Richardson elements for parabolic subgroups of classical groups
in positive characteristic
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 $\p = \l \oplus \u$, where $\u$ is the Lie algebra of
the unipotent radical of $P$ and $\l$ is a Levi complement. Thanks
to a fundamental theorem of R.~W.~Richardson (\cite{Ri}), $P$ 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 (\cite{Ba}), 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.
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