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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)].

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