On invariant domains of holomorphy. (English) Zbl 0827.32029

Jakóbczak, Piotr (ed.) et al., Topics in complex analysis. Proceedings of the semester on complex analysis, held in autumn of 1992 at the International Banach Center in Warsaw, Poland. Warszawa: Polish Academy of Sciences, Institute of Mathematics, Banach Cent. Publ. 31, 349-357 (1995).
This paper is an exposition of recent results of P. Heizner and the author concerning \(K\)-invariant domains of holomorphy. Here \(K\) is a compact Lie group acting by biholomorphic automorphisms on the domain \(D \subset \mathbb{C}^n\). \({\mathcal O}^K(D)\) denotes the class of \(K\)- invariant holomorphic functions.
The domain \(D\) is orbit convex if for any \(z \in D\) and any \(b \in ik\)- Lie algebra of \(K\), the inclusion \(\exp (b) \cdot z \in D\) implies \(\exp (tb) \cdot z \in D\) for all \(0 \leq t \leq 1\). Here \(\exp : k^\mathbb{C} \to K^\mathbb{C}\) is the exponential mapping of the corresponding complexifications. Complexification of the domain \(D\) is a domain \(D_\mathbb{C} = K^\mathbb{C} \cdot D\) i.e. the image of \(D\) under \(K^\mathbb{C}\) action. One of the results is the following.
Theorem 1. Let \(D\) be a \(K\)-invariant orbit convex domain in \(\mathbb{C}^n\). Then any \(K\)-invariant holomorphic function \(f\) on \(D\) can be extended to a \(K^\mathbb{C}\)-invariant holomorphic function \(\widehat f\) on \(D_\mathbb{C}\).
For the entire collection see [Zbl 0816.00022].


32M05 Complex Lie groups, group actions on complex spaces
32D05 Domains of holomorphy