Equivariant $\mathbb{R}$-test configurations of polarized spherical varieties

Let $G$ be a connected, complex reductive Lie group and $G\mathrm{/}H$ a spherical homogenous space. Let $(X,L)$ be a polarized $G$-variety which is a spherical embedding of $G\mathrm{/}H$. In this paper we classify $G$-equivariant normal $\mathbb{R}$-test configurations of $(X,L)$ via combinatory data. In particular we classify the special ones, and prove a finiteness theorem of central fibres of $G$-equivariant special $\mathbb{R}$-test configurations. Also, as an application we study the semistable degeneration problem of a $\mathbb{Q}$-Fano spherical variety.