Orbit misbehavior, isotropy discontinuity, and large isotypic components

Abstract

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.