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

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Unitary cocycle-corrected left action

Statement

Assume AC. For $g \in G$ and $F \in C_{c} (G, H; V)$ , define $(Π_{ρ} (g) F) (x) = D_{g} (x H)^{1 / 2} F (g^{- 1} x) = [ρ (g^{- 1} x) / ρ (x)]^{1 / 2} F (g^{- 1} x)$ . This is covariant, preserves the inner product, satisfies $Π_{ρ} (g_{1}) Π_{ρ} (g_{2}) = Π_{ρ} (g_{1} g_{2})$ , and extends to a unitary on the completion.

Facts & Assumptions

Given: The induced model, its quotient measure, and $g, g_{1}, g_{2} \in G$ .

[A1]

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

[F1]

$D_{g}$ is positive, representative-independent, and satisfies the density cocycle (Continuous quotient translation cocycle).

[F2]

Covariant functions and their norm are defined by the induced model (Continuous covariant model and measurable completion).

[F3]

The integrated inner product is positive definite (Well-defined induced inner product).

Proof

technique · direct

1.1F1F2A1

Since $D_{g}$ is a function on $G / H$ , it is right $H$ -invariant. Thus $F (g^{- 1} x h) = σ (h)^{- 1} F (g^{- 1} x)$ proves covariance of $Π_{ρ} (g) F$ . Its quotient support is the translate by $g$ of the compact support of $F$ .

2.1F1step 1.1

Applying twice gives $Π_{ρ} (g_{1}) Π_{ρ} (g_{2}) F (x) = (D_{g_{1}} (x H) D_{g_{2}} (g_{1}^{- 1} x H))^{1 / 2} F (g_{2}^{- 1} g_{1}^{- 1} x) = Π_{ρ} (g_{1} g_{2}) F (x)$ by the cocycle identity [F1]. Also $Π_{ρ} (e) = I$ , so $Π_{ρ} (g^{- 1})$ is the inverse.

3.1A1F1F2F3step 1.1step 2.1

The cocycle identity with $g_{1} = g^{- 1}, g_{2} = g$ gives $D_{g} (g r H) D_{g^{- 1}} (r H) = 1$ . The change-of-measure formula $d ((g^{- 1})_{*} μ_{ρ}) / d μ_{ρ} = D_{g^{- 1}}$ then yields $∥ Π_{ρ} (g) F ∥_{2}^{2} = \int D_{g} (q) ∥ F (g^{- 1} q) ∥^{2} d μ_{ρ} (q) = \int D_{g} (g r) D_{g^{- 1}} (r) ∥ F (r) ∥^{2} d μ_{ρ} (r) = ∥ F ∥_{2}^{2} .$ Thus the operator is an isometry on the dense continuous model, and its inverse from step 2.1 makes its extension unitary on the completion. ∎

Sources

Bekka–de la Harpe–Valette, Kazhdan’s Property (T), Appendix E §E.1, Proposition E.1.4, PDF pp. 413–414. Full relevant text was inspected.

Depends on

Used by

Strong continuity of unitary induction Lemma
Independence of rho and equivalent quotient representative Theorem
Induction in stages for closed subgroup chains Theorem
Unitary induction from a closed subgroup Theorem

Dependency tree · two levels

14 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)