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 ν ( K , k ) \nu (K,k) of conjugacy classes of K consisting of elements x satisfying x k = 1 {x^k} = 1 is given by \[ ν ( K , k ) = ∏ i = 1 l m i + k m i + 1 , \nu (K,k) = \prod \limits _{i = 1}^l {\frac {{{m_i} + k}}{{{m_i} + 1}},} \] where l is the rank of K and m 1 , … , m l {m_1}, \ldots ,{m_l} are the exponents of W. If G is the complexification of K then we have ν ( G , k ) = ν ( K , k ) \nu (G,k) = \nu (K,k) without any restriction on k.