Eigenvalue selectors for representations of compact connected groups
Alexandru Chirvasitu
TL;DR
The paper classifies when a continuous eigenvalue selector exists for representations of a compact connected group by proving that an irreducible representation selects eigenvalues iff it annihilates $Z_0(G)\cap G'$. It develops a framework based on maximal pro-tori, weights, and coherence to analyze selectors and their behavior under quotients and products, and connects this to the topology of adjoint-orbit spaces via a fiber-bundle description of $G/\mathrm{Ad}$ over the abelianization $G_{ab}$. It establishes a precise criterion, descent and coherence properties for selectors, and a detailed topological description of $G/\mathrm{Ad}$ for compact connected Lie groups, including explicit descriptions for many simple types (e.g., $E_7: D^4*\mathbb{RP}^2$, $E_6: D^2*L^3$). The results illuminate when continuous eigenvalue selectors exist (notably for $\mathrm{SU}(n)$ but not for $\mathrm{U}(n)$) and tie representation-theoretic phenomena to the geometry of adjoint orbits, with concrete implications for classical groups.
Abstract
A representation $ρ$ of a compact group $\mathbb{G}$ selects eigenvalues if there is a continuous circle-valued map on $\mathbb{G}$ assigning an eigenvalue of $ρ(g)$ 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 $\mathrm{U}(n)$.