Orbit misbehavior, isotropy discontinuity, and large isotypic components

Let $\mathbb{G}$ be a compact Hausdorff group acting on a compact Hausdorff space $X$, $\alpha$ an irreducible $\mathbb{G}$-representation, and $C(X)$ the $C^{\ast }$-algebra of complex-valued continuous functions on $X$. We prove that the isotypic component $C(X{)}_{\alpha }$ is finitely generated as a module over the invariant subalgebra $C(X\mathrm{/}\mathbb{G})\subseteq C(X)$ precisely when the map sending $x\in X$ to the dimension of the space of vectors in $\alpha$ invariant under the isotropy group ${\mathbb{G}}_x$ is locally constant. This (a) specializes back to an observation of De Commer-Yamashita equating the finite generation of all $C(X{)}_{\alpha }$ with the Vietoris continuity of $x\mapsto {\mathbb{G}}_x$, and (b) recovers and extends Watatani’s examples of infinite-index expectations resulting from non-free finite-group actions. We also show that the action of a compact group $\mathbb{G}$ on the maximal equivariant compactification on the disjoint union of its Lie-group quotients has tubes about all orbits precisely when $\mathbb{G}$ is Lie. This is the converse (via a canonical construction) of the well-known fact that actions of compact Lie groups on Tychonoff spaces admit tubes.