DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-6.1-sol)audited 2026-10-02

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Rho-function for a closed subgroup

Definition

Fix left Haar measures on a locally compact Hausdorff group $G$ and its closed subgroup $H$ . Use the convention $\int_{G} f (x h) d x = Δ_{G} (h)^{- 1} \int_{G} f (x) d x$ for right translation by $h \in H$ . A rho-function for $(G, H)$ is a positive continuous function $ρ : G \to (0, \infty)$ satisfying

$ρ (x h) = Δ_{H} (h) Δ_{G} (h)^{- 1} ρ (x) (x \in G, h \in H) .$

The ratio is fixed by the left-Haar and right- $H$ averaging conventions used in the Weil formula. In particular, a later ratio such as $ρ (g^{- 1} x) / ρ (x)$ is constant on each right $H$ -fiber.

Depends on

Used by

Uniform lattice quotient and quasi-regular action Example
Composition of Weil quotient integrals Lemma
Continuous quotient translation cocycle Lemma
Criterion for an invariant quotient measure Proposition
Existence of rho-functions and quotient measure classes Theorem
Weil formula with a rho-function Theorem

Dependency tree · two levels

9 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)