beezer.tex

\subsubsection*{The geometric theta correspondence}

We first explain the term "geometric theta correspondence". The Weil (or oscillator) representation gives us a method to construct closed differential forms on locally
symmetric spaces associated to groups which belong to dual pairs. Let $V$ be a rational quadratic space of signature $(p,q)$ with for simplicity even dimension. Then the Weil representation induces an action of $\SL_2(\R) \times \Orth(V_\R)$ on $\mathcal{S}(V_\R)$, the Schwartz functions on $V_\R$. Let $G = \SO_0(V_\R)$ and let $K$ be a maximal compact subgroup. We let $\mathfrak{g}$ and $\mathfrak{k}$ be their respective Lie algebras and let $\mathfrak{g} = \mathfrak{p} \oplus \mathfrak{k}$ be the associated Cartan decomposition. Suppose 
\[
\varphi\in (\mathcal{S}(V_\R) \otimes \wedge^r \mathfrak{p}^{\ast})^{K}
\]
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is a cocycle in the relative
Lie algebra complex for $G$ with values in $\mathcal{S}(V)$. Then $\varphi$ corresponds to a closed differential $r$-form $\tilde{\varphi}$ on the symmetric space $D= G/K$ of dimension $pq$ with values in $\mathcal{S}(V)$.
For a coset of a lattice $\mathcal{L}$ in $V$, we define the theta distribution $\Theta=\Theta_{\mathcal{L}}$ by $\Theta = \sum_{\ell \in \mathcal{L}} \delta_{\ell}$, where $ \delta_{\ell}$ is the delta measure concentrated at $\ell$. It is obvious that $\Theta$ is invariant under $\G = \Stab(\mathcal{L}) \subset G$. There is also a congruence subgroup $\Gamma'$ of $\SL(2,\Z)$) such that $\Theta$ is also invariant under $\Gamma'$. Hence we can apply the theta distribution to $\tilde{\varphi}$ to obtain a closed $r$-form $\theta_{\varphi}$ on $X = \Gamma \backslash D$ given by 
\[
\theta_{\varphi}(\mathcal{L})= \langle \Theta_{\mathcal{L}}, \tilde{\varphi} \rangle.
\]
Assume now in addition that $\varphi$ has weight $k$ under the maximal compact subgroup $\SO(2) \subset \SL_2(\R)$. Then $\theta_{\varphi}$ also gives rise to a (in general) non-holomorphic function on the upper half place $\h$ which is modular of weight $k$ for $\G'$. We may then use $\theta_{\varphi}$ as the kernel of a pairing of modular forms $f$ with (closed) differential $(pq-r)$-forms $\eta$ or $r$-chains (cycles) $C$ in $X$. The resulting pairing in $f$, $\eta$ (or $C$), and $\varphi$ as these objects vary, we call the {\bf geometric theta correspondence}.