Timeline for Metric on Siegel upper half space
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Apr 26, 2017 at 14:46 | comment | added | Zhiwei Zheng | I know that one can use the Killing form on $G$ to define a $G$-invariant metric on $G/K$ when $G$ is semi-simple and $K$ is a maximal compact subgroup. Our case is $G=PSL(2g, \mathbb{R})$, acting on siegel upper half space $S_g$ with $K$ the stabilizer of a point $p$ in $S_g$. Be precise, we should have a decomposition of the tangent spaces (lie algebras) $\mathfrak{g}_G=\mathfrak{g}_K\oplus T_p S_g$, and the restriction of the Killing form on $T_p S_g$ is positive definite. My feeling is by making this relation explicitly one should be able to get the formula of the metric. | |
| Apr 26, 2017 at 14:20 | comment | added | Zhiwei Zheng | @Venkataramana Thank you! I think I should look at the book you recommended, it must be helpful. | |
| Apr 26, 2017 at 14:08 | comment | added | Zhiwei Zheng | @AmirSagiv Thank you! I'll remember this. | |
| Apr 26, 2017 at 13:58 | vote | accept | Zhiwei Zheng | ||
| Apr 26, 2017 at 13:58 | vote | accept | Zhiwei Zheng | ||
| Apr 26, 2017 at 13:58 | |||||
| Apr 26, 2017 at 9:29 | answer | added | David Loeffler | timeline score: 2 | |
| Apr 26, 2017 at 6:35 | review | Close votes | |||
| Apr 26, 2017 at 6:55 | |||||
| S Apr 26, 2017 at 6:33 | history | suggested | Amir Sagiv | CC BY-SA 3.0 |
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| Apr 26, 2017 at 6:25 | comment | added | Venkataramana | I do not know why this was downvoted; it seems like a legitimate question (maybe easy for some, but not for all). In any case, to find the invariant distance on $G/K$, it is enough to find the distance from the identity coset $o$ to any other point $p$. By again changing by an element of $K$ and using $G=KAK$ (Cartan decomposition), you may assume that $p=a(o)$ for some $a\in A$. Then the formulae involve logs in the entries od the diagonal $a$. I am sure this is done in Helgason's book, but cannot find the precise chapter and verse. | |
| Apr 26, 2017 at 6:23 | review | Suggested edits | |||
| S Apr 26, 2017 at 6:33 | |||||
| Apr 26, 2017 at 6:22 | comment | added | Amir Sagiv | Hi, welcome to MO. I think you should add more details about your question, some definitions. It'll help everyone to better help you. | |
| Apr 26, 2017 at 6:12 | review | First posts | |||
| Apr 26, 2017 at 6:23 | |||||
| Apr 26, 2017 at 6:10 | history | asked | Zhiwei Zheng | CC BY-SA 3.0 |