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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

$ k$-regular elements in semisimple algebraic groups


Author: Peter P. Andre
Journal: Trans. Amer. Math. Soc. 201 (1975), 105-124
MSC: Primary 20G30; Secondary 20G20
DOI: https://doi.org/10.1090/S0002-9947-1975-0357637-8
MathSciNet review: 0357637
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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$-regular element in a semisimple algebraic group defined over an arbitrary field of characteristic zero. The existence of semisimple and unipotent $ k$-regular elements in a semisimple algebraic group defined over a field of characteristic zero is proved. The structure of all $ k$-regular unipotent elements is given. The number of minimal parabolic subgroups containing a $ k$-regular element is given. The number of conjugacy classes of $ R$-regular unipotent elements is given, where $ R$ is the real field. The number of conjugacy classes of $ {Q_p}$-regular unipotent elements is shown to be finite, where $ {Q_p}$ is the field of $ p$-adic numbers.


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DOI: https://doi.org/10.1090/S0002-9947-1975-0357637-8
Keywords: Algebraic group, Lie algebra, minimal parabolic subgroup, maximal $ k$-split torus, $ k$-regular element
Article copyright: © Copyright 1975 American Mathematical Society