Eigenvalue selectors for representations of compact connected groups

A representation $\rho$ of a compact group $\mathbb{G}$ selects eigenvalues if there is a continuous circle-valued map on $\mathbb{G}$ assigning an eigenvalue of $\rho(g)$ to every $g\in \mathbb{G}$. For every compact connected $\mathbb{G}$, we characterize the irreducible $\mathbb{G}$-representations which select eigenvalues as precisely those annihilating the intersection $Z_0(\mathbb{G})\cap \mathbb{G}'$ of the connected center of $\mathbb{G}$ with its derived subgroup. The result applies more generally to finite-spectrum representations isotypic on $Z_0(\mathbb{G})$, and recovers as applications (noted in prior work) the existence of a continuous eigenvalue selector for the natural representation of $\mathrm{SU}(n)$ and the non-existence of such a selector for $\mathrm{U}(n)$.