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Conjugacy Classes of Maximal Tori in Simple Real Algebraic Groups and Applications

Published online by Cambridge University Press:  20 November 2018

Dragomir Ž. Doković  and
Nguyêñ Quôć Thăńg
Dragomir Ž. Doković
Affiliation:
Department of Pure Mathematics University of Waterloo Waterloo, Ontario N2L 3G1
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Abstract

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Let G be an almost simple complex algebraic group defined over R, and let G(R) be the group of real points of G. We enumerate the G(R)-conjugacy classes of maximal R-tori of G. Each of these conjugacy classes is also a single G(R)˚-conjugacy class, where G(R)˚ is the identity component of G(R), viewed as a real Lie group. As a consequence we also obtain a new and short proof of the Kostant-Sugiura's theorem on conjugacy classes of Cartan subalgebras in simple real Lie algebras.

A connected real Lie group P is said to be weakly exponential (w.e.) if the image of its exponential map is dense in P. This concept was introduced in [HM] where also the question of identifying all w.e. almost simple real Lie groups was raised. By using a theorem of A. Borel and our classification of maximal R-tori we answer the above question when P is of the form G(R)˚.

Keywords

20G2022E4522E46
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 46 , Issue 4 , 01 August 1994 , pp. 699 - 717
Copyright
Copyright © Canadian Mathematical Society 1994

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