On the boundedness of invariant hyperbolic domains (Q784373)

The main result is a generalization of a result of \textit{A. Kodama} [Proc. Japan Acad., Ser. A 58, 227--230 (1982; Zbl 0515.32011)]. Specifically, let \(K\) be a compact Lie group with a Lie group homomorphism \(\rho\colon K\to\mathrm{GL}({\mathbb C}^n)\), which naturally defines a representation of \(K\) in the space of entire functions \(\mathcal{O}({\mathbb C}^n)\). The authors prove that if every \(K\)-invariant entire function is constant and \(\Omega\subseteq {\mathbb C}^n\) is a \(K\)-invariant orbit convex domain with \(0\in\Omega\), then \(\Omega\) is bounded if and only if it is Kobayashi hyperbolic.