On bounded homogeneous connected domains of C^n - MathOverflow

On bounded homogeneous connected domains of C^n

2011-09-08T22:30:56Z

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)?

Answer by Gjergji Zaimi for On bounded homogeneous connected domains of C^n

2011-09-08T23:46:28Z

It is a theorem of H. Cartan that $Hol(D)$ for any bounded such $D\subset \mathbb C^n$ is a finite dimensional real Lie group. See for example chapter 9 of "Several complex variables" by R. Narasimhan.

Answer by algori for On bounded homogeneous connected domains of C^n

2011-09-09T00:40:26Z

Re question 3: a bounded homogeneous domain is biholomorphic to a Siegel domain, which is contractible. See e.g. Siegel domain and references therein (those references probably answer question 2 as well). Another useful link is 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.