Irreducibility in algebraic groups and regular unipotent elements

We study (connected) reductive subgroups $G$ of a reductive algebraic group $H$, where $G$ contains a regular unipotent element of $H$. The main result states that $G$ cannot lie in a proper parabolic subgroup of $H$. This result is new even in the classical case $H = \mathrm {SL}(n,F)$, the special linear group over an algebraically closed field, where a regular unipotent element is one whose Jordan normal form consists of a single block. In previous work, Saxl and Seitz (1997) determined the maximal closed positive-dimensional (not necessarily connected) subgroups of simple algebraic groups containing regular unipotent elements. Combining their work with our main result, we classify all reductive subgroups of a simple algebraic group $H$ which contain a regular unipotent element.