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I am a little bit confused about the definition of hyperbolic $n$-space. How do we see $\mathbb{H}^n$ as a homogeneous space model?

We can think, the Poincare upper half plane $\mathfrak{H}$ as $\mathbb{H}^2$ and also think as $SL(2, \mathbb{R})/SO(2,\mathbb{R}).$ By the help of Iwasawa decomposition we can bring up a coordinate system of $\mathbb{H}^2$.

1) Real hyperbolic space has a symmetric space model of the form $SO^+(n, \mathbb{R})/O(n,\mathbb{R})$. I am confused about the connection of this definition of the previous one. More over, what do we mean by 'symmetric model' and 'homogeneous model'?

2) What is the analogous homogeneous space model of $\mathbb{H}^n$?

3) Can we also think $\mathbb{H}^2=PGL(2,\mathbb{R})/PO(2, \mathbb{R})$?

4) Instead of taking $\mathbb{R}$ if one takes ring of adeles $\mathbb{A}$, will then $SL(2, \mathbb{A})/SO(2,\mathbb{A})$ also be considered as $\mathbb{H}^2$ (because of Iwasawa decomposition we can have exactly similar cordinate system)?

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Real hyperbolic $n$-space is $SO(n,1)/S(O(n)\times O(1))$, and complex hyperbolic $n$-space is $SU(n,1)/S(U(n)\times U(1))$.

There are several anomalous isogenies (finite-to-one) maps from other classical groups to orthogonal groups, which accounts for $SL_1(\mathbb R)\to SO(2,1)$ and $SL_2(\mathbb C)\to SO(3,1)$.

Your point 1) has some problem in it, since the quotient you wrote may not make sense (the alleged subgroup may not really be so...), not to mention that it would be compact, which hyperbolic $n$-space is not. You need "signatures"...

The question about the distinction between "homogeneous" and "symmetric-space" models is that I don't know of any serious distinction, except that "symmetric-space" may require more, e.g., that the subgroup $H$ in a quotient $G/H$ is compact.

2) The models of hyperbolic $n$-spaces are as above.

3) Yes, hyperbolic $2$-space is also expressible in terms of $PGL_2(\mathbb R)$, partly because of the anomalous $SL_2(\mathbb R)\to SO(2,1)$.

4) The adelic analogue is not what you want, because (in part) $SO_2(\mathbb A)$ is not at all a maximal compact subgroup of $SL_2(\mathbb A)$. It is the adelization of the algebraic subgroup $SO(2)$ of $SL_2$, but that's another story.

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  • $\begingroup$ Also, can you kindly provide some reference for hyperbolic n-space in adelic picture? Thanks! $\endgroup$ – Kunnysan Dec 11 '13 at 1:42

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