Complete reducibility in good characteristic

Abstract

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