论文标题
定量不稳定性的几何解释
Geometric interpretation of quantitative instability
论文作者
论文摘要
给定一个真实的代数组$ g $在线性空间$ v $上作用，如果$ 0 \ in \ in \ overline {gv} -gv $，vector $ v \在v $中称为不稳定，在该{gv} -gv $中，在zariski topology方面进行了关闭。 KEMPF在几何不变理论中的基本定理指出，$ v $是不稳定的，并且只有当时有一个参数子组$ a $ a $ g $的$ g $，因此$ av $是不稳定的。假设$ g $是一个半imple真实代数$ \ mathbb {q} $ - 组，我们使用设置的几何解释为此结果提供了新的证明。在此过程中，我们还提供了Shah和Yang的有效版本的新证明。我们的解释涉及将线性动作下的向量的长度联系起来，以传达在某些$ \ cat $空间上的功能，并通过Busemann函数从下面绑定了函数。
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.
