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

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 \[ \nu (K,k) = \prod \limits _{i = 1}^l {\frac {{{m_i} + k}}{{{m_i} + 1}},} \] 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.