Some remarks on complex Lie groups

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 the real Lie algebra of a maximal compact real Lie subgroup K of a complex Lie group G. Put q := dimC 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.