  Let $G$ be a complex, connect reductive Lie group which is the
complexification of a compact Lie group $K$. Let $M$ be a $\mathbb Q$-Fano
$G$-compactification. In this paper, we first prove the uniqueness of $K\times
K$-invariant (singular) K\"ahler-Einstein metric. Then we show the existence of
(singular) K\"ahler-Einstein metric implies properness of the reduced Ding
functional. Finally, we show that the barycenter condition is also necessary of
properness.
