Orbit misbehavior, isotropy discontinuity, and large isotypic components

Abstract

Let G be a compact Hausdorff group acting on a compact Hausdorff space X, α 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)α 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 α 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)α 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 Tikhonov spaces admit tubes.