  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
$C^*$-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 $C(X/\mathbb{G})\subseteq C(X)$ 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.
