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Some remarks on complex Lie groups. (English) Zbl 0957.32010
Two main results are shown in this paper. First it is shown that there exists a complete Kähler metric on any complex Lie group. Second one obtains a plurisubharmonic exhaustion function on any complex Lie group as follows. Let \(k\) the real Lie algebra of a maximal compact real Lie subgroup \(K\) of a complex Lie group \(G\). Put \(q:=\dim_\mathbb{C} k\cap \sqrt{-1}k\). Then one obtains a plurisubharmonic, strongly \((q+1)\)-pseudoconvex – in the sense of Andreotti-Grauert – and \(K\)-invariant exhaustion function on \(G\), using an integral method with respect to Haar measure on \(G\).
Reviewer: H.Kazama (Fukuoka)

MSC:
32M05 Complex Lie groups, group actions on complex spaces
32U10 Plurisubharmonic exhaustion functions
32F10 \(q\)-convexity, \(q\)-concavity
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