"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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* $g\in \mathbb{Z}$, $g\geq 1$ | * $g\in \mathbb{Z}$, $g\geq 1$ | ||
* Heisenberg group $H(2g, \mathbb{R})$ and its Lie algebra | * 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 smooth vector $f_{\Omega}\in \mathcal{H}_{\infty}$, Schwarz space | ||
* a functional $\mu_{\mathbb{Z}}\in \mathcal{H}_{-\infty}$ | * a functional $\mu_{\mathbb{Z}}\in \mathcal{H}_{-\infty}$ | ||
| 11번째 줄: | 13번째 줄: | ||
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* modular transformation properties follows from the action of $Mp(2g,\mathbb{R})$ on $\mathfrak{h}_g$ and $H$ | * modular transformation properties follows from the action of $Mp(2g,\mathbb{R})$ on $\mathfrak{h}_g$ and $H$ | ||
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==related items== | ==related items== | ||
2015년 5월 19일 (화) 09:31 판
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 $H$