Published paper
Role in dependence graphs
Proof-critical source
Winning property of counterexamples to Uniform Littlewood's Conjecture
This paper is included only for the following marked statement:
- Theorem 1.1 · VOR p. 156 / PDF p. 3; proof pp. 158–165 / PDF pp. 5–12An eigenvector for a one-parameter subgroup is fixed by the associated ad-compact normal subgroup.
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Exact reviewed source
Exact Mathematical Sciences Publishers version-of-record scan · Pacific Journal of Mathematics 86 (1980), 155–169
Calvin C. Moore. The Mautner phenomenon for general unitary representations. Pacific Journal of Mathematics 86 (1980), no. 1, 155–169.
Open audited source ↗01Statements1 reported findingNo statement issues found
Theorem 1.1 is the general Mautner statement used in Shah's Proposition 2.7.
Theorem 1.1No issue found
An eigenvector for a one-parameter subgroup is fixed by its ad-compact normal subgroup
Version of record p. 156 · PDF p. 3 · Theorem 1.1; proof pp. 158–165 · PDF pp. 5–12
For in the Lie algebra of a connected Lie group, let be the smallest connected normal subgroup for which the image of in the quotient is ad-compact. Theorem 1.1 proves that every eigenvector for the unitary action of is fixed by . Taking eigenvalue zero gives the Mautner property for fixed vectors. Shah applies this separately to the unipotent one-parameter subgroups generating and then combines the resulting normal subgroups.
02Proofs2 reported findingsFurther proof verification needed
Most reductions are written out, but the decisive final reduction imports exact arguments from Dani 1977 and Moore 1966.
Internal proof architectureCorrect and complete modulo represented sources
The spectral and Lie-algebra reductions are coherent
Version of record pp. 158–164 · PDF pp. 5–11 · Theorems 2.3, 4.1–4.2
Theorem 2.3 separates expanding and contracting root subgroups and obtains Lebesgue spectrum on their non-fixed subspaces. Section 3 removes the compact semisimple part of . Theorem 4.1 handles the nilradical case through Mackey–Kirillov decomposition, and Theorem 4.2 turns it into the normal-subgroup Mautner statement used in the final reduction. These steps consistently imply the fixed-vector conclusion once Theorems 5.1–5.2 are available.
Recursive source expansionClosed with represented article branch
Dani 1977 and Moore 1966 are proof-critical non-book parents
Version of record pp. 164–165 · PDF pp. 11–12 · Theorems 5.1–5.2
The proof of Theorem 5.1 explicitly says that Dani 1977, Proposition 4.1, carries over word for word after replacing Dani's geometric preliminary lemma by Moore's Theorem 4.2. Theorem 5.2 likewise invokes Dani 1977, Proposition 5.1, and gives Moore 1966's semisimple result as an alternative justification. Both papers are now represented with exact metadata and consumer restatements, but their original proofs remain inaccessible and explicitly unverified. Mackey and Kirillov theory are used as standard representation-theoretic foundations in the fully written internal reductions; they do not replace the two omitted Section 5 arguments.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.