Singular Kähler-Einstein metrics on Q-Fano compactifications of Lie groups

In this paper, we prove an existence result for Kähler-Einstein metrics on Q-Fano compactifications of Lie groups by the variational method, provided their moment polytopes satisfy a fine condition. As an application, we prove that there is no Q-Fano SO₄( C)-compactification which admits a Kähler-Einstein metric with the same volume as that of a smooth K-unstable Fano SO₄( C)-compactification.