Richardson elements for parabolic subgroups of classical groups in positive characteristic

Details Richardson elements for parabolic subgroups of classical groups in positive characteristic Richardson elements for parabolic subgroups of classical groups in positive characteristic

2006-08-31

arxiv.org/abs/math/0608775v1

Algebr. Represent. Theory 11 (2008), no. 3, 275--297

Mathematics

Representation Theory

20 pages

Scientific paper

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, $P$ 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.

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