Some remarks on complex Lie groups

Abstract

First we show that any complex Lie group is complete Kähler. Moreover we obtain a plurisubharmonic exhaustion function on a complex Lie group as follows. Let ${\frak k}$ the real Lie algebra of a maximal compact real Lie subgroup $K$ of a complex Lie group $G$. Put $q:=\dim_ {\Bbb C} {\frak k} \cap \sqrt{-1} {\frak k}$. Then we obtain that there exists a plurisubharmonic, strongly $(q + 1)$-pseudoconvex in the sense of Andreotti-Grauert and $K$-invariant exhaustion function on $G$.