On conjugacy classes of elements of finite order in complex semisimple lie groups

Abstract The main theme of this paper is the study of the number, v ( G , k ), of conjugacy clases of a complex semisimple group G which consist of elements whose orders divide k . We define an involution ∨ on the set of isomorphism classes of such groups and show that ν ( G ∨ , k ) = ν ( G , k ). Explicit formulas for ν ( G , k ) are obtained for all simply connected or adjoint groups, as well as for all groups of type A l . We obtain also some new partition identities which may be of interest to number-theorists.