K\"ahler-Einstein metrics and Ding functional on $\mathbb Q$-Fano group compactifications

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.