Complete reducibility in good characteristic

Litterick, Alastair; Thomas, Adam

doi:10.1090/tran/7085

Complete reducibility in good characteristic
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by Alastair J. Litterick and Adam R. Thomas PDF

Trans. Amer. Math. Soc. 370 (2018), 5279-5340 Request permission

Abstract:

Let $G$ be a simple algebraic group of exceptional type, over an algebraically closed field of characteristic $p \ge 0$. A closed subgroup $H$ of $G$ is called $G$-completely reducible ($G$-cr) if whenever $H$ is contained in a parabolic subgroup $P$ of $G$, it is contained in a Levi subgroup of $P$. In this paper we determine the $G$-conjugacy classes of non-$G$-cr simple connected subgroups of $G$ when $p$ is good for $G$. For each such subgroup $X$, we determine the action of $X$ on the adjoint module $L(G)$ and the connected centraliser of $X$ in $G$. As a consequence we classify all non-$G$-cr connected reductive subgroups of $G$, and determine their connected centralisers. We also classify the subgroups of $G$ which are maximal among connected reductive subgroups, but not maximal among all connected subgroups.

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