On overgroups of distinguished unipotent elements in reductive groups and finite groups of Lie type
Abstract
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 using so-called good $A_1$ subgroups of $G$, introduced by Seitz. 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- July 2024
- DOI:
- 10.48550/arXiv.2407.16379
- arXiv:
- arXiv:2407.16379
- Bibcode:
- 2024arXiv240716379B
- Keywords:
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- Mathematics - Group Theory;
- Mathematics - Representation Theory;
- 20G15;
- 14L24
- E-Print:
- 17 pages