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 ${\mathfrak p} = \text{Lie } P$ be the Lie algebra of P with Levi decomposition ${\mathfrak p} = {\mathfrak l} \oplus {\mathfrak u}$ , where ${\mathfrak u}$ is the Lie algebra of the unipotent radical of P and ł is a Levi complement. Thanks to a fundamental theorem of Richardson (Bull. London Math. Soc. 6:21–24, 1974), P acts on ${\mathfrak u}$ with an open dense orbit; this orbit is called the Richardson orbit and its elements are called Richardson elements. Recently Baur (J. Algebra 297(1):168–185, 2006), the first author gave constructions of Richardson elements in the case $k = {\mathbb C}$ 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.