First we show that any complex Lie group is complete Kahler. Moreover we ob
tain a plurisubharmonic exhaustion function on a complex Lie group as follo
ws. Let the real Lie algebra of a maximal compact real Lie subgroup K of a
complex Lie group G. Put q := dimC boolean AND root-1. 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.