𝑘-regular elements in semisimple algebraic groups

Abstract

In this paper, Steinberg’s concept of a regular element in a semisimple algebraic group defined over an algebraically closed field is generalized to the concept of a k k -regular element in a semisimple algebraic group defined over an arbitrary field of characteristic zero. The existence of semisimple and unipotent k k -regular elements in a semisimple algebraic group defined over a field of characteristic zero is proved. The structure of all k k -regular unipotent elements is given. The number of minimal parabolic subgroups containing a k k -regular element is given. The number of conjugacy classes of R R -regular unipotent elements is given, where R R is the real field. The number of conjugacy classes of Q p {Q_p} -regular unipotent elements is shown to be finite, where Q p {Q_p} is the field of p p -adic numbers.