Geometric interpretation of quantitative instability
File(s)
Published version
Author(s)
Solan, Omri N
Tamam, Nattalie
Type
Journal Article
Abstract
Given a real algebraic group action on a linear space G ü V, a vector v ∈ V is called
unstable if 0 ∈ Gv − Gv, where the closure is taken with respect to the Zariski topology. A
fundamental theorem of Kempf [21] in geometric invariant theory states that v is unstable
if and only if there is a one-parameter subgroup A of G such that v is unstable with respect
to it, i.e., 0 ∈ Av − Av. Assuming G is a semisimple real algebraic group defined over
Q, 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 [30,
Prop. 2.2]. Our interpretation involves relating the length of vectors under a linear action to
convex functions on certain CAT(0)-spaces, and bound the latter from below by Busemann
functions.
unstable if 0 ∈ Gv − Gv, where the closure is taken with respect to the Zariski topology. A
fundamental theorem of Kempf [21] in geometric invariant theory states that v is unstable
if and only if there is a one-parameter subgroup A of G such that v is unstable with respect
to it, i.e., 0 ∈ Av − Av. Assuming G is a semisimple real algebraic group defined over
Q, 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 [30,
Prop. 2.2]. Our interpretation involves relating the length of vectors under a linear action to
convex functions on certain CAT(0)-spaces, and bound the latter from below by Busemann
functions.
Date Issued
2025-10-01
Date Acceptance
2025-08-12
Citation
Geometriae Dedicata, 2025, 219 (5)
ISSN
0046-5755
Publisher
Springer
Journal / Book Title
Geometriae Dedicata
Volume
219
Issue
5
Copyright Statement
© The Author(s) 2025. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Subjects
DIVERGENT TRAJECTORIES
FLOWS
HOMOGENEOUS SPACES
Mathematics
ORBITS
Physical Sciences
Science & Technology
Publication Status
Published
Article Number
78
Date Publish Online
2025-09-08