(\mathbb{R})$ to $M / \{ \pm 1 \}$. Surjectivity is clear since $M$ is by definition generated by $\{ {\mathbf{W}}, {\mathbf{E}}_+(b) : b \in \mathbb{R}\}$. For injectivity, we can use the fact that the only proper nontrivial normal subgroup of $SL_2(\mathbb{R})$ is $\{ \pm 1 \}$, and

$$\iota \left( \begin{array}{cc} -1 & 0 \\0 & -1 \\\end{array} \right) = \iota (h(-1)) = [ {\mathbf{H}}(-1) ] \neq [ \pm 1 ].$$

In this way, one can constructively define the metaplectic group $\operatorname {Mp}_2(\mathbb{R})$ to be the subgroup of $\operatorname {U}({\mathcal{H}})$ generated by ${\mathbf{W}}$ and $\{ {\mathbf{E}}_+(b) : b \in \mathbb{R}\}$. The topological characterization of the first section is replaced by an analytic construction. This construction extends from $\operatorname {Mp}_2(\mathbb{R})$ to $\operatorname {Mp}_{2n}(\mathbb{R})$, when the Hilbert space ${\mathcal{L}}^2(\mathbb{R})$ is replaced by ${\mathcal{L}}^2(\mathbb{R}^n)$.

More dramatically, this construction extends from $\mathbb{R}$ to any field where one may perform harmonic analysis, e.g., the fields $\mathbb{R}$, $\mathbb{Q}_p$, $\mathbb{F}_p(\!(t)\!)$, and their finite extensions (avoiding characteristic two). The Hilbert space ${\mathcal{L}}^2(\mathbb{R})$ is replaced by ${\mathcal{L}}^2(F)$ for such a local field $F$. The function $x \mapsto e^{2 \pi i x}$ is replaced by a suitable additive character $x \mapsto \mathbf{e}(x) \in \operatorname {U}(1)$, to define a Fourier transform ${\mathbf{F}}$ and operator ${\mathbf{E}}_+(b)$ in this context. This defines metaplectic groups $\operatorname {Mp}_{2n}(F)$ for every local field (avoiding characteristic two), following Weil Wei64 and Shalika’s 1966 PhD thesis (published in Sha04).

Beyond just constructing the metaplectic group for its own sake, Weil’s operator theoretic approach set the stage for decades of results in number theory. Constructing the metaplectic group over all local fields, and putting the local fields together, Weil’s construction defines an adelic metaplectic group $\operatorname {Mp}_{2n}(\mathbb{A})$. Here $\mathbb{A}$ is the ring of adeles, and $\operatorname {Mp}_{2n}(\mathbb{A})$ is a double-cover of the adelic symplectic group $\operatorname {Sp}_{2n}(\mathbb{A})$.

As is customary in number theory (since Tate’s Thesis) and automorphic forms, one considers the subgroup $\operatorname {Sp}_{2n}(\mathbb{Q}) \subset \operatorname {Sp}_{2n}(\mathbb{A})$. The metaplectic cover $\operatorname {Mp}_{2n}(\mathbb{A})$ naturally “splits” over $\operatorname {Sp}_{2n}(\mathbb{Q})$, meaning that the inclusion map $\operatorname {Sp}_{2n}(\mathbb{Q}) \hookrightarrow \operatorname {Sp}_{2n}(\mathbb{A})$ lifts to realize $\operatorname {Sp}_{2n}(\mathbb{Q})$ as a subgroup of $\operatorname {Mp}_{2n}(\mathbb{A})$.

$$\vcenter{\img[][223pt][48pt][{\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{tikzcd} 1 \arrow[r] &amp; \mu_2 \arrow[r]{} &amp; \Mp_{2n}(\AA) \arrow[r] &amp; \Sp_{2n}(\AA) \arrow[r] &amp; 1 \\ &amp; &amp; &amp; \Sp_{2n}(\QQ) \arrow{u}[swap]{\text{inclusion}} \ar{ul}{\text{splits!}} &amp; \end{tikzcd}}]{Images/img6afd018ca92257f4cf3f4fad3bf5ce14.svg}}$$

This natural splitting carries precisely the information of quadratic reciprocity, Thus one finds a basic connection between harmonic analysis (on local fields and the adeles) and reciprocity laws in number theory. The splitting above also set the stage for Gelbart’s analysis Gel76 of automorphic forms and representations of metaplectic groups, e.g., the spectral decomposition of ${\mathcal{L}}^2(\operatorname {SL}_2(\mathbb{Q}) \backslash \operatorname {Mp}_{2}(\mathbb{A}))$. This in turn led to the full development of the theta correspondence by Waldspurger. It is too big a subject for treatment here, so we refer to the beautiful survey of Dipendra Prasad Pra98 and more recent state of the art by Wee Teck Gan and Wen-Wei Li GL18.

3. Steinberg and Matsumoto

Proposition 2.4 describes $\operatorname {SL}_2(F)$ by generators and relations, and Steinberg generalizes this to Chevalley groups in Ste16, §6. While Shalika used relations like (i-iv) to prove that the group of operators $\operatorname {Mp}_2(F)$ is a double-cover of $\operatorname {SL}_2(F)$, one may use generators and relations to directly construct the metaplectic group – without any reference to operators at all! Here we describe this last approach to metaplectic groups as it leads to important generalizations.

Forget about topology and analysis, and consider a simple simply-connected Chevalley group over any field $F$. These are classified according to Cartan type: the type $A_\ell$ groups are $\operatorname {SL}_{\ell +1}(F)$, and the type $C_\ell$ are the symplectic groups $\operatorname {Sp}_{2 \ell }(F)$, and the type $B_\ell$ and $D_\ell$ are the “split” spin groups $\operatorname {Spin}_{2 \ell + 1}(F)$ and $\operatorname {Spin}_{2 \ell }(F)$ respectively. One has exceptional Chevalley groups too, with types $E_6, E_7, E_8, F_4, G_2$. Each Chevalley group comes with a root system $\Phi$, which Steinberg uses to present the group: the generators are elements $e_\alpha (x)$ for each root $\alpha \in \Phi$ and all $x \in F$. The relations (see Ste16, §6) are essentially those of Proposition 2.4, but for all roots in $\Phi$:

A.

$e_\alpha (x+y) = e_\alpha (x) e_\alpha (y)$ for all $x,y \in F$, $\alpha \in \Phi$.

B.

A description of the commutator of $e_\alpha (x)$ and $e_\beta (y)$, for all $\alpha , \beta \in \Phi$ with $0 \neq \alpha + \beta$.

$\tilde{\mathrm{B}}$.

$w_\alpha (t) e_\alpha (x) w_\alpha (-t) = e_{-\alpha }(- t^{-2} x)$ for all $t \in F^\times$, $x \in F$. Here $w_\alpha (t) \coloneq e_\alpha (t) e_{-\alpha }(t^{-1}) e_\alpha (t)$.

C.