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Compact lifts and averaging onto C_c(G/H)

Statement

Assume AC. If $H$ is closed in a locally compact Hausdorff group $G$ , then $X = G / H$ is locally compact Hausdorff and the quotient map $p : G \to X$ is open. Every compact $Q \subseteq X$ lies in $p (K)$ for some compact $K \subseteq G$ . For fixed left Haar measure $d h$ on $H$ , $T_{H} f (x H) = \int_{H} f (x h) d h$ maps $C_{c} (G)$ onto $C_{c} (X)$ .

Facts & Assumptions

Given: The LCH group $G$ , its closed subgroup $H$ , and AC.

[F1]

AC implies DC, and under DC a compact set inside an open LCH set admits a $C_{c}$ cutoff (AC implies DC implies countable choice, LCH Urysohn cutoff).

[F2]

For compactly supported continuous kernels on two LCH spaces, the two positive Radon integrations commute and the partial integrals are continuous with compact support (Compactly supported kernels admit commuting radon integrals).

[A1]

AC is the choice-function principle (The Axiom of Choice).

Proof

technique · direct

1.1givenconstruct

The quotient map is open because $p^{- 1} (p (O)) = O H$ is open whenever $O \subseteq G$ is open. To separate distinct cosets $x H, y H$ , note $y^{- 1} x \notin H$ . Closedness gives an open neighborhood $W$ of $y^{- 1} x$ disjoint from $H$ . Continuity of $(v, u) \mapsto v^{- 1} y^{- 1} x u$ gives identity neighborhoods $V, U$ with $V^{- 1} y^{- 1} x U \subseteq W$ ; hence $p (x U)$ and $p (y V)$ are disjoint. Thus $X$ is Hausdorff. If $U$ is a relatively compact open neighborhood of $x$ , then $p (U)$ is open and its closure lies in the compact, hence closed, set $p (\overline{U})$ , so $X$ is locally compact.

2.1step 1.1choose

For compact $Q \subseteq X$ , cover $Q$ by sets $p (U_{q})$ where each $U_{q}$ is relatively compact and open. A finite subcover exists, and the union $K$ of the corresponding finitely many compact closures satisfies $Q \subseteq p (K)$ .

2.2F2step 1.1construct

For each $f \in C_{c} (G)$ , the function $x \mapsto \int_{H} f (x h) d h$ is continuous locally on $G$ : around any $x_{0}$ choose a compact neighborhood $K$ ; the kernel $(x, h) \mapsto f (x h)$ on $K \times H$ is supported in the compact set ${(x, h) \in K \times H : x h \in supp f}$ , so [F2] gives continuity there. Left invariance of $d h$ makes this function right $H$ -invariant, and openness of $p$ makes its descended function $T_{H} f$ continuous. Its support lies in the compact set $p (supp f)$ , so $T_{H} f \in C_{c} (X)$ .

3.1A1F1F2step 1.1step 2.1step 2.2chooseconstruct

Let $ϕ \in C_{c} (X)$ and $Q = supp ϕ$ . By [A1], choose a lift $x_{q}$ of each $q \in Q$ . For each lift apply [F1] to the singleton and a relatively compact open neighborhood to obtain a nonnegative $u_{q} \in C_{c} (G)$ with $u_{q} (x_{q}) = 1$ . A nonzero left Haar measure has full support: its support is a nonempty closed set invariant under every left translation, hence is all of $H$ . Thus $u_{q} (x_{q} \cdot)$ has positive integral on $H$ , so $T_{H} u_{q} (q) > 0$ . By continuity from step 2.2, $T_{H} u_{q}$ stays positive on a neighborhood of $q$ . A finite subcover of $Q$ gives $u = \sum_{i} u_{q_{i}}$ with $T_{H} u > 0$ on an open neighborhood $W$ of $Q$ . The function $ψ = ϕ / (T_{H} u)$ on $W$ , extended by zero, is in $C_{c} (X)$ because its support is contained in the compact set $Q \subset W$ . Set $f = (ψ \circ p) u$ . Then $f \in C_{c} (G)$ and $T_{H} f = ψ T_{H} u = ϕ$ , also when $ϕ = 0$ (use $f = 0$ ). Thus $T_{H} : C_{c} (G) \to C_{c} (X)$ is onto; compact lifts were proved in step 2.1. ∎

Sources

Bekka, de la Harpe, and Valette, Kazhdan’s Property (T), Appendix B §B.1, Lemma B.1.1 (compact lifts) and Lemma B.1.2 (surjectivity of subgroup averaging), PDF pp. 349–352. Full relevant text was inspected; this proof supplies the local quotient-topology and compact-kernel details.
Bruhat, Lectures on Lie Groups and Representations of Locally Compact Groups, Chapter 7 §3.3, Proposition 2, PDF pp. 72–74. Bruhat writes the opposite coset convention; the displayed formulas here use left cosets $G / H$ and $f (x h)$ .

Depends on

Used by

Bruhat cutoff normalized along H-fibers Lemma
Composition of Weil quotient integrals Lemma
Continuous quotient translation cocycle Lemma
Density of averaged covariant generators Lemma
Local densities for equivalent Radon quotient measures Lemma
Strong continuity of unitary induction Lemma
Well-defined induced inner product Lemma
Criterion for an invariant quotient measure Proposition
Existence of rho-functions and quotient measure classes Theorem
Independence of rho and equivalent quotient representative Theorem
Induction in stages for closed subgroup chains Theorem
Weil formula with a rho-function Theorem

Dependency tree · two levels

16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Bekka–de la Harpe–Valette, Kazhdan’s Property (T), Appendices B and E (standard reference, not scraped)
Bruhat, Lectures on Lie Groups and Representations of Locally Compact Groups, Chapters 1 and 7 (standard reference, not scraped)