"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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| 6번째 줄: | 6번째 줄: | ||
** but this is only a projective representation | ** 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 | * we can turn it into a genuine representation of the metaplectic group and we call it the Weil representation | ||
| − | * a smooth vector $f_{\Omega}\in \mathcal{H}_{\infty}$, | + | * a smooth vector $f_{\Omega}\in \mathcal{H}_{\infty}$, Schwartz space |
* a functional $\mu_{\mathbb{Z}}\in \mathcal{H}_{-\infty}$ | * 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 $H$ | * modular transformation properties follows from the action of $Mp(2g,\mathbb{R})$ on $\mathfrak{h}_g$ and $H$ | ||
2015년 5월 19일 (화) 09:36 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
- 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 $H$