Published paper
Abstract
The article classifies finite ergodic invariant measures for maximal horospherical flows on reductive homogeneous spaces and derives consequences for minimal sets.
Role in dependence graphs
Proof-critical source
The Dirichlet spectrum
This paper is included only for the following marked statement:
- Theorem 9.1 · Inventiones Mathematicae 64 (1981), pp. 379–383Classification of finite ergodic measures invariant under a maximal horospherical subgroup.
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Exact reviewed source
Exact IAS repository Springer version-of-record scan · Inventiones Mathematicae 64 (1981), 357–385
S. G. Dani. Invariant measures and minimal sets of horospherical flows. Inventiones Mathematicae 64 (1981), no. 2, 357–385.
Open audited source ↗01Statements3 reported findingsNo statement issues found
The finite-measure classification for maximal horospherical actions, including the selected reductive-group Theorem 9.1, is correct. Remark 8.3 accurately isolates a defect in the earlier 1978 proof without importing that defect here. Corollary 9.5 has one uniquely repairable missing star in its final subgroup name.
Selected Theorem 9.1Correct · focal dependency verified
Finite ergodic measures are homogeneous on closed finite-volume orbits
Printed pages 381–382 · PDF pages 25–26 · exact Springer version of record
After reducing to connected , the theorem projects through the smallest normal subgroup for which is semisimple with trivial center and no compact factors. The projected finite ergodic measure is classified by Theorem 8.2, and the exact 1978 Proposition 9.3 lifts it through the proper quotient to a connected subgroup containing the original horospherical group. This gives exactly the asserted closed finite-volume orbit and invariant measure.
1981 IAS repository Springer version of record ↗Remark 8.3Correct correction notice · no inherited gap
The earlier general-rank proof defect is stated accurately and bypassed
Printed page 381 · PDF page 25 · Remark 8.3 and the preceding proof of Theorem 8.2
The remark says that the 1978 proof of Theorem 10.1 is valid for lattices in real-rank-one groups but not in its full stated product generality because the 1978 Lemma 10.3 is incorrect. The present paper does not reuse that reduction: it proves the rank-one case directly, handles higher-rank arithmetic lattices through Theorem 6.1, and handles products through Proposition 8.1 and the rectangular-set argument. The final lift uses the unaffected 1978 Proposition 9.3.
Corollary 9.5Suspected typo · uniquely repairable
The final subgroup name is missing its star
Printed page 382 · PDF page 26; proof on printed page 383 · PDF page 27
The corollary defines only but concludes that every compact minimal set is an orbit of . The proof explicitly obtains , so the unique correction is to replace the terminal by . No mathematical conclusion changes.
02Proofs4 reported findingsNo proof issues found
Theorem 9.1 closes through the independently proved semisimple classification and the exact 1978 lifting proposition. The arithmetic, rank-one, and product branches of Theorem 8.2 are explicit, and the paper's own Remark 8.3 correctly prevents reliance on the invalid earlier reduction.
Proof of selected Theorem 9.1Correct and complete with explicit imported input
Proper projection, semisimple classification, and lifting fit exactly
Printed pages 381–382 · PDF pages 25–26
The connected-component reduction is justified by the finitely many component orbits. Properness preserves finiteness under projection. Theorem 8.2 supplies a homogeneous projected measure together with the normal subgroup required by the lifting hypothesis, and 1978 Proposition 9.3 returns a connected subgroup containing with a closed finite-volume orbit and invariant measure. The checked proof of Proposition 9.3 does not invoke the faulty 1978 Lemma 10.3 or Theorem 10.1.
Theorem 8.2No issue found in the reviewed proof
The three irreducible cases and the product step close
Printed pages 379–381 · PDF pages 23–25
For an irreducible lattice, the compact, total-real-rank-one, and arithmetic cases are exhausted by the cited arithmeticity inputs. The compact branch gives Haar measure, the arithmetic branch uses Theorem 6.1, and the rank-one branch uses the Bruhat decomposition and cusp lattice to obtain either Haar measure or a compact horospherical orbit. For a finite-index product decomposition, the rectangular-set argument upgrades homogeneous marginals to invariance under each factor subgroup and hence under their product.
Theorem 6.1 and Section 7Correct with explicit imported inputs
The arithmetic core preserves its invariance and finiteness hypotheses
Printed pages 373–379 · PDF pages 17–23
A minimal positive-measure Bruhat stratum gives invariance under its connected subgroup. Arithmetic finiteness rules out a nontrivial opposite-unipotent factor because its Haar measure would be infinite, forcing the relevant Weyl cell into the chosen parabolic. Section 7 then proves the deferred invariance on the reductive Levi quotient and lifts it through the unipotent radical. The imported finiteness theorem is used with the hypotheses stated in its source.
Theorem 4.1No issue found in the reviewed proof
Invariant quotient measure lifts through the nilpotent fiber
Printed pages 367–369 · PDF pages 11–13
Lemma 4.3 reduces exceptional conditional measures to countably many rational proper normal subgroups. Positive measure for one exceptional set would make an exterior-power analytic function vanish identically and force the normal closure into a proper subgroup, a contradiction. Almost every fiber conditional is therefore Haar, and the invariant quotient measure lifts to a -invariant measure.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.