Hurwitz-Radon numbers and proper actions of semisimple Lie groups
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917337187221504 |
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| author | Kannaka, Kazuki Tojo, Koichi |
| author_facet | Kannaka, Kazuki Tojo, Koichi |
| contents | We study proper isometric actions of non-compact semisimple Lie groups on pseudo-Riemannian symmetric spaces. Motivated by Okuda's classification of semisimple symmetric spaces admitting proper $SL(2,\mathbb{R})$-actions [J. Differential Geom., 2013], we focus on symmetric spaces lying on the boundary of the existence of proper $SL(2,\mathbb{R})$-actions. As a rigidity result, we show that any connected non-compact semisimple Lie group acting properly on these symmetric spaces must be globally isomorphic to $Spin(n,1)$ up to compact factors. Moreover, the Hurwitz-Radon number arises as the largest value of $n$ for the existence of $Spin(n,1)$-proper actions. Our symmetric spaces include the pseudo-Riemannian hyperbolic space $\mathbf{H}_{+}^{N,N-1}$ of signature $(N,N-1)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_04544 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hurwitz-Radon numbers and proper actions of semisimple Lie groups Kannaka, Kazuki Tojo, Koichi Differential Geometry Representation Theory Primary: 57S30. Secondary: 15A66, 22E40, 53C35, 53C50, 57S20 We study proper isometric actions of non-compact semisimple Lie groups on pseudo-Riemannian symmetric spaces. Motivated by Okuda's classification of semisimple symmetric spaces admitting proper $SL(2,\mathbb{R})$-actions [J. Differential Geom., 2013], we focus on symmetric spaces lying on the boundary of the existence of proper $SL(2,\mathbb{R})$-actions. As a rigidity result, we show that any connected non-compact semisimple Lie group acting properly on these symmetric spaces must be globally isomorphic to $Spin(n,1)$ up to compact factors. Moreover, the Hurwitz-Radon number arises as the largest value of $n$ for the existence of $Spin(n,1)$-proper actions. Our symmetric spaces include the pseudo-Riemannian hyperbolic space $\mathbf{H}_{+}^{N,N-1}$ of signature $(N,N-1)$. |
| title | Hurwitz-Radon numbers and proper actions of semisimple Lie groups |
| topic | Differential Geometry Representation Theory Primary: 57S30. Secondary: 15A66, 22E40, 53C35, 53C50, 57S20 |
| url | https://arxiv.org/abs/2602.04544 |