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On conjugacy classes of elements of finite order in compact or complex semisimple Lie groups


Author: Dragomir Ž. Djoković
Journal: Proc. Amer. Math. Soc. 80 (1980), 181-184
MSC: Primary 20G20; Secondary 22E10
DOI: https://doi.org/10.1090/S0002-9939-1980-0574532-7
MathSciNet review: 574532
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Abstract: If K is a connected compact Lie group with simple Lie algebra and if k is an integer relatively prime to the order of the Weyl group W of K then the number $ \nu (K,k)$ of conjugacy classes of K consisting of elements x satisfying $ {x^k} = 1$ is given by

$\displaystyle \nu (K,k) = \prod\limits_{i = 1}^l {\frac{{{m_i} + k}}{{{m_i} + 1}},} $

where l is the rank of K and $ {m_1}, \ldots ,{m_l}$ are the exponents of W. If G is the complexification of K then we have $ \nu (G,k) = \nu (K,k)$ without any restriction on k.

References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1980-0574532-7
Keywords: Semisimple complex Lie group, compact semisimple group, Weyl group, conjugacy classes, orbits, exponents of the Weyl group
Article copyright: © Copyright 1980 American Mathematical Society