A geometric take on Kostant's Convexity Theorem
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866929638375161856 |
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| author | Mendes, Ricardo A. E. |
| author_facet | Mendes, Ricardo A. E. |
| contents | Given a compact Lie group $G$ and an orthogonal $G$-representation $V$, we give a purely metric criterion for a closed subset of the orbit space $V/G$ to have convex pre-image in $V$. In fact, this also holds with the natural quotient map $V\to V/G$ replaced with an arbitrary submetry $V\to X$.
In this context, we introduce a notion of "fat section" which generalizes polar representations, representations of non-trivial copolarity, and isoparametric foliations.
We show that Kostant's Convexity Theorem partially generalizes from polar representations to submetries with a fat section, and give examples illustrating that it does not fully generalize to this situation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_12966 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A geometric take on Kostant's Convexity Theorem Mendes, Ricardo A. E. Metric Geometry Differential Geometry Representation Theory 57S15, 51F99 Given a compact Lie group $G$ and an orthogonal $G$-representation $V$, we give a purely metric criterion for a closed subset of the orbit space $V/G$ to have convex pre-image in $V$. In fact, this also holds with the natural quotient map $V\to V/G$ replaced with an arbitrary submetry $V\to X$. In this context, we introduce a notion of "fat section" which generalizes polar representations, representations of non-trivial copolarity, and isoparametric foliations. We show that Kostant's Convexity Theorem partially generalizes from polar representations to submetries with a fat section, and give examples illustrating that it does not fully generalize to this situation. |
| title | A geometric take on Kostant's Convexity Theorem |
| topic | Metric Geometry Differential Geometry Representation Theory 57S15, 51F99 |
| url | https://arxiv.org/abs/2202.12966 |