"Jacobi's theta function from a representation theoretic viewpoint"의 두 판 사이의 차이
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imported>Pythagoras0 |
imported>Pythagoras0 |
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| 1번째 줄: | 1번째 줄: | ||
==introduction== | ==introduction== | ||
* $g\in \mathbb{Z}$, $g\geq 1$ | * $g\in \mathbb{Z}$, $g\geq 1$ | ||
| − | * Heisenberg | + | * Heisenberg group $H(2g, \mathbb{R})$ and its Lie algebra |
* action of $Sp(2g,\mathbb{R})$ | * action of $Sp(2g,\mathbb{R})$ | ||
* Weil representation on $L^2(\mathbb{R}^g)$ | * Weil representation on $L^2(\mathbb{R}^g)$ | ||
| − | * a smooth vector $f_{\Omega}\in \mathcal{H}_{\infty}$ | + | * a smooth vector $f_{\Omega}\in \mathcal{H}_{\infty}$, Schwarz space |
* a functional $\mu_{\mathbb{Z}}\in \mathcal{H}_{-\infty}$ | * a functional $\mu_{\mathbb{Z}}\in \mathcal{H}_{-\infty}$ | ||
* then $\theta(\mathbf{x},\Omega)$ appears as pairing | * then $\theta(\mathbf{x},\Omega)$ appears as pairing | ||
2015년 5월 19일 (화) 07:52 판
introduction
- $g\in \mathbb{Z}$, $g\geq 1$
- Heisenberg group $H(2g, \mathbb{R})$ and its Lie algebra
- action of $Sp(2g,\mathbb{R})$
- Weil representation on $L^2(\mathbb{R}^g)$
- 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 $H$