Jacobi's theta function from a representation theoretic viewpoint

introduction

$g\in \mathbb{Z}$, $g\geq 1$
Heisenberg group $H(2g, \mathbb{R})$ and its Lie algebra
Schrodinger representation of $H(2g, \mathbb{R})$ on $\mathcal{H}=L^2(\mathbb{R}^g)$
Stone-von Neumann theorem induces an action of $Sp(2g,\mathbb{R})$ on $\mathcal{H}$
but this is only a projective representation
but we can obtain a genuine representation of the metaplectic group and we call it the Weil representation
a smooth vector $f_{\Omega}\in \mathcal{H}_{\infty}$, Schwarz space
a functional $\mu_{\mathbb{Z}}\in \mathcal{H}_{-\infty}$
then $\theta(\mathbf{x},\Omega)$ appears as pairing

$$ \theta(\mathbf{x},\Omega)=\langle U_{(1,x)}f_{\Omega}, \mu_{\mathbb{Z}}\rangle $$

modular transformation properties follows from the action of $Mp(2g,\mathbb{R})$ on $\mathfrak{h}_g$ and $H$