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Geometric interpretation of quantitative instability

Omri N. Solan, Nattalie Tamam

TL;DR

The paper recasts instability for real G-actions on a representation V into a geometric problem on the symmetric space M = K ackslash G, framing the growth of vector norms as convex functions and identifying a fastest shrinking geodesic whose associated Busemann function bounds these growth rates. It establishes that the shrink-rate for a vector v is bounded below by a positive multiple of a Busemann function, with equality when v is a highest weight vector, and proves a precise correspondence between geometric fastest shrinking geodesics and Kempf's algebraic instability data. The authors provide equivalent descriptions of the Busemann framework via parabolic homomorphisms and fundamental representations, thereby connecting geometric and algebraic viewpoints and yielding a geometric proof of Kempf-type results along with an effective version. This geometric interpretation offers intuition and tools for understanding orbit closures, with potential extensions to broader fields and representations, and informs a second proof of a quantitative Kempf theorem by Nimish-Pengyu. Overall, the work bridges convex geometric analysis on Hadamard spaces with classical invariant theory to illuminate instability phenomena in real algebraic group actions.

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.

Geometric interpretation of quantitative instability

TL;DR

The paper recasts instability for real G-actions on a representation V into a geometric problem on the symmetric space M = K ackslash G, framing the growth of vector norms as convex functions and identifying a fastest shrinking geodesic whose associated Busemann function bounds these growth rates. It establishes that the shrink-rate for a vector v is bounded below by a positive multiple of a Busemann function, with equality when v is a highest weight vector, and proves a precise correspondence between geometric fastest shrinking geodesics and Kempf's algebraic instability data. The authors provide equivalent descriptions of the Busemann framework via parabolic homomorphisms and fundamental representations, thereby connecting geometric and algebraic viewpoints and yielding a geometric proof of Kempf-type results along with an effective version. This geometric interpretation offers intuition and tools for understanding orbit closures, with potential extensions to broader fields and representations, and informs a second proof of a quantitative Kempf theorem by Nimish-Pengyu. Overall, the work bridges convex geometric analysis on Hadamard spaces with classical invariant theory to illuminate instability phenomena in real algebraic group actions.

Abstract

Given a real algebraic group acting on a linear space , a vector is called unstable if , where the closure is taken with respect to the Zariski topology. A fundamental theorem of Kempf in geometric invariant theory states that is unstable if and only if there is a one-parameter subgroup of such that is unstable. Assuming is a semisimple real algebraic -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 -spaces, and bound the later from below by Busemann functions.
Paper Structure (29 sections, 34 theorems, 119 equations)

This paper contains 29 sections, 34 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Geometric interpretation
  4. Algebraic properties
  5. On the proof
  6. Further research
  7. Preliminaries
  8. Homomorphisms
  9. Hadamard spaces
  10. Symmetric spaces of non-compact type
  11. Overview
  12. Metric properties
  13. Explicit construction
  14. Example: $G=\operatorname{SL}_n(\mathbb{R})$
  15. Representations of G and short vectors
  16. ...and 14 more sections

Key Result

Theorem 1.1

Assume $G$ is semisimple and let $\varrho:G\to \operatorname{GL}(V)$ be a $\mathbb{Q}$-representation and $0\neq v\in V(\mathbb{Q})$ be an unstable vector, i.e., $0\in \overline{Gv}-Gv$. Then, there exists a $\mathbb{Q}$-highest weight representation $\tilde{\varrho}:G\rightarrow\operatorname{GL}(W)

Theorems & Definitions (115)

  • Theorem 1.1: NimishPengyu
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Conjecture 1.5
  • Definition 2.1: Geodesic
  • Lemma 2.2
  • proof
  • Definition 2.3: Convex functions
  • Definition 2.4: Flats, geometric description
  • ...and 105 more