  In this paper, we study Mabuchi's K-energy on a compactification M of a
reductive Lie group G, which is a complexification of its maximal compact
subgroup K. We give a criterion for the properness of K-energy on the space of
K \times K-invariant Kahler potentials. In particular, it turns to give an
alternative proof of Delcroix's theorem for the existence of Kahler-Einstein
metrics in case of Fano manifolds M . We also study the existence of minimizers
of K-energy for general Kahler classes of M.
