Jacobi's theta function from a representation theoretic viewpoint

abstract

Jacobi introduced his theta functions to develop the theory of elliptic functions. Jacobi's theta function has two notable properties : quasi-periodicity and modularity. Weil's approach to theta functions opened up the way to understand these classical transformation properties from a representation theoretic viewpoint, which paved the way to the theory of Howe duality. This involves the Heisenberg group, the Stone-Von Neumann theorem and Weil representations of metapletic groups. In this talk, I will give an introduction to this topic.

$g\in \mathbb{Z}$, $g\geq 1$
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
we can turn it into a genuine representation of the metaplectic group and we call it the Weil representation
a smooth vector $f_{\Omega}\in \mathcal{H}_{\infty}$, Schwartz space
a functional $\mu_{\mathbb{Z}}\in \mathcal{H}_{-\infty}$
let $\mathbf{x}=(x_1,x_2)$ and $\underline{x}=\Omega x_1+x_2$
$\theta(\underline{x},\Omega)$ appears as pairing

$$ \langle U_{(1,x)}f_{\Omega}, \mu_{\mathbb{Z}}\rangle=c\exp(\pi i ^tx_1 \mathbf{x})\theta(\underline{x},\Omega) $$

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