Richardson elements for parabolic subgroups of classical groups in positive characteristic

Abstract

Let $G$ be a simple algebraic group of classical type over an algebraically closed field $k$. Let $P$ be a parabolic subgroup of $G$ and let $\p = \Lie P$ be the Lie algebra of $P$ with Levi decomposition $\p = ł\oplus \u$, where $\u$ is the Lie algebra of the unipotent radical of $P$ and $ł$ is a Levi complement. Thanks to a fundamental theorem of R. W. Richardson, $P$ acts on $\u$ with an open dense orbit; this orbit is called the {\em Richardson orbit} and its elements are called {\em Richardson elements}. Recently, the first author gave constructions of Richardson elements in the case $k = \C$ for many parabolic subgroups $P$ of $G$. In this note, we observe that these constructions remain valid for any algebraically closed field $k$ of characteristic not equal to 2 and we give constructions of Richardson elements for the remaining parabolic subgroups.