So let $D\subseteq \mathbb{C}^n$ be a **bounded** connected open set with a transitive action of its group of biholomorphisms (which we denote by $Hol(D)$). Note that **I'm not** assuming that $D$ is symmetric. We thus have that $D$ is "homeomorphic" to $Hol(D)/K$ where $K=Stab(d\_0)$ for some $d\_0\in D$.


In the special case where $Hol(D)$ is a real Lie group and that
$K$ is a maximal compact of $Hol(D)$ then by a theorem of Elie Cartan we have that
$Hol(D)/K$ is homeomoprphic to $\mathbb{R}^m$ and thus contractible. 


Under my assumptions: 


(1) Is $Hol(D)$ always a Lie group?


(2) Is $K$ always a maximal compact?


(3) In general is $D$ always contractible (or simply connected)?



On bounded homogeneous connected domains of C^n
Re question 3: a bounded homogeneous domain is biholomorphic to a Siegel domain, which is contractible. See e.g. [Siegel domain](http://www.encyclopediaofmath.org/index.php?title=Siegel_domain) and references therein (those references probably answer question 2 as well). Another useful link is [Homogeneous bounded domain](http://www.encyclopediaofmath.org/index.php?title=Homogeneous_bounded_domain).


upd: Another Google search gave the following references:


"Homogeneous Bounded Domains and Siegel Domains" by Soji Kaneyuki, Springer LNM 241.


"Theory of complex homogeneous bounded domains" by Yichao Xu, Mathematics and its applications 569.


