How to determine all conjugacy classes in the complex orthogonal group $O(n,\mathbb{C})$ of finite order $k$

Question

I was wondering how one would go about determining the conjugacy classes of the complex orthogonal group $O(n,\mathbb{C})$ of some finite order $k$. That is, if $[A]$ is the conjugacy class of $A\in O(n,\mathbb{C})$, then the cyclic group $\langle A\rangle$ generated by $A$ is of order $k$.

Answer 1

I do not have a complete answer but a suggestion. Every finite order element is diagonalizable, and, hence, up to conjugation is contained in a maximal torus $T<G=O(n,C)$. Furthermore, the homomorphism $Z_k\to T$ will extend to a homomorphism $C^*\to T$. Such homomorphisms are called cocharacters. I think, you are looking for the description of all cocharacters up to the action of the Weyl group $W$ of $G$. Such classification is given by: Every cocharacter is equivalent to a dominant cocharacter, the latter are described by dominant coweights of $G$. The issue I am not certain about is how to classify monomorphisms $Z_k\to G$ which come from inequivalent cocharacters.