On conjugacy classes of elements of finite order in complex semisimple Lie groups
Let G be a connected semi-simple complex Lie group, and k be an integer. The main theme of this paper is the study of the number \(\nu\) (G,k) of conjugacy classes of G consisting of elements whose orders divide k. The author first shows that \(\nu (G^{\vee},k)=\nu (G,k)\) for the dual group \(G^{\vee}\) of G. Thus the case of adjoint groups reduces to the case of simply connected groups. Then he obtains a simple formula for the generating function \(F(G,t)=\sum_{k\geq 0}\nu (G,k)t^ k\) in the case when G is simply connected. In a previous paper, the author shows that \(\nu (G,k)=\prod^{\ell}_{i=1}(m_ i+k)/(m_ i+1),\) where \(m_ i\) (1\(\leq i\leq \ell)\) are exponents of the Weyl group W of G, under the condition that k and the order of W are relatively prime [Proc. Am. Math. Soc. 80, 181-184 (1980; Zbl 0436.22008)]. He shows that for simply connected almost simple G, this formula is also valid under some weaker conditions. Further, when G is a group of type \(A_{\ell}\), an explicit formula for \(\nu\) (G,k) and also some new partition identities are given.
scientific article; zbMATH DE number 3899198
scientific article