English

Complete reducibility in bad characteristic

Group Theory 2023-04-18 v1

Abstract

Let GG be a simple algebraic group of exceptional type over an algebraically closed field of characteristic p>0p > 0. This paper continues a long-standing effort to classify the connected reductive subgroups of GG. Having previously completed the classification when pp is sufficiently large, we focus here on the case that pp is bad for GG. We classify the connected reductive subgroups of GG which are not GG-completely reducible, whose simple components have rank at least 33. For each such subgroup XX, we determine the action of XX on the adjoint module L(G)L(G) and on a minimal non-trivial GG-module, and the connected centraliser of XX in GG. As corollaries we obtain information on: subgroups which are maximal among connected reductive subgroups; products of commuting GG-completely reducible subgroups; subgroups with trivial connected centraliser; and subgroups which act indecomposably on an adjoint or minimal module for GG.

Keywords

Cite

@article{arxiv.2304.08388,
  title  = {Complete reducibility in bad characteristic},
  author = {Alastair J. Litterick and Adam R. Thomas},
  journal= {arXiv preprint arXiv:2304.08388},
  year   = {2023}
}

Comments

41 pages, comments welcome

R2 v1 2026-06-28T10:08:34.605Z