Timeline for Realizing a subgroup of a Lie group as a stabilizer subgroup
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Oct 27, 2022 at 19:37 | comment | added | Nicolò Cavalleri | I know this is false in case $G$ is not compact, but can we still say something in that case? Like further conditions that $H$ needs to satisfy? | |
| Oct 5, 2019 at 20:15 | history | edited | Alex M. | CC BY-SA 4.0 |
Fixed a URL containing an invalid character
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| Nov 27, 2013 at 20:51 | vote | accept | Slava Rychkov | ||
| Nov 27, 2013 at 19:03 | history | edited | Francois Ziegler | CC BY-SA 3.0 |
Add link to Palais
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| Nov 27, 2013 at 18:52 | comment | added | Slava Rychkov | Thanks now I see how it works. It would be nice to know an efficient and explicit way to find $E$ in concrete situations. | |
| Nov 27, 2013 at 18:36 | comment | added | Francois Ziegler | (Proof:) Apply Cor. 1 to the canonical operation of $G$ on the compact manifold $G/H$. This gives an analytic linear representation $\rho : G\to GL(E)$ and an embedding $\varphi : G/H \to E$ such that $\varphi(gx) = \rho(g)\varphi(x)$, $g \in G$, $x \in G/H$. Let $\bar e\in G/H$ be the class of $e \in G$, and $v = \varphi(\bar e)$ its image. For all $g\in G$, we have $\rho(g)v = v \Leftrightarrow \varphi(g\bar e) = \varphi(\bar e) \Leftrightarrow g\bar e = \bar e \Leftrightarrow g \in H$. | |
| Nov 27, 2013 at 18:31 | comment | added | Francois Ziegler | That's because Google will not show the next page, which says: COROLLARY 2. Let $H$ be a closed subgroup of $G$. There exists an analytic linear representation of $G$ on a finite dimensional vector space $E$ and a point $v\in E$ with fixer $H$. | |
| Nov 27, 2013 at 18:26 | comment | added | Slava Rychkov | Could you be a bit more specific how to get the result I need from the Mostow-Palais theorem? I've looked at the Mostow paper and the page from the book that you mention, but I don't see how to apply these results to the group and its subgroup simultaneously ensuring the stabilizer condition. | |
| Nov 27, 2013 at 17:40 | history | edited | Francois Ziegler | CC BY-SA 3.0 |
Add link to Bourbaki
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| Nov 27, 2013 at 17:30 | history | answered | Francois Ziegler | CC BY-SA 3.0 |