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Fonctions plurisousharmoniques sur un groupe de Lie complexe invariantes par une forme réelle. (Invariant plurisubharmonic functions by a real form on a complex Lie group). (French) Zbl 0616.31006
We give a condition on a real form $$G_{{\mathbb{R}}}$$ of a complex Lie group $$G_{{\mathbb{C}}}$$ for the existence on $$G_{{\mathbb{C}}}$$ of a regular strictly plurisubharmonic function which is invariant by the right action of $$G_{{\mathbb{R}}}$$, and which is exhaustive on $$G_{{\mathbb{C}}}/G_{{\mathbb{R}}}$$ (Theorem 1). In the non-existence case, there are stronger results (Propositions 1 and 2, Theorem 2,i). As an application, we show the existence of non-compact complex solve-manifolds without Kählerian structure. This last result has some relation with A. T. Huckleberry and B. Gilligan’s work [Math. Ann. 238, 39- 49 (1978; Zbl 0405.32009)].

##### MSC:
 31C10 Pluriharmonic and plurisubharmonic functions 32U05 Plurisubharmonic functions and generalizations 22E10 General properties and structure of complex Lie groups