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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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##### References:
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