A Hilbert--Mumford criterion for polystability in Kaehler geometry

Consider a Hamiltonian action by biholomorphisms of a compact Lie group $K$ on a Kaehler manifold $X$, with moment map $\mu :X\to {\text{klie}}^{\ast }$. We characterize which orbits of the complexified action of $G=K^{\text{CC}}$ in $X$ intersect ${\mu }^{-1}(0)$ in terms of the maximal weights $\lim_{t\to\infty}\la\mu(e^{\imag ts}\cdot x),s\ra$,where$s$belongstotheLiealgebraof$K$. We do not impose any a priori restriction on the stabilizer of $x$. Assuming some mild growth conditions on the action of $K$ on $X$, we view the maximal weights as defining a maps ${\lambda }_x$ from the boundary at infinity of the symmetric space $K\mathrm{\backslash }G$ to $\text{RR}\cup \{\mathrm{\infty }\}$. We prove that $G\cdot x$ meets ${\mu }^{-1}(0)$ 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 $K\mathrm{\backslash }G$ to another boundary point $y^{\mathrm{\prime }}$ satisfying ${\lambda }_x(y^{\mathrm{\prime }})=0$. We also prove that ${\lambda }_{g\cdot x}(y)={\lambda }_x(y\cdot g)$ for any $g\in G$ and $y\in {\mathrm{\partial }}_{\mathrm{\infty }}(K\mathrm{\backslash }G)$.