Invariant plurisubharmonic functions (Q2731415)

Kiselman's minimum principle [\textit{C. O. Kiselman}, Invent. Math. 49, 137-148 (1978;NEWLINENEWLINENEWLINEZbl 0388.32009)] is extended to plurisubharmonic functions on \(M \times G\), \(M\) being a complex variety and \(G\) a connected complex Lie group admitting a real form \(H\). Let \(\Omega\) be a domain in \(M\times G\), invariant by the right action of \(H:(x,g)\in \Omega\to (x,gh)\in \Omega\) \(\forall h\in H\); let \(\omega\) be the projection of \(\Omega\) to \(M\), \(\dot\Omega_x\) be the image of the section \(\Omega_x\) of \(\Omega\) over \(x\in\omega\) under the canonical map \(\pi:G \to G/H\), and \(\dot\Omega\) be the image of \(\Omega\) under the map \((id_M, \pi)\). Assuming each \(\dot\Omega_x\) connected, the following is proved. If \(u\) is a smooth plurisubharmonic function on \(\Omega\) such that \(u(x,\cdot)\) is strictly plurisubharmonic and invariant by \(H\) on \(\Omega_x\) and the induced function \(\dot u(x,\cdot)\) is exhausting on \(\dot\Omega_x\), then \(v(x)= \inf_gu(x,g)\) is a smooth plurisubharmonic function on \(\omega\). As a consequence, the existence of a continuous, strictly plurisubharmonic and invariant function \(\varphi\) on \(\Omega\) such that \(\dot\varphi\) exhausts \(\dot\Omega\), implies that \(\omega\) is a pseudoconvex domain in \(M\).NEWLINENEWLINENEWLINERelated results were obtained by \textit{J. J. Loeb} [Ann. Inst. Fourier 35, No. 4, 59-97 (1985; Zbl 0563.32013)] and \textit{H. Kazama} and \textit{T. Umeno} [Math. Rep. Coll. Gen. Educ., Kyushu Univ. 19, 9-16 (1993; Zbl 0808.32019)].