Equivariant $\mathbb{R}$-test configurations and semistable limits of $\mathbb{Q}$-Fano group compactifications

Let $G$ be a connected, complex reductive group. In this paper, we classify $G\times G$-equivariant normal $\mathbb{R}$-test configurations of a polarized $G$-compactification. Then for $\mathbb{Q}$-Fano $G$-compactifications, we express the H-invariant of its equivariant normal $\mathbb{R}$-test configurations in terms of the combinatory data. Based on \cite{Han-Li}, we compute the semistable limit of a K-unstable Fano $G$-compactification. As an application, we show that for the two K-unstable Fano $SO_4(\mathbb C)$-compactifications, the corresponding semistable limits are indeed the limit spaces of the normalized K"ahler-Ricci flow.