Eigenvalue selectors for representations of compact connected groups

TL;DR

This paper characterizes which irreducible representations of compact connected groups admit a continuous eigenvalue selector, linking this property to the group's algebraic structure, with applications to special unitary and unitary groups.

Contribution

It provides a complete characterization of eigenvalue-selecting irreducible representations for compact connected groups based on their algebraic properties.

Findings

Eigenvalue selectors exist precisely for representations annihilating the intersection of the connected center and derived subgroup.

The result applies to finite-spectrum representations that are isotypic on the connected center.

A continuous eigenvalue selector exists for the natural representation of SU(n) but not for U(n).

Abstract

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 $\mathrm{U}(n)$.