K-energy on polarized compactifications of Lie groups

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×K-invariant Kähler potentials. In particular, it turns to give an alternative proof of Delcroix's theorem for the existence of Kähler-Einstein metrics in case of Fano manifolds M. We also study the existence of minimizers of K-energy for general Kähler classes of M.