{"id":"4907a926-9f5e-47c9-bdf1-2a6da284dace","arxiv_id":"1908.07812","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every stationary character on an irreducible lattice of a higher-rank semisimple Lie group is a genuine character, yielding new rigidity and URS finiteness results.","lead":"For rigid higher-rank lattices, probability measures that are stationary for a random walk must actually be conjugation invariant, meaning they are genuine characters. This unlocks rigidity results: every weakly mixing representation contains the regular representation, and every uniformly recurrent subgroup is finite.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No internal gap found; confidence rests on the imported splitting theorem [SZ98, Thm 4.2] used in Claim 5.10 of Theorem 5.1.","rationale":"The reader identified the same load-bearing premise: the correctness and applicability of the imported operator-algebraic splitting theorems in the proof of Theorem 5.1. My independent read of the full manuscript did not uncover an internal gap beyond this. The proof structure is coherent: Theorem 4.3 reduces the lattice case to the Lie group case by induction; Lemma 5.4 embeds M into the induced algebra; Lemmas 5.5–5.6 handle the non-faithfulness of ψ via Mautner's phenomenon and the projection q0; Claims 5.7–5.9 construct the reduced algebra M0 and the subalgebra Q0; Claim 5.10 uses [SZ98, Theorem 4.2]; Claim 5.11 uses [GK95, Theorem B]; and the final application of the commutative [NZ00, Theorem 1] is legitimate after proving that Z(M0) is ergodic and non-invariant. The later proofs of Theorems A, C, D, E, and Corollary F are straightforward consequences of Theorem B and standard rigidity results, with no further fragile assumptions. Because the concern is about verification of an external theorem rather than an identified error, the reader's ACCEPT verdict with moderate confidence remains appropriate; I would only emphasize that a formal check of the [SZ98] application is the single most useful next step.","tokens_in":40771,"tokens_out":27210,"duration_ms":282323,"concrete_test":"Extract the exact statement of [SZ98, Theorem 4.2] and instantiate it in the setting of Claim 5.10 with A = ι0(M0), B = L∞(Vθ), and D = Q0. Check explicitly the two hypotheses: (i) C1 ⊗ Z(Q0) ⊂ Z(A) and (ii) A ∩ (B ⊗ Z(D)) = C1 ⊗ Z(D), and verify that the conclusion A = B ⊗ D follows. Then test the case where ψ is non-faithful on N0, as in Lemmas 5.5–5.6, and where ψ0 becomes faithful only later in Claim 5.11; if the theorem requires a normal conditional expectation, a separating modular action, or a tracial state, confirm those conditions hold here. A failure or an unmet hypothesis would invalidate the contradiction in Claim 5.10.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of the paper is Theorem A, and its proof is a direct consequence of Theorem B, which in turn is derived from the noncommutative Nevo–Zimmer theorem, Theorem 5.1. The most delicate point in the proof of Theorem 5.1 is Claim 5.10, where the authors prove that Z(M0) is strictly larger than C1 ⊗ Z(Q0). The argument assumes that the equality ι0(M0) ∩ (L∞(Vθ) ⊗ Z(Q0)) = C1 ⊗ Z(Q0) is exactly the hypothesis under which [SZ98, Theorem 4.2] forces ι0(M0) = C1 ⊗ Q0, and that no additional faithfulness or traciality condition on ψ0 is silently required. This is genuinely load-bearing: if the splitting theorem does not apply in this noncommutative, non-faithful-state setting, the contradiction with faithfulness of the G-action on M0 fails, and with it the dichotomy in Theorem B and hence Theorem A. The paper also invokes [GK95, Theorem B] in Claim 5.11 in a similar way. I found no circularity, no fitted parameters, and no obvious internal inconsistency in the surrounding argument; the only real risk is whether the imported splitting result has been applied with all its hypotheses verified.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves Theorem A: for a connected semisimple Lie group G with finite center, no nontrivial compact factors, and all simple factors of real rank at least two, any stationary character on an irreducible lattice Γ < G is a genuine character. The proof passes through a new structure theorem, Theorem B, for ergodic (Γ, μ0)-von Neumann algebras, and its core technical engine is Theorem 5.1, a noncommutative analogue of the Nevo–Zimmer theorem: any ergodic (G, μ)-von Neumann algebra either has a G-invariant state or admits a G-equivariant normal unital ∗-embedding of L∞(G/Q, ν_Q) for some proper parabolic Q. The authors also obtain a new proof of Peterson’s character rigidity (Theorem C), and applications to weak containment of the regular representation (Corollary D), essential freeness of stationary actions (Theorem E), and finiteness of uniformly recurrent subgroups (Corollary F).","tokens_in":40943,"tokens_out":13824,"duration_ms":154462,"significance":"If the arguments are correct, Theorem A is a significant advance: it shows that the stationary analogue of characters collapses to ordinary characters for higher-rank irreducible lattices, and the noncommutative Nevo–Zimmer theorem (Theorem 5.1) is a new tool likely to have further applications. The paper’s applications, especially the resolution of the Glasner–Weiss URS question and the new proof of Peterson’s character rigidity, are notable. The manuscript is generally careful and self-contained in its new parts: the induction of stationary states (Section 4) and the absolute-continuity results (Section 3) are proved in detail, with no hidden fitted parameters and no circularity.","major_comments":[{"comment":"The proof of the inequality C1_{Vθ} ⊗ Z(Q0) ≠ Z(ι0(M0)) hinges on the assertion that the equality ι0(M0) ∩ (L∞(Vθ) ⊗ Z(Q0)) = C1_{Vθ} ⊗ Z(Q0) 'splits' and hence, by [SZ98, Theorem 4.2], that ι0(M0) = C1_{Vθ} ⊗ Q0. The paper does not state [SZ98, Theorem 4.2], does not define what 'splits' means, and does not verify the hypotheses of the theorem in this noncommutative setting, where the state ψ0 need not be faithful. Because the contradiction with faithfulness of the G-action depends entirely on this implication, the proof is incomplete as written. Please provide the precise theorem statement and check all hypotheses, including whether a conditional expectation or a centrality condition is required.","section":"Section 5.2, Claim 5.10"},{"comment":"The conclusion that ι0(Z(M0)) = C1_{Vθ} ⊗ Z(Q0) is obtained by invoking [GK95, Theorem B] after showing density of {bψ0 | b ∈ Z(Q0)} in Z(Q0)_*. As in Claim 5.10, the theorem is not stated and the hypotheses are not checked; in particular, it is not clear that the conditions verified (namely that (id_{Vθ} ⊗ ρ)(ι0(x)) ∈ C1_{Vθ} for all ρ ∈ Z(Q0)_*) are exactly those required by [GK95, Theorem B]. This is load-bearing because Claim 5.11 is what allows the reduction to the commutative Nevo–Zimmer theorem. Please state the theorem and verify its hypotheses explicitly.","section":"Section 5.2, Claim 5.11"}],"minor_comments":[{"comment":"There are typos in the notation paragraph: 'reak rank' should be 'real rank' and 'leat st two' should be 'least two'.","section":"Introduction, Notation"},{"comment":"The term 'splits' is used in Claims 5.10 and 5.11 without a definition; please define it at first use, for example as 'a von Neumann subalgebra A ⊂ B1 ⊗ B2 splits if A = A1 ⊗ A2 for some subalgebras Ai ⊂ Bi'.","section":"Section 5.1"},{"comment":"The measure on Uθ is denoted m_{U_θ} in the statement of Lemma 5.5 and ν_{U_θ} later in the same proof; please use consistent notation.","section":"Lemma 5.5"},{"comment":"The map obtained from Theorem 5.1 is initially denoted Θ : C(G/Q) → M, although Theorem 5.1 gives a map on L∞(G/Q); the text later extends Θ, but the notation should be clarified to avoid confusion.","section":"Proof of Theorem B, Section 6.1"},{"comment":"The word 'greatful' should be 'grateful'.","section":"Acknowledgments"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the main risk is the unverified application of the Ge–Kadison / Strătilă–Zsidó splitting theorems in Claims 5.10–5.11. I believe the authors can address this by stating the theorems and verifying their hypotheses; if they cannot, the proof of Theorem 5.1 would be incomplete. The rest of the paper appears sound."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Rémi and Cyril have a strong paper here. The headline result is that stationary characters on irreducible higher-rank lattices are ordinary characters, and the engine is a noncommutative Nevo–Zimmer theorem for von Neumann algebras. That theorem is the real new thing: Nevo–Zimmer's commutative rigidity is extended to noncommutative actions without a stabilizer map, using Ge–Kadison/Strătilă–Zsido splitting instead. The applications are substantial: URS finiteness (answering Glasner–Weiss), weak containment of the regular representation in weakly mixing reps, and a new route to Peterson's character rigidity.\n\nThe paper is well organized and carefully written. The induction of stationary states in Section 4 is neat, and the absolute continuity results in Section 3 are solid. I went looking for circularity or fitted parameters and found none. The statements are clean and the dependencies are explicit.\n\nThe soft spot, as your stress-test correctly identifies, is Claim 5.10. The proof of Theorem 5.1 reduces to the splitting of ι₀(M₀) inside L∞(Vθ)⊗Q₀, and the authors invoke [SZ98, Thm 4.2] after checking that the intersection with the 'central' part is C1_{Vθ}⊗Z(Q₀). That is the whole substantive step. The cited theorem is heavy and its hypotheses are not spelled out in the paper. In particular, it isn't obvious that no extra condition on the state ψ₀ or the support projection is required in this noncommutative, non-faithful setting. If the splitting theorem doesn't apply here, the dichotomy collapses and so does Theorem A. This is a genuine load-bearing point. I don't claim it's wrong—the stress-test found no internal gap—but the verification is terse, and a referee needs to check it against [SZ98] line by line. The authors should expand that passage in any revision.\n\nOverall: this is a significant paper that deserves a serious referee. The possible issue is real but localized. If the splitting step checks out, the paper is a clear accept. My recommendation: send it to peer review, and tell the referee to spend time on Claim 5.10.","headline":"Strong paper with a genuine noncommutative Nevo–Zimmer theorem; the one real spot to press is the terse invocation of [SZ98] in Claim 5.10.","tokens_in":41521,"tokens_out":3861,"would_cite":true,"duration_ms":71503,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22D10","22D25","22E40","37A15","46L10","46L30","46L45","60J50"],"pacs":[],"model":"deepseek-v4-flash","headline":"Any stationary character of an irreducible higher-rank lattice is conjugation invariant, hence a genuine character.","keywords":["Boundary theory","C*-algebras","Characters","Lattices in semisimple Lie groups","Stationary measures","Uniformly recurrent subgroups","von Neumann algebras"],"falsifier":"Build an ergodic $(\\Gamma,\\mu_0)$-von Neumann algebra for an irreducible lattice in a higher-rank group, with a $\\mu_0$-stationary faithful normal state $\\varphi$ that is not $\\Gamma$-invariant and for which no proper parabolic subgroup $Q$ admits a $\\Gamma$-equivariant normal unital embedding of $L^\\infty(G/Q,\\nu_Q)$ into the algebra with $\\varphi\\circ\\theta = \\nu_Q$; this would directly falsify Theorem B. Alternatively, exhibit a $\\mu_0$-character on such a lattice that is not conjugation invariant; this would falsify Theorem A and, because the proof of Theorem A derives non-invariance of characters from a violation of Theorem B, it would locate the failure in the dichotomy.","tokens_in":40499,"feed_emoji":"🧩","tokens_out":13797,"duration_ms":120408,"temperature":0.7,"pith_summary":"The paper proves that stationarity cannot be a weaker substitute for invariance in the character theory of higher-rank lattices. For an irreducible lattice $\\Gamma$ in a connected semisimple Lie group whose simple factors all have real rank at least two, any positive definite function that is stationary under the random walk driven by a Furstenberg measure — a 'stationary character' — is automatically conjugation invariant, i.e. a genuine character. This matters because characters encode unitary representations and factors, and the genuine-character conclusion is exactly the rigidity phenomenon behind operator-algebraic superrigidity. The proof introduces a noncommutative structure theorem: ergodic stationary actions of such lattices on von Neumann algebras either have an invariant state or factor through a proper parabolic boundary. If the theorem is right, the same rigidity passes to weak containment of the regular representation, to essential freeness of stationary actions, and to finiteness of uniformly recurrent subgroups.","feed_headline":"Stationary characters on higher-rank lattices are genuine characters","feed_subtitle":"Forces the regular representation into every weakly mixing representation; makes uniformly recurrent subgroups finite.","key_machinery":"The central object is the noncommutative Nevo–Zimmer dichotomy: an ergodic stationary action of a higher-rank semisimple Lie group on a von Neumann algebra either has a $G$-invariant state or admits a $G$-equivariant normal unital embedding of a boundary von Neumann algebra $L^\\infty(G/Q,\\nu_Q)$ for a proper parabolic subgroup $Q$. The proof of Theorem A feeds a GNS representation of an extreme stationary character into this dichotomy, and a lemma based on the boundary embedding and the uniqueness of the stationary measure on $G/Q$ forces the character to vanish off the center. The transfer from lattice to ambient Lie group uses a new induction theorem: an ergodic $(\\Gamma,\\mu_0)$-von Neumann algebra with a $\\mu_0$-stationary faithful normal state induces a $\\mu$-stationary faithful normal state on the induced $G$-von Neumann algebra, where $\\mu$ is a $K$-invariant admissible measure; this observation is new even in the commutative setting. The noncommutative proof invokes tensor-slice maps, the essential range of measurable functions with values in a von Neumann algebra, and imported splitting theorems to reduce to the commutative case on the center.","core_discovery":"On its own terms, the central claim is Theorem A: if $G$ is a connected semisimple Lie group with finite center, no compact factor, and all simple factors of real rank at least two, $\\Gamma$ is any irreducible lattice, and $\\mu_0$ is a Furstenberg probability measure on $\\Gamma$, then every $\\mu_0$-character on $\\Gamma$ is conjugation invariant. The engine is Theorem B, a dichotomy for ergodic $(\\Gamma,\\mu_0)$-von Neumann algebras: either the stationary normal state is $\\Gamma$-invariant, or there is a proper parabolic subgroup $Q$ and a $\\Gamma$-equivariant normal unital embedding of $L^\\infty(G/Q,\\nu_Q)$ into the algebra pulling the state back to $\\nu_Q$. The authors prove the $G$-version of this dichotomy (a noncommutative Nevo–Zimmer theorem) and then transfer it to lattices by an induction construction for stationary states. From the dichotomy they also recover a proof of the existing character rigidity theorem for such lattices, show that the left regular representation is weakly contained in every weakly mixing representation, and show that every uniformly recurrent subgroup is finite.","pith_inferences":["Inference: the induction theorem relating $\\Gamma$-stationary states to $G$-stationary states on the induced algebra is a transfer principle that should apply beyond von Neumann algebras, for example to stationary actions on compact convex spaces and to equivariant operator systems.","Inference: the proof suggests that for higher-rank lattices, stationarity with respect to a Furstenberg measure is as rigid as invariance for many boundary-based conclusions; stationary versions of other rigidity theorems may hold without new assumptions.","Inference: the noncommutative Nevo–Zimmer dichotomy is the most portable part of the paper, and its correctness in full generality is the point most worth testing independently of the lattice applications."],"forward_implications":["For any irreducible lattice $\\Gamma$ in such a group $G$ with trivial center, the left regular representation $\\lambda_\\Gamma$ is weakly contained in every weakly mixing representation $\\pi$ of $\\Gamma$; equivalently, $C^*_\\pi(\\Gamma)$ has a unique tracial state and a unique maximal ideal.","Every uniformly recurrent subgroup of $\\Gamma$ is finite, so every minimal action of $\\Gamma$ on a compact metrizable space is either finite or topologically free.","Every ergodic stationary action of $\\Gamma$ on a probability space is either measure-preserving or factors onto a proper parabolic boundary $G/Q$; faithful properly ergodic stationary actions are essentially free.","The known character rigidity theorem for such lattices is recovered: every extreme character is either almost periodic or the Dirac character."],"supporting_citations":[{"why":"Supplies the commutative Nevo–Zimmer structure theorem, applied to the center of the reduced subalgebra in the noncommutative proof.","marker":"[NZ00, Theorem 1]"},{"why":"Guarantees the existence of a Furstenberg probability measure $\\mu_0$ supported on $\\Gamma$ for which $G/P$ is the Poisson boundary.","marker":"[Fu67, Theorem 3]"},{"why":"Establishes uniqueness of the $\\mu_0$-stationary measure on each $G/Q$, used to identify the pulled-back state in the dichotomy.","marker":"[Fu73, GM89]"},{"why":"Provides the Poisson-boundary map for stationary points in compact convex affine $G$-spaces, used to construct boundary maps from stationary states.","marker":"[BS04, Theorem 2.16]"},{"why":"Ge–Kadison splitting theorem used to prove that the center of the reduced subalgebra splits, a key step in the noncommutative Nevo–Zimmer theorem.","marker":"[GK95, Theorem B]"},{"why":"Strătilă–Zsidó generalization of the splitting theorem invoked in the same step as the Ge–Kadison result.","marker":"[SZ98, Theorem 4.2]"},{"why":"Used to show that non-central lattice elements act essentially freely on proper homogeneous spaces $G/H$, which forces stationary characters to vanish off the center.","marker":"[Oz16, Remark 13]"}],"fun_headline_variants":["Higher-rank lattices: all stationary characters are genuine","Stationary characters on lattices are conjugation invariant","Forces regular rep into all weakly mixing reps of lattice","New structure theorem for lattice actions on von Neumann algebras","Every uniformly recurrent subgroup of higher-rank lattice is finite"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that a handful of imported results — the splitting theorems for tensor products of von Neumann algebras and the known boundary rigidity theorem for the ambient Lie group — apply to the specific algebras constructed in the proof; if any of those imports fails, the dichotomy in Theorem B collapses and Theorem A goes with it.","fun_headline_variants_meta":{"raw":{"variants":["Higher-rank lattices: all stationary characters are genuine","Stationary characters on lattices are conjugation invariant","Forces regular rep into all weakly mixing reps of lattice","New structure theorem for lattice actions on von Neumann algebras","Every uniformly recurrent subgroup of higher-rank lattice is finite"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000333,"raw_usage":{"total_tokens":1838,"prompt_tokens":919,"completion_tokens":919,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":535,"completion_tokens_details":{"reasoning_tokens":843}},"tokens_in":535,"tokens_out":919,"duration_ms":9029,"temperature":1.0,"reasoning_tokens":843,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:55:42.933626+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Build an ergodic $(\\Gamma,\\mu_0)$-von Neumann algebra for an irreducible lattice in a higher-rank group, with a $\\mu_0$-stationary faithful normal state $\\varphi$ that is not $\\Gamma$-invariant and for which no proper parabolic subgroup $Q$ admits a $\\Gamma$-equivariant normal unital embedding of $L^\\infty(G/Q,\\nu_Q)$ into the algebra with $\\varphi\\circ\\theta = \\nu_Q$; this would directly falsify Theorem B. Alternatively, exhibit a $\\mu_0$-character on such a lattice that is not conjugation invariant; this would falsify Theorem A and, because the proof of Theorem A derives non-invariance of characters from a violation of Theorem B, it would locate the failure in the dichotomy.","supporting_citations":[],"review_version":1}