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Jean-Jacques Loeb
Action d'une forme réelle d'un groupe de Lie complexe sur les fonctions plurisousharmoniques
Annales de l'institut Fourier, 35 no. 4 (1985), p. 59-97, doi: 10.5802/aif.1028
Article PDF | Reviews MR 87c:32035 | Zbl 0563.32013 | 1 citation in Cedram

Résumé - Abstract

Let $G_{{\bf C}}$ be a complex Lie group and $G_{{\bf R}}$ a closed real form of $G_{{\bf C}}$. By definition, a pair $(G_{{\bf C}},G_{{\bf R}})$ is pseudo-convex, if there exists on $G_{{\bf C}}$ a regular function, strictly p.s.h., invariant by $G_{{\bf R}}$, and exhaustive on $G_{{\bf C}}/G_{{\bf R}}$. By definition, $G_{{\bf R}}$ has a purely imaginary specter, if for all $X$ in $(G_{{\bf R}})$, the eigenvalues of ad$\ X$ are purely imaginary. When $G_{{\bf C}}$ has a simply connected radical, this last property is the same as pseudo-convexity of $(G_{{\bf C}},G_{{\bf R}})$. For $(G_{{\bf C}},G_{{\bf R}})$ pseudo-convex and under a discrete subgroup hypothesis, there exists on an invariant open subset $\Omega $, a strictly p.s.h. invariant function, exhaustive on $\Omega /G_{{\bf R}}$. With the same hypothesis, we have the following theorem: ``Let be $\Omega $ a $G_{{\bf R}}$-invariant open subset of de $X\times G_{{\bf C}}$, with connected fibers upon $X$. His protection on $X$ is Stein, when $X$ is Stein". We prove also the non existence of an invariant kählerian metric on $G_{{\bf C}}$, when the specter of $G_{{\bf R}}$ is not purely imaginary. We deduce the non existence of a kählerian metric on some non compact complex nilmanifolds.

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