Stationary characters on lattices of semisimple Lie groups

Cyril Houdayer; R\'emi Boutonnet

REVIEW 2 major objections 5 minor 13 references

Stationary characters on lattices of semisimple Lie groups

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Any stationary character of an irreducible higher-rank lattice is conjugation invariant, hence a genuine character.

desk verdict 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. read the letter →

arxiv 1908.07812 v2 pith:4VKMV4MW submitted 2019-08-21 math.GR math.DSmath.OAmath.RT

Rémi Boutonnet , Cyril Houdayer This is my paper · ORCID

classification math.GRmath.DSmath.OAmath.RT MSC 22D1022D2522E4037A1546L1046L3046L4560J50

keywords BoundarytheoryC*-algebrasCharactersLatticesinsemisimpleLiegroupsStationarymeasuresUniformlyrecurrentsubgroupsvonNeumannalgebras

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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.

The 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.

Overall: 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.

Referee Report

2 major / 5 minor

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).

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 (2)

[Section 5.2, Claim 5.10] 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 5.2, Claim 5.11] 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.

minor comments (5)

[Introduction, Notation] There are typos in the notation paragraph: 'reak rank' should be 'real rank' and 'leat st two' should be 'least two'.
[Section 5.1] 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'.
[Lemma 5.5] 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.
[Proof of Theorem B, Section 6.1] 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.
[Acknowledgments] The word 'greatful' should be 'grateful'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation chain is self-contained and the cited self-work is not load-bearing.

full rationale

The paper's central claim, Theorem A, is not an input renamed as an output. A mu0-character is defined by the stationarity equation sum_gamma mu0(gamma) phi(gamma^{-1} g gamma) = phi(g), and the conclusion that phi is conjugation invariant is not contained in that definition; no parameter of the problem is fitted to the target result. The derivation chain is: Theorem 4.3 induces a stationary state from a Furstenberg lattice action; Theorem 5.1 is a noncommutative Nevo-Zimmer dichotomy proved by reducing to the commutative case through Ge-Kadison and Stratila-Zsido splitting results; Theorem B follows by induction and disintegration; and Theorem A follows from Theorem B plus the GNS construction and Lemma 6.3. The only self-citations are [BBHP20] in Section 4, described as a more synthetic presentation, and Remark 5.12, described as an alternative route from the simple to the semisimple case; in both places the main text supplies its own construction or iteration ('Since G has only finitely many simple factors, we can continue a finite number of steps'), so the self-citation is not load-bearing. The delicate application of [SZ98, Theorem 4.2] in Claim 5.10 and [GK95, Theorem B] in Claim 5.11 is a dependence on independent external theorems, not a circular reduction; if those theorems were inapplicable in this setting the proof would fail, but that would be a correctness risk rather than circularity. No fitted-input-called-prediction, uniqueness-imported-from-authors, or ansatz-via-citation pattern appears. Score 0 reflects the absence of circularity, and the non-load-bearing self-citations do not raise the score.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim rests on structural hypotheses, namely irreducible lattices in semisimple Lie groups all of whose simple factors have real rank at least two, and on standard theorems in boundary theory and operator algebras. No free numerical parameters are introduced. The most consequential imported inputs are the existence and uniqueness of Furstenberg stationary measures on flag manifolds, the Nevo-Zimmer commutative structure theorem, the Ge-Kadison/Stratătilă-Zsidó splitting theorem, and Stuck-Zimmer/Margulis rigidity results. These are all literature theorems; none are invented for this paper.

assumptions (7)

domain assumption Existence of a Furstenberg measure mu0 on Gamma with support Gamma and Poisson boundary (G/P, nu_P).
Section 1, Definition and discussion; from [Fu67, Theorem 3] and [Fu00, Theorem 2.21]. This fixes the notion of stationary character.
domain assumption Uniqueness of nu_Q as the mu0-stationary measure on each G/Q for P subset Q subset G.
Used in Theorems B and E and Lemma 6.3 to identify phi composed with theta with nu_Q; sourced to [Fu73, GM89].
standard math Nevo-Zimmer structure theorem [NZ00, Theorem 1] for ergodic stationary actions of G on Lebesgue spaces.
Invoked at the end of Theorem 5.1 on Z(M0) after Claim 5.11.
standard math Ge-Kadison and Stratătilă-Zsidó splitting theorems for von Neumann subalgebras ([GK95], [SZ98, Theorem 4.2]).
Claim 5.10 uses splitting to force iota0(M0) to split when its center splits.
standard math Stuck-Zimmer stabilizer rigidity [SZ92, Corollary 4.4].
Used in Theorem E for the Gamma-invariant case and as the measure-theoretic analogue being extended.
standard math Margulis normal subgroup theorem [Ma91, Theorem IV.4.10].
Used in Corollary F and Lemma 6.2 to control intersections of Gamma with normal subgroups.
standard math Bader-Furman theorem on boundary maps and factors [BF14, Theorem 2.5].
Used in Lemma 6.4(i) to prove ergodicity of the conjugation action on the noncommutative Poisson boundary B.

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Pith. "Pith review of Stationary characters on lattices of semisimple Lie groups." pith.science (2026). https://pith.science/paper/4VKMV4MW

@misc{pith2026190807812,
  author       = {Pith},
  title        = {Pith review of: Stationary characters on lattices of semisimple Lie groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4VKMV4MW}},
  note         = {Machine review of arXiv:1908.07812}
}

abstract

We show that stationary characters on irreducible lattices $\Gamma < G$ of higher-rank connected semisimple Lie groups are conjugation invariant, that is, they are genuine characters. This result has several applications in representation theory, operator algebras, ergodic theory and topological dynamics. In particular, we show that for any such irreducible lattice $\Gamma < G$, the left regular representation $\lambda_\Gamma$ is weakly contained in any weakly mixing representation $\pi$. We prove that for any such irreducible lattice $\Gamma < G$, any uniformly recurrent subgroup (URS) of $\Gamma$ is finite, answering a question of Glasner-Weiss. We also obtain a new proof of Peterson's character rigidity result for irreducible lattices $\Gamma < G$. The main novelty of our paper is a structure theorem for stationary actions of lattices on von Neumann algebras.

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Works this paper leans on

13 extracted references · 12 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

On invariant random positive definite functions

[7s12] M. Abert, N. Bergeron, I. Biringer, T. Gelander, N. Nikolov, J. Raimbault, I. Samet , On the growth of L2-invariants for sequences of lattices in Lie groups. Ann. of Math. 185 (2017), 711–790. [AGV12] M. Abert, Y. Glasner, B. Virag , Kesten ’s theorem for invariant random subgroups. Duke Math. J. 163 (2014), 465–488. [AB18] V. Alekseev, R. Brugger ...

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Invent. Math. 169 (2007), 401–425. [Be19] B. Bekka , Character rigidity of simple algebraic groups. Math. Ann. 378 (2020), 1223–1243. [BCH94] B. Bekka, M. Cowling, P. de la Harpe , Some groups whose reduced C∗ -algebra is simple. Publ. Math. Inst. Hautes ´Etudes Sci. 80 (1994), 117–134. [BF20] B. Bekka, C. Francini , Characters of algebraic groups over nu...

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[PT13] J. Peterson, A. Thom , Character rigidity for special linear groups. J. Reine Angew. Math. 716 (2016), 207–228. [SZ98] S ¸. Str˘atil˘a, L. Zsid ´o, The commutation theorem for tensor products over von Neuman n algebras. J. Funct. Anal. 165 (1999), 293–346. [SZ92] G. Stuck, R.J. Zimmer , Stabilizers for ergodic actions of higher rank semisimple g ro...

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[GK95] L. Ge, R. Kadison , On tensor products for von Neumann algebras. Invent. Math. 123 (1996), 453–

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[NZ97] A

x+388 pp. [NZ97] A. Nevo, R.J. Zimmer , Homogenous projective factors for actions of semi-simple L ie groups. Invent. Math. 138 (1999), 229–252. [NZ00] A. Nevo, R.J. Zimmer , A structure theorem for actions of semisimple Lie groups. Ann. of Math. 156 (2002), 565–594. [NZ02] A. Nevo, R.J. Zimmer , Actions of semisimple Lie groups with stationary measure. R...

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[7] [2000]

Characters of the group $\mathrm{EL}_d (R)$ for a commutative Noetherian ring $R$

[KK14] M. Kalantar, M. Kennedy , Boundaries of reduced C∗ -algebras of discrete groups. J. Reine Angew. Math. 727 (2017), 247–267. STATIONARY CHARACTERS ON LATTICES OF SEMISIMPLE LIE GROUPS 33 [Ke15] M. Kennedy , An intrinsic characterization of C∗ -simplicity. Ann. Sci. ´Ec. Norm. Sup´ er.53 (2020), 1105–1119. [LL20] Omer Lavi, Arie Levit , Characters of...

work page Pith review arXiv 2017

[8] [2002]

Furstenberg , A Poisson formula for semi-simple Lie groups

[Fu62a] H. Furstenberg , A Poisson formula for semi-simple Lie groups. Ann. of Math. 77 (1963), 335–386. [Fu62b] H. Furstenberg , Non commuting random products. Trans. Amer. Math. Soc. 108 (1963), 377–428. [Fu67] H. Furstenberg , Poisson boundaries and envelopes of discrete groups. Bull. Amer. Math. Soc. 73 (1967), 350–356. [Fu73] H. Furstenberg , Boundar...

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