July | 2022 | Theories and Theorems

In Siegel modular forms, we mentioned that one could construct Siegel modular forms from elliptic modular forms (see Modular Forms) via a process called “lifting”. In this post, we discuss a more general method that produces new automorphic forms (which generalize modular forms, and are also more properly a part of representation theory, see also Automorphic Forms) out of old ones. There is also a local version that deals with representations of p-adic Lie groups. Both of these form the theory of the (global and local) theta correspondence.

We begin with the local theory. Let $F$ be a nonarchimedean local field of characteristic zero (e.g. $\mathbb{Q}_{p}$ or a finite extension of $\mathbb{Q}_{p}$ ). Let $E$ be a quadratic etale $F$ -algebra. Let $V$ be a vector space over $E$ equipped with a Hermitian form $\langle-,-\rangle_{V}$ , and let $W$ be a vector space over $E$ equipped with a skew-Hermitian form $\langle -,-\rangle_{W}$ . Their respective groups of isometries are the unitary groups $\mathrm{U}(V)$ and $\mathrm{U}(W)$ . These two groups form an example of a reductive dual pair. The theory of the local theta correspondence relates representations of one of these groups to representations of the other.

Now the tensor product $V\otimes_{E} W$ can be viewed as a vector space over $F$ and we can equip it with a symplectic form $(-,-)=\mathrm{Tr}(\langle-,-\rangle_{V}\otimes\langle -,-\rangle_{W})$ . We have a map

$\displaystyle \mathrm{U}(V)\times\mathrm{U}(W)\to\mathrm{Sp}(V\otimes_{E} W)$

The symplectic group $\mathrm{Sp}(V\otimes_{E} W)$ has a cover called the metaplectic group; we will describe it in more detail shortly, but we say now that the importance of it for our purposes is that it has a special representation called the Weil representation, which as we shall shortly see will be useful in relating representations of $\mathrm{U}(V)$ to $\mathrm{U}(W)$ , and vice-versa.

We first need to construct the Heisenberg group $H(V\otimes_{E} W)$ . Its elements are given by $(V\otimes_{E} W)\oplus F$ , and we give it the group structure

$\displaystyle (x_{1},t_{1})\cdot (x_{2},t_{2})=\left(x_{1}+x_{2},t_{1}+t_{2}+\frac{1}{2}(x_{1},x_{2})\right)$

The Stone-von Neumann theorem tells us that, for every nontrivial character $\psi:F\to\mathbb{C}^{\times}$ the Heisenberg group has a unique irreducible representation $\omega_{\psi}$ with central character $\psi$ . Furthermore, the representation $\omega_{\psi}$ is unitary.

If $V\otimes_{E} W=X\oplus Y$ is a Lagrangian decomposition, we can realize the representation $\omega_{\psi}$ on the vector space of Schwarz functions on either $X$ or $Y$ . Let us take it to be $Y$ . In particular, we can express $\omega_{\psi}$ as follows. We first extend the character $\psi$ to $H(X)$ (defined to be the subgroup $X\oplus F$ of $H(V\otimes_{E}W)$ ) and then define $\omega_{\psi}$ as the induced representation (see also The Local Langlands Correspondence for General Linear Groups for another example of an induced representation)

$\displaystyle \omega_{\psi}=\mathrm{c-Ind}_{H(X)}^{H(V\otimes_{E}W)}\psi$

The symplectic group $\mathrm{Sp}(V\otimes_{E} W)$ acts on the Heisenberg group $H(V\otimes_{E}W)$ by $g\cdot (x,t)=(g\cdot x,t)$ for $g\in \mathrm{Sp}(V\otimes_{E} W)$ and $(x,t)\in H(V\otimes_{E}W)$ . We can compose this action with the representation $\omega_{\psi}$ to get another representation ${}^{g}\omega_{\psi}=\omega_{\psi}\circ g^{-1}$ of $H(W)$ . Now since the action of $\mathrm{Sp}(V\otimes_{E} W)$ on $H(V\otimes_{E}W)$ has trivial center, the central characters of ${}^{g}\omega_{\psi}$ and $\omega_{\psi}$ are the same. By the Stone-von Neumann theorem, ${}^{g}\omega_{\psi}$ and $\omega_{\psi}$ have to be isomorphic.

What this means now, is that for every $g\in \mathrm{Sp}(V\otimes_{E}W)$ , we have a linear transformation $A_{\psi}(g)$ of the underlying vector space $\mathcal{S}$ of the representation $\omega_{\psi}$ , so that

$\displaystyle A_{\psi}(g)\circ {}^{g}\omega_{\psi}=\omega_{\psi}\circ A_{\psi}(g)$

This action however is only defined up to a factor of $\mathbb{C}^{\times}$ . Since $\omega_{\psi}$ is unitary, we can also require $A_{\psi}$ to be unitary, and so the action becomes well-defined up to $S^{1}$ . All in all, this means that we have a representation

$\displaystyle A_{\psi}:\mathrm{Sp}(V\otimes W)\to \mathrm{GL}(\mathcal{S})/S^{1}$

Now if we pull back the map $\mathrm{GL}(\mathcal{S})\to\mathrm{GL}(\mathcal{S})/S^{1}$ by the map $A_{\psi}:\mathrm{Sp}(V\otimes_{E} W)\to \mathrm{GL}(\mathcal{S})/S^{1}$ , we get a map $\widetilde{A}_{\psi}:\mathrm{Mp}(V\otimes_{E} W)\to \mathrm{GL}(\mathcal{S})$ , where the group $\mathrm{Mp}(V\otimes_{E} W)$ is an $S^{1}$ -cover of $\mathrm{Sp}(V\otimes_{E} W)$ . This group $\mathrm{Mp}(V\otimes_{E} W)$ is the metaplectic group mentioned earlier.

Our construction allows us to extend the representation $\omega_{\psi}$ of $H(V\otimes_{E }W)$ to the semidirect product $\mathrm{Mp}(V\otimes_{E} W)\ltimes H(V\otimes_{E} W)$ . This representation of $\mathrm{Mp}(V\otimes_{E} W)\ltimes H(V\otimes_{E} W)$ is called the Heisenberg-Weil representation. The representation of $\mathrm{Mp}(V\otimes_{E} W)$ obtained by restriction is called the Weil representation.

Recall that we have a map $\mathrm{U}(V)\times \mathrm{U}(W)\to \mathrm{Sp}(V\otimes_{E}W)$ . If we could lift this to a map $\mathrm{U}(V)\times \mathrm{U}(W)\to \mathrm{Mp}(V\otimes_{E}W)$ , then we could obtain a representation of $\mathrm{U}(V)\times \mathrm{U}(W)$ by restricting the Weil representation $\omega_{\psi}$ from $\mathrm{Mp}(V\otimes_{E}W)$ to $\mathrm{U}(V)\times \mathrm{U}(W)$ . It turns out such a lifting can be defined and is determined by a pair $(\chi_{V},\chi_{W})$ of characters of $E^{\times}$ satisfying certain conditions. Once we have this lifting, we denote the resulting representation of $\mathrm{U}(V)\times \mathrm{U}(W)$ by $\Omega$ .

Now let $\pi$ be an irreducible representation of $V$ . We consider the maximal $\pi$ -isotypic quotient of $\Omega$ , which is its quotient by the intersection of all the kernels of morphisms of representations of $U(V)$ from $\Omega$ to $\pi$ . This quotient is of the form $\pi\otimes\theta(\pi)$ , where $\Theta(\pi)$ is a representation of $U(W)$ called the big theta lift of $\pi$ . The maximal semisimple quotient of $\Theta(\pi)$ is denoted $\theta(\pi)$ , and is called the small theta lift of $\pi$ .

Let us now look at the global picture. Let $k$ be a number field and let $k_{v}$ be the completion of $k$ at one of its places $v$ . Let $E$ be a quadratic extension of $k$ . Now we let $V$ and $W$ be vector spaces over $E$ equipped with Hermitian and skew-Hermitian forms $\langle-,-\rangle_{B}$ and $\langle--\rangle_{W}$ , as in the local case, and consider the tensor product $V\otimes_{E} W$ as a vector space over $k$ , and equip it with the symplectic form $\mathrm{Tr}(\langle-,-\rangle_{V}\otimes\langle-,-\rangle_{W})$ . We have localizations $(V\otimes_{E} W)_{v}$ for every $v$ , and we have already seen that in this case we can construct the metaplectic group $\mathrm{Mp}((V\otimes_{E} W)_{v})$ . We want to put each of these together for every $v$ to construct an “adelic” metaplectic group.

First we take the restricted product $\prod_{v}'\mathrm{Mp}((V\otimes_{E} W)_{v})$ . “Restricted” means that all but finitely many of the factors in this product belong to the hyperspecial maximal compact subgroup $K_{v}$ of $\mathrm{Sp}((V\otimes_{E} W)_{v})$ , which is also a compact open subgroup of $\mathrm{Mp}((V\otimes_{E} W)_{v})$ . This restricted product contains $\bigoplus_{v}S^{1}$ as a central subgroup. Now if we quotient out the restricted product $\prod_{v}'\mathrm{Mp}((V\otimes_{E} W)_{v})$ by the central subgroup $Z$ given by the set of all $(z_{v})\in\bigoplus_{v}S^{1}$ such that $\prod_{v}z_{v}=1$ , the resulting quotient is the “adelic” metaplectic group $\mathrm{Mp}(V\otimes_{E} W)(\mathbb{A})$ that we are looking for.

We have a representation $\bigotimes_{v}'\omega_{\psi_{v}}$ of $\prod_{v}'\mathrm{Mp}((V\otimes_{E} W)_{v})$ which acts trivially on the central subgroup $Z$ defined above and therefore gives us a representation $\omega_{\psi}$ of $\mathrm{Mp}(V\otimes_{E} W)(\mathbb{A})$ .

What is the underlying vector space of the representation $\omega_{\psi}$ ? If $V\otimes_{E}W=X\oplus Y$ is a Lagrangian decomposition, we have seen that we can realize the local Weil representation $\omega_{\psi_{v}}$ on $\mathcal{S}(Y_{v})$ , the vector space of Schwarz functions of $Y_{v}$ (the corresponding localization of $Y$ ). Likewise we can also realize the global Weil representation $\omega_{\psi}$ as functions on the vector space $\mathcal{S}(Y_{\mathbb{A}})$ , defined to be the restricted product $\bigotimes'\mathcal{S}(Y_{v})$ .

So now we have the global Weil representation $\omega_{\psi}$ , which is a representation of the group $\mathrm{Mp}(V\otimes_{E}W)(\mathbb{A})$ on the vector space $\mathcal{S}(Y_{\mathbb{A}})$ . But suppose we want an automorphic representation, i.e. one realized on the vector space of automorphic forms for $\mathrm{Mp}(V\otimes_{E}W)(\mathbb{A})$ (recall that one of our motivations in this post is to “lift” automorphic forms from one group to another). This is accomplished by the formation of theta functions $\theta(f)(g)$ , so-called because it is a generalization of the Jacobi theta function discussed in Sums of squares and the Jacobi theta function. Let $f$ be a vector in the underlying vector space of the Weil representation. Then the theta function $\theta(f)(g)$ is obtained by summing the evaluations of the output of the action of Weil representation on $f$ over all rational points $y\in Y(k)$ :

$\displaystyle \theta(f)(g)=\sum_{y\in Y(k)}(\omega_{\psi}(g)\cdot f)(y)$

Now suppose we have a pair of characters $\chi_{1},\chi_{2}$ of $E^{\times}$ , so that we have a lifting of $U(V)(\mathbb{A})\times U(W)(\mathbb{A})$ to $\mathrm{Mp}(V\otimes_{E}W)(\mathbb{A})$ . This lifting sends $U(V)(k)\times U(W)(k)$ to $\mathrm{Mp}(V\otimes_{E}W)(k)$ , which means that we can consider $\theta(f)(g)$ as an automorphic form for $U(V)(\mathbb{A})\times U(W)(\mathbb{A})$ .

Now we can perform our lifting. Let $f$ be a cuspidal automorphic form for $U(V)$ , and let $\varphi$ be a vector in the underlying vector space of the Weil representation. We can now obtain an automorphic form $\theta(\varphi,f)(g)$ on $U(W)$ as follows:

$\displaystyle \theta(\varphi,f)(g)=\int_{[\mathrm{U}(V)]}\theta(\varphi)(g,h)\cdot \overline{f(h)}dh$

The space generated in this way, for all vectors $f$ in a cuspidal automorphic representation $\pi$ of $U(V)$ , and all vectors $\varphi$ in the in the underlying vector space of the Weil representation, is called the global theta lift of $\pi$ , denoted $\Theta(\pi)$ . It is an automorphic representation of $U(W)$ .

There is also an analogue of all that we discussed for the reductive dual pair $\mathrm{SO}(V)\times \mathrm{Sp}(W)$ when $V$ and $W$ are vector space over some field, equipped with a quadratic form and symplectic form respectively.

Many cases of “lifting”, for instance the Saito-Kurokawa lift from elliptic modular forms to Siegel modular forms, can be considered special cases of the global theta lift (in particular for the reductive dual pair $\mathrm{SO}(V)\times \mathrm{Sp}(W)$ ). The theory of theta lifting is itself part of the theory of Langlands functoriality (see also Trace Formulas). More aspects and examples of the theta correspondence will be discussed in future posts on this blog.

References:

Theta correspondence on Wikipedia

Heisenberg group on Wikipedia

Metaplectic group on Wikipedia

Saito-Kurokawa lift on Wikipedia

Automorphic forms and the theta correspondence by Wee Teck Gan

A brief survey of the theta correspondence by Dipendra Prasad

Non-tempered Arthur packets of G2 by Wee Teck Gan and Nadia Gurevich

A quaternionic Saito-Kurokawa lift and cusp forms on G2 by Aaron Pollack