Orbit misbehavior, isotropy discontinuity, and large isotypic components

Let G be a compact Hausdorff group acting on a compact Hausdorff space X, an irreducible G-representation, and C (X) the C^*-algebra of complex-valued continuous functions on X. We prove that the isotypic component C (X) _ is finitely generated as a module over the invariant subalgebra C (X/G) C (X) precisely when the map sending x X to the dimension of the space of vectors in invariant under the isotropy group Gₓ is locally constant. This (a) specializes back to an observation of De Commer-Yamashita equating the finite generation of all C (X) _ with the Vietoris continuity of x Gₓ, 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 G on the maximal equivariant compactification on the disjoint union of its Lie-group quotients has tubes about all orbits precisely when 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.