Geometric interpretation of quantitative instability

Abstract

Given a real algebraic group $G$ acting on a linear space $V$, a vector $v\in V$ is called unstable if $0\in \overline{Gv}-Gv$, where the closure is taken with respect to the Zariski topology. A fundamental theorem of Kempf in geometric invariant theory states that $v$ is unstable if and only if there is a one-parameter subgroup $A$ of $G$ such that $Av$ is unstable. Assuming $G$ is a semisimple real algebraic $\mathbb{Q}$-group, we give a new proof to this result using a geometric interpretation of the setting. In the process, we also give a new proof of an effective version of this result by Shah and Yang. Our interpretation involves relating the length of vectors under a linear action to convex functions on certain $\cat$-spaces, and bound the later from below by Busemann functions.