An invariant version of Cartan's lemma and complexification of invariant domains of holomorphy
Assume that a connected complex reductive Lie group \(G\) acts holomorphically on a Stein manifold \(X\) and let \(D \subset X\) be a Stein domain which is invariant and orbit connected with respect to some connected closed real form \(G_{R}\) of \(G\). The main assertions of the article are: i) \(G \cdot D\) is Stein if there is a \(G_R\)-invariant plurisubharmonic function on \(G\) which is exhausting on \(G/G_R\). ii) Suppose that \(G \cdot D\) is saturated with respect to the quotient \(X \to X // G\). Then \(G \cdot D\) is Stein if and only if every \(G_{R}\)-invariant holomorphic function defined on a \(G_R\)-invariant analytic subset of \(D\) admits a \(G_{R}\)-invariant holomorphic extension to \(D\). The extended future tube is discussed as an example. It must be said that the paper (or at least the present translation) contains unclear formulations (e.g. the theorem on p.~461) and mistakes (e.g. the proposition on p.~462). For work on related questions and a proof of the extendend future tube conjecture see \textit{P. Heinzner} [Doc. Math. 3, 1-14 (1998; Zbl 0939.32021)].
complexification of invariant Stein domains
scientific article; zbMATH DE number 1557722