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Orbit misbehavior, isotropy discontinuity, and large isotypic components

Alexandru Chirvasitu

Abstract

Let $\mathbb{G}$ be a compact Hausdorff group acting on a compact Hausdorff space $X$, $α$ an irreducible $\mathbb{G}$-representation, and $C(X)$ the $C^*$-algebra of complex-valued continuous functions on $X$. We prove that the isotypic component $C(X)_α$ 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 $α$ 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 $C(X)_α$ 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.

Orbit misbehavior, isotropy discontinuity, and large isotypic components

Abstract

Let be a compact Hausdorff group acting on a compact Hausdorff space , an irreducible -representation, and the -algebra of complex-valued continuous functions on . We prove that the isotypic component is finitely generated as a module over the invariant subalgebra precisely when the map sending to the dimension of the space of vectors in invariant under the isotropy group is locally constant. This (a) specializes back to an observation of De Commer-Yamashita equating the finite generation of all with the Vietoris continuity of , 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 on the maximal equivariant compactification on the disjoint union of its Lie-group quotients has tubes about all orbits precisely when 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.
Paper Structure (2 sections, 8 theorems, 33 equations)

This paper contains 2 sections, 8 theorems, 33 equations.

Table of Contents

  1. Orbits, isotropy jumps, and (failure of) finite generation
  2. Tube existence as a Lie-marking property

Key Result

Theorem 1

Let ${\mathbb G}$ be a compact group acting on a compact Hausdorff space $X$. The isotypic component $C(X)_{\alpha}$, $\alpha\in\mathop{\mathrm{Irr}}\nolimits({\mathbb G})$ is finitely generated as a $C(X)^{{\mathbb G}}$ module if and only if the function is locally constant. $\blacksquare$

Theorems & Definitions (19)

  • Theorem 1
  • Theorem 2
  • Example 1.1
  • Example 1.3
  • Definition 1.4
  • Theorem 1.5
  • Proof 1
  • Corollary 1.6
  • Proof 2
  • Corollary 1.7
  • ...and 9 more