LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-02

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Continuous quotient translation cocycle

Statement

Assume AC. For the rho-derived $μ_{ρ}$ and $g \in G$ , $d (g_{*} μ_{ρ}) / d μ_{ρ} (x H) = D_{g} (x H) = ρ (g^{- 1} x) / ρ (x) > 0$ . This is independent of representative, jointly continuous, and satisfies $D_{g_{1} g_{2}} (q) = D_{g_{1}} (q) D_{g_{2}} (g_{1}^{- 1} q)$ .

Facts & Assumptions

Given: Closed $H \leq G$ , a rho-function $ρ$ , its Weil measure $μ_{ρ}$ , and elements $g, g_{1}, g_{2} \in G$ .

[A1]

AC is assumed as stated (The Axiom of Choice).

[F1]

Rho-functions satisfy $ρ (x h) = Δ_{H} (h) Δ_{G} (h)^{- 1} ρ (x)$ (Rho-function for a closed subgroup).

[F2]

The rho-derived measure satisfies the Weil formula (Weil formula with a rho-function).

[F3]

Every $ϕ \in C_{c} (G / H)$ equals $T_{H} f$ for some $f \in C_{c} (G)$ (Compact lifts and averaging onto C_c(G/H)).

[F4]

Radon measures agreeing on $C_{c}$ agree on Borel sets (Assuming Dependent Choice, uniqueness of the RMK representing measure among Radon measures).

[F5]

The quotient map is open and $G / H$ is LCH (Compact lifts and averaging onto C_c(G/H)).

[A2]

AC implies DC, so the Radon-measure uniqueness supplier applies (AC implies DC implies countable choice).

Proof

technique · direct

1.1F1F5construct

Define $D_{g} (x H) = ρ (g^{- 1} x) / ρ (x)$ . Replacing $x$ by $x h$ multiplies numerator and denominator by the same factor from [F1], so the ratio is well-defined and positive. The continuous function $(g, x) \mapsto ρ (g^{- 1} x) / ρ (x)$ is constant on fibers in the second coordinate; [F5] makes its descent through $G \times G \to G \times G / H$ continuous.

2.1A1A2F2F3F4step 1.1

For $ϕ = T_{H} f$ , Weil’s formula and left invariance give $\int_{G / H} ϕ (g q) d μ_{ρ} (q) = \int_{G} f (g x) ρ (x) d x = \int_{G} f (y) ρ (g^{- 1} y) d y = \int_{G / H} ϕ (q) D_{g} (q) d μ_{ρ} (q) .$ The positive continuous density is locally bounded, so it defines a Radon measure relative to the Radon measure $μ_{ρ}$ . By [F3] the equality holds on every $C_{c} (G / H)$ function; [F4] identifies $g_{*} μ_{ρ} = D_{g} μ_{ρ}$ . This proves the derivative formula.

3.1A1A2step 1.1step 2.1algebra

For $q = x H$ , the ratios telescope: $D_{g_{1}} (q) D_{g_{2}} (g_{1}^{- 1} q) = \frac{ρ (g_{1}^{- 1} x)}{ρ (x)} \frac{ρ (g_{2}^{- 1} g_{1}^{- 1} x)}{ρ (g_{1}^{- 1} x)} = D_{g_{1} g_{2}} (q) .$ Together with step 1.1, this proves the stated cocycle identity and continuity. ∎

Sources

Bekka–de la Harpe–Valette, Kazhdan’s Property (T), Appendix B §B.1, Theorem B.1.4 and its quotient-measure density calculation, PDF pp. 352–354. Full relevant text was inspected.

Depends on

Used by

Uniform lattice quotient and quasi-regular action Example
Unitary cocycle-corrected left action Lemma
Independence of rho and equivalent quotient representative Theorem
Unitary induction from a closed subgroup Theorem

Dependency tree · two levels

22 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)
David Vogan, Unitary Representations of Locally Compact Groups and Induced Representations (standard reference, not scraped)