A Hilbert-Mumford criterion for polystability in Kaehler geometry

Consider a Hamiltonian action of a compact Lie group $K$ on a Kaehler manifold $X$ with moment map $\mu:X\to\mathfrak{k}^*$. Assume that the action of $K$ extends to a holomorphic action of the complexification $G$ of $K$. We characterize which $G$-orbits in $X$ intersect $\mu^{-1}(0)$ in terms of the maximal weights $\lim_{t\to\infty}\langle\mu(e^{\mathbf{i} ts}\cdot x),s\rangle$, where $s\in\mathfrak{k}$. We do not impose any a priori restriction on the stabilizer of $x$. Under some mild restrictions on the action $K\circlearrowright X$, we view the maximal weights as defining a collection of maps: for each $x\in X$,

$$\lambda_x:\partial_{\infty}(K\backslash G)\to\mathbb{R}\cup\{\infty\},$$

where $\partial_{\infty}(K\backslash G)$ is the boundary at infinity of the symmetric space $K\backslash G$. We prove that $G\cdot x\cap\mu^{-1}(0)\neq\emptyset$ if: (1) $\lambda_x$ is everywhere nonnegative, (2) any boundary point $y$ such that $\lambda_x(y)=0$ can be connected with a geodesic in $K\backslash G$ to another boundary point $y'$ satisfying $\lambda_x(y')=0$. We also prove that the maximal weight functions are $G$-equivariant: for any $g\in G$ and any $y\in \partial_{\infty}(K\backslash G)$ we have $\lambda_{g\cdot x}(y)=\lambda_x(y\cdot g)$.