Jacobi's theta function from a representation theoretic viewpoint
imported>Pythagoras0님의 2015년 5월 19일 (화) 07:42 판
introduction
- $g\in \mathbb{Z}$, $g\geq 1$
- Heisenberg algebra and group $H$
- Weil representation on $L^2(\mathbb{R}^g)$
- a smooth vector $f_{\Omega}\in \mathcal{H}_{\infty}$
- 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 $H$