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Let G be a simple linear algebraic group over an algebraically closed field K of characteristic p≥ 0, let H be a proper closed subgroup of G and let V be a nontrivial finite dimensional irreducible rational KG-module. We say that (G,H, V) is an irreducible triple if V is irreducible as a KH-module. Determining these triples is a fundamental problem in the representation theory of algebraic groups, which arises naturally in the study of the subgroup structure of classical groups. In the 1980s, Seitz and Testerman extended earlier work of Dynkin on connected subgroups in characteristic zero to all algebraically closed fields. In this article we will survey recent advances towards a classification of irreducible triples for all positive dimensional subgroups of simple algebraic groups.
be a simple exceptional algebraic group of adjoint type over an algebraically closed field of characteristic
be a subgroup of
containing a regular unipotent element
. By a theorem of Testerman,
is contained in a connected subgroup of
. In this paper we prove that with two exceptions,
itself is contained in such a subgroup (the exceptions arise when
). This extends earlier work of Seitz and Testerman, who established the containment under some additional conditions on
and the embedding of
. We discuss applications of our main result to the study of the subgroup structure of finite groups of Lie type.