A Frobenius-Type Formula for Compact Lie Groups

Let $G$ be a group and $\alpha :G\times G\to G$ denote the commutator map. In the case of finite groups, Frobenius gave the formula to compute the cardinalities of the fibres ${\alpha }^{-1}(g)$ in terms of the character values $\chi (g)$ for irreducible characters $\chi$ of $G$. We generalise this formula to compact Lie groups. Further, we connect this generalised formula to the commutator probability of the concerned groups.