On good $A_1$ subgroups, Springer maps, and overgroups of distinguished unipotent elements in reductive groups
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
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2024
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| _version_ | 1866914979886661632 |
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| author | Bate, Michael Böhm, Sören Martin, Benjamin Roehrle, Gerhard |
| author_facet | Bate, Michael Böhm, Sören Martin, Benjamin Roehrle, Gerhard |
| contents | Suppose $G$ is a simple algebraic group defined over an algebraically closed field of good characteristic $p$. In 2018 Korhonen showed that if $H$ is a connected reductive subgroup of $G$ which contains a distinguished unipotent element $u$ of $G$ of order $p$, then $H$ is $G$-irreducible in the sense of Serre. We present a short and uniform proof of this result under an extra hypothesis using so-called good $A_1$ subgroups of $G$, introduced by Seitz. In the process we prove some new results about good $A_1$ subgroups of $G$ and their properties. We also formulate a counterpart of Korhonen's theorem for overgroups of $u$ which are finite groups of Lie type. Moreover, we generalize both results above by removing the restriction on the order of $u$ under a mild condition on $p$ depending on the rank of $G$, and we present an analogue of Korhonen's theorem for Lie algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_16379 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On good $A_1$ subgroups, Springer maps, and overgroups of distinguished unipotent elements in reductive groups Bate, Michael Böhm, Sören Martin, Benjamin Roehrle, Gerhard Group Theory Representation Theory 20G15, 14L24 Suppose $G$ is a simple algebraic group defined over an algebraically closed field of good characteristic $p$. In 2018 Korhonen showed that if $H$ is a connected reductive subgroup of $G$ which contains a distinguished unipotent element $u$ of $G$ of order $p$, then $H$ is $G$-irreducible in the sense of Serre. We present a short and uniform proof of this result under an extra hypothesis using so-called good $A_1$ subgroups of $G$, introduced by Seitz. In the process we prove some new results about good $A_1$ subgroups of $G$ and their properties. We also formulate a counterpart of Korhonen's theorem for overgroups of $u$ which are finite groups of Lie type. Moreover, we generalize both results above by removing the restriction on the order of $u$ under a mild condition on $p$ depending on the rank of $G$, and we present an analogue of Korhonen's theorem for Lie algebras. |
| title | On good $A_1$ subgroups, Springer maps, and overgroups of distinguished unipotent elements in reductive groups |
| topic | Group Theory Representation Theory 20G15, 14L24 |
| url | https://arxiv.org/abs/2407.16379 |