# A bounded homogeneous space which fails to be symmetric?

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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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