Geometric interpretation of quantitative instability

TL;DR

This paper provides a new geometric proof of Kempf's instability theorem for real algebraic groups, relating vector instability to convex functions on $ ext{CAT}$-spaces and Busemann functions.

Contribution

It introduces a geometric interpretation of instability, offering an alternative proof and an effective version of Kempf's theorem for semisimple real algebraic groups.

Findings

New geometric proof of Kempf's instability theorem

Relation of vector lengths to convex functions on $ ext{CAT}$-spaces

Bounded convex functions by Busemann functions

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.