I want to calculate the distance between two points in Siegel upper half space, but I do not know the explicit formula of the metric (which is invariant under the action of the real symplectic group). Can anyone tell me about that or just tell where I can find it? Thanks a lot!
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$\begingroup$ Hi, welcome to MO. I think you should add more details about your question, some definitions. It'll help everyone to better help you. $\endgroup$– Amir SagivCommented Apr 26, 2017 at 6:22
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6$\begingroup$ 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. $\endgroup$– VenkataramanaCommented Apr 26, 2017 at 6:25
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$\begingroup$ @AmirSagiv Thank you! I'll remember this. $\endgroup$– Zhiwei ZhengCommented Apr 26, 2017 at 14:08
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$\begingroup$ @Venkataramana Thank you! I think I should look at the book you recommended, it must be helpful. $\endgroup$– Zhiwei ZhengCommented Apr 26, 2017 at 14:20
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$\begingroup$ 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. $\endgroup$– Zhiwei ZhengCommented Apr 26, 2017 at 14:46
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An explicit formula is given at the bottom of page 21 of this PhD thesis.
answered Apr 26, 2017 at 9:29