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Do we have examples of a contractible bounded open set $D\subseteq\mathbf{C}^n$ such that $Hol(D)$ (the group of biholomorphisms $f:D\rightarrow D$) acts transitively on $D$ but such that there exists no symmetry at a given point $x\in D$ (so at all points by homogeneity). By a symmetry at $x$ I mean an element $s\in Hol(D)$ such that in a small neighborhood of $x$ only $x$ is fixed and $s^2=1$.

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up vote 7 down vote accepted

There are some examples given by Pjateckiĭ-Šapiro in Classification of bounded homogeneous regions in n-dimensional complex space. Dokl. Akad. Nauk SSSR 141 1961 316–319. and On bounded homogeneous domains in an n-dimensional complex space. Izv. Akad. Nauk SSSR Ser. Mat. 26 1962 107–124.

(I have a vague memory that the smallest examples are 4-dimensional, but might have misremembered.)

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Thank you so much for the reference! So it seems that in complex dimension less than or equal to $3$ all bounded homogeneous domains $D$ as defined above are automatically symmetric. One may probably list all such domains and then give a proof by inspection,I guess. –  Hugo Chapdelaine Dec 23 '10 at 3:57
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E.Cartan proved in 1936 that for dimension 1 and 2 bounded homogeneous spaces are symmetric. For dimension 3 he did not publish the proof considering it loo long in comparison to the interest of the result. This has now changed with P-Sapiro's example for dimension 4. So the proof for dimension 3 is presumably somewhere in E.Cartan's Nachlass, unpublished.

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Indeed, (via Marcel Berger) I begged a copy of this unpublished manuscript of Cartan père from Henri Cartan and have spent some time looking at it, though I can't claim to have gone through it in detail. As you might imagine, since É. Cartan regarded it as too long in comparison with the interest of the result, it contains a large number of nontrivial, complicated calculations. –  Robert Bryant Feb 19 '12 at 2:45
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Welcome to MO Professor Helgason! –  Suvrit Feb 19 '12 at 3:27
    
Thanks Prof. Helgason for the short genealogy of the problem. By the way, I discovered a nice paper of Borel: Les fonctions automorphes de plusieurs variables complexes, where on p. 177 he mentions Cartan's result in the case n=3 and points out that the problem was still open (in 1952) for n=4. –  Hugo Chapdelaine Feb 23 '12 at 12:25
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