Projective representations of almost unimodular groups

Aldo Garcia Guinto

REVIEW 2 major objections 5 minor 30 references

Projective representations of almost unimodular groups

T0 review · 2 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read The paper establishes a projective analogue of the Atiyah–Schmid formula for almost unimodular groups, extending it to non-unimodular groups, 2-cocycle twists, and finite-covolume subgroups.

desk verdict A well-written projective analogue of the almost unimodular program; the main formula is asserted without proof and leans on an unpublished companion preprint. read the letter →

arxiv 2509.09065 v1 pith:EVRT57JF submitted 2025-09-11 math.OA

Aldo Garcia Guinto This is my paper · ORCID

classification math.OA MSC 22D2546L1022D10

keywords projectiverepresentationsalmostunimodulargroupsAtiyah–SchmidformulatwistedgroupvonNeumannalgebrasPlancherelweightformaldegreeMurray–vondimension2-cocycles

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 extends the Atiyah–Schmid dimension formula—a classical identity relating formal degrees of square-integrable representations to covolumes of lattice subgroups—to projective representations of almost unimodular groups. An almost unimodular group is a locally compact group whose modular function has open kernel; the paper treats groups that are not necessarily unimodular and subgroups that only have finite covolume, rather than being lattices. The argument goes through twisted group von Neumann algebras: the algebra generated by the left regular projective representation with respect to a 2-cocycle, equipped with its twisted Plancherel weight. The central new result is Theorem 4.6, which gives a formula for the Murray–von Neumann dimension of an induced square-integrable projective representation in terms of the formal degree on the unimodular kernel and the covolume of the subgroup. Along the way, the paper shows that almost unimodularity is preserved under passing to the central extension associated to a 2-cocycle, and that the basic construction for the Plancherel-weight inclusion is itself a twisted group von Neumann algebra.

What carries the argument

The central object is the twisted group von Neumann algebra L^ω(G)—the von Neumann algebra generated by the left regular ω-projective representation—together with the twisted Plancherel weight φ^ω_G. Almost unimodularity is equivalent to φ^ω_G being almost periodic, which makes the modular operator diagonalizable and identifies the centralizer with L^ω(ker Δ_G). The mechanism that carries the argument is the correspondence between ω-projective representations of G and ordinary representations of the central extension T ⋊_{(1,ω)} G whose restriction to T is one-dimensional. Via this correspondence, the paper transfers structure theorems and the Atiyah–Schmid formula for untwisted almost unimo

What would settle it

Compute both sides of the formula in a concrete non-unimodular example, e.g., G = ax+b group R ⋊ R_+, H = Z ⋊ p^Z, with a non-trivial continuous 2-cocycle and an irreducible square-integrable projective representation induced from the unimodular kernel. If the Murray–von Neumann dimension over (L^ω(H), φ^ω_H) does not equal d_{π_1} (1/|Δ| Σ_{δ∈Δ} δ) [μ_G:μ_H], the theorem fails. A simpler check: set ω ≡ 1; then the formula must reduce to the companion paper's untwisted Atiyah–Schmid formula, so any counterexample to that companion result falsifies this paper's main theorem.

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Extended reading notes

Core claim

The core claim is Theorem 4.6: for a second countable almost unimodular group G, a finite covolume subgroup H, and a Borel 2-cocycle ω, if (π,H) is an irreducible square-integrable ω-projective representation induced from (π_1,H_1) on ker Δ_G, then the Murray–von Neumann dimension of the extension π̃ followed by the inclusion θ^ω_Δ of the basic construction for H into that for G equals d_{π_1} (1/|Δ| Σ_{δ∈Δ} δ) [μ_G:μ_H]. Here d_{π_1} is the formal degree of the inducing representation, [μ_G:μ_H] is the covolume, and the average over Δ accounts for the modular image of H being a proper subgroup of that of G. When H is a lattice in a unimodular group and ω is trivial, this reduces to the clas

Load-bearing premise

The main theorem leans on the companion paper's untwisted Atiyah–Schmid formula for finite-covolume subgroups of almost unimodular groups, and Theorem 4.6's proof is omitted, so the central formula is asserted rather than derived in this text.

Editorial extensions

If this is right

The classical Atiyah–Schmid formula becomes a special case of a formula that applies to non-unimodular groups and finite-covolume subgroups, with a 2-cocycle twist.
Formal degree operators of irreducible square-integrable projective representations are diagonalizable, and formal degree weights of factorial ones are almost periodic; the centralizer is the algebra of the unimodular kernel.
The basic construction for the Plancherel-weight inclusion is a twisted group von Neumann algebra, yielding concrete groups where the twisted algebra is a semifinite factor while the untwisted group algebra is a purely infinite non-factor.
The dimension scaling identity (Theorem 4.3) gives a projective analogue of covolume scaling for Murray–von Neumann dimensions, with the scaling factor reducing to the covolume when the modular images of H and G coincide.

Reading between the lines

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

If the formula holds, it likely enables a projective version of L²-index theory for non-unimodular groups, where the formal degree operator plays the role of a density and the covolume provides the natural normalization.
The explicit continuous 2-cocycle on Δ_G(G)^× G may be a canonical cohomological invariant; computing its cohomology class in the paper's examples could reveal whether it detects changes in factoriality type.
Since the reduction to the central extension is the main tool, any new permanence property of almost unimodular groups should automatically transfer to twisted group von Neumann algebras, offering a test bed in concrete groups like the ax+b group.
The theorem is stated for irreducible representations; a plausible extension, not written in the paper, is that the same dimension formula holds for factorial square-integrable projective representations with the formal degree weight replacing d_{π_1}.

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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

This paper is a coherent, useful extension of the almost unimodular program to twisted/projective representations. The genuinely new items—the basic-construction identification (Theorem 2.2), the examples of twisted semifinite factors with non-factor group von Neumann algebras (Example 2.3), and the twisted Atiyah–Schmid formula (Theorem 4.6)—are natural and, as far as the written proofs go, internally consistent. The paper deserves a serious referee.

What works: the proof of Theorem 2.2 is written out completely and I could not find a gap. Proposition 2.5's decomposition of L(T ⋊ G) into twisted group algebras is handled carefully and will be a useful reference. Theorem 4.3's dimension formula is derived in detail through the central extension and the corner isomorphisms. The bookkeeping of the 2-cocycles and the weight identifications is done honestly.

Where it is soft: nearly every main theorem is proved by reduction to [GGN25], an unpublished preprint by the same authors. That is not by itself a crime—the untwisted results are the natural input—but the load-bearing weight of that reliance is high. Theorem 4.6, which is the headline, is stated with no proof at all. The note after it says the argument is similar to Theorem 4.3 and uses [GGN25, Theorem 5.12]; the route through the central extension is plausible, but the specific steps (formal degree invariance under the lift, covolume transfer, identification of the compressed θ map) are asserted, not shown. If [GGN25] is wrong anywhere in its structural theorems, this paper inherits the error uncorrected. Also, Proposition 2.1 is given without proof—minor, since it is a routine adaptation of GGN25's Theorem 2.1.

The math that is written down checks out. The paper is not trying to hide anything; it says exactly where the omitted proof is and what it depends on. The citation pattern is fine.

Who this is for: people working on group von Neumann algebras, formal degrees of non-unimodular groups, and the almost unimodular program. It would be a solid addition to the literature once the companion preprint is verified or the transfer argument is written out in a revision.

Recommendation: send it to peer review. The referee should have access to [GGN25] and be asked to verify that the untwisted theorems used here are correct, or the editors should ask for a revision that expands the proof of Theorem 4.6. Conditional acceptance is the right shape.

Referee Report

2 major / 5 minor

Summary. This paper develops the projective/twisted version of the theory of almost unimodular groups from [GGN25]. It proves that for an almost unimodular group G the basic construction for L(G)^{φ_G} ≤ L(G) is isomorphic to a twisted group von Neumann algebra L^ω(Δ_G(G)^̂ × G) with an explicit continuous 2-cocycle (Theorem 2.2), and it gives examples where the twisted algebra is a factor while the untwisted one is not. It establishes an equivalence between almost unimodularity of G and of the central extension T ⋊_{(1,ω)} G (Proposition 2.4) and a decomposition of L(T ⋊_{(1,ω)} G) into twisted group algebras (Proposition 2.5). In Section 3 it proves diagonalizability of formal degree operators and almost-periodicity of formal degree weights for square-integrable ω-projective representations. Section 4 proves a Murray–von Neumann dimension scaling formula for finite-covolume subgroups (Theorem 4.3) and states a twisted Atiyah–Schmid formula (Theorem 4.6). The paper is heavily dependent on the unpublished companion preprint [GGN25], and Theorem 4.6 is explicitly not proved in the manuscript.

Significance. If the results are correct, the paper gives a natural projective representation analogue of the Atiyah–Schmid formula for non-unimodular groups with finite-covolume subgroups, and it supplies concrete new examples of twisted group von Neumann algebras that are factors while the corresponding untwisted algebras are not. Strengths include the explicit construction in Theorem 2.2, the detailed proof of Proposition 2.5, and the relatively complete proof of Theorem 4.3. The written proofs are internally consistent. However, the central Theorem 4.6 is not proved and depends on an unpublished companion preprint, so the significance is conditional until that gap is filled.

major comments (2)

[Theorem 4.6 (§4)] The headline Atiyah–Schmid formula is stated without proof. The text says: 'We omit the details of the following proof since it is similar to how we computed the dimension in Theorem 4.3 and uses the Atiyah–Schmidt formula [GGN25, Theorem 5.12].' This is a self-declared missing proof. The conclusion is not a direct consequence of Theorem 4.3: the intended route via H(ω) ≤ G(ω) requires (ii) the formal degree of π_1 equals that of its lift π_{1,ω}; (iii) [μ_{T×G} : μ_{T×H}] = [μ_G : μ_H]; and (iv) θ^ω_Δ is the precise corner restriction of θ_Δ from [GGN25]. None of (ii)–(iv) is written out, and [GGN25] is an unpublished preprint. The central claim is therefore conditional.
[Overall dependence on [GGN25]] Nearly every new theorem is reduced to [GGN25]: Proposition 2.1, the proof of Theorem 2.2's first equality, Theorem 3.1, Theorem 3.2, Theorem 3.7, Theorem 3.8, Theorem 4.3, and Theorem 4.6 all import structural results from [GGN25] (Thms 2.1, 4.1, 4.2, 4.5, 4.6, 5.1, 5.4, 5.8, 5.12). The present paper does not state the needed [GGN25] results in enough detail for a reader to check them, and [GGN25] is not peer-reviewed. This is not circularity—the projective statements are not the untwisted theorems—but it makes the paper's validity conditional. The authors should either include the necessary statements/proofs or carefully document them in an appendix once [GGN25] is available in final form.

minor comments (5)

[§4 (notation)] The letter H is used both for a closed subgroup of G and for the Hilbert space of a representation, sometimes in the same theorem (e.g., Theorem 4.3). This makes statements hard to parse; renaming the Hilbert space or subgroup would help.
[Throughout] The name is inconsistent: 'Atiyah–Schmidt' appears in the introduction, in the proof of Theorem 4.6, and elsewhere, while the standard reference [AS77] and Theorem 4.6/Corollary 4.7 use 'Atiyah–Schmid'.
[Theorem 4.6 / Corollary 4.7] In the statements, φ^ω_G and φ^ω_H are called 'Plancherel weight', but the paper elsewhere correctly calls them 'twisted Plancherel weight'. This should be corrected for consistency.
[Example 2.3] In the second example the text becomes garbled: 'L(Δ_{G_2}(G_2)ˆ×L(G_2)' should presumably read 'L(Δ_{G_2}(G_2)ˆ×G_2)'.
[Proposition 2.1] The proof is omitted. Even if this is a routine adaptation of [GGN25, Thm 2.1], a short argument or a precise reference to the relevant line in [GGLN25, Lemma 1.4] would improve self-containedness.

Circularity Check

2 steps flagged · score 4.0 of 10

Twisted results are mostly a translation of the same author group's untwisted companion [GGN25]; Theorem 4.6's proof is explicitly omitted and deferred to [GGN25, Thm 5.12], making the capstone load-bearing self-citation rather than independent derivation.

self citation load bearing [Theorem 4.6 and the paragraph preceding it (Section 4, p. 15)]
"In [GGN25, Theorem 5.12], we generalize the Atiyah–Schmidt formula for finite covolume subgroups of almost unimodular groups. We provide a slight generalization to the case when the almost unimodular group admits a 2-cocycle ω. ... We omit the details of the following proof since it is similar to how we computed the dimension in Theorem 4.3 and uses the Atiyah–Schmidt formula [GGN25, Theorem 5.12]."

The central new formula is asserted without proof. The only indicated route is to cite the same authors' unpublished [GGN25, Theorem 5.12] and to claim similarity with Theorem 4.3. To arrive at dim = d_{π1}(1/|Δ|∑δ)[μ_G:μ_H] one needs unstated transfer facts: the lift π_{1,ω} has formal degree d_{π1}, the covolume passes through T⋊_{(1,ω)} unchanged, and θ^ω_Δ is the compressed θ_Δ. None are derived; the theorem is therefore a 'slight generalization' of the authors' own untwisted theorem whose proof is skipped. This is load-bearing self-citation: if [GGN25, Thm 5.12] fails, Theorem 4.6 has no support in this paper.
self citation load bearing [Theorem 3.2 proof (Section 3.1, p. 10)]
"By [GGN25, Theorem 4.2], there exists an irreducible square integrable representation (ρ,H_1) of G_1(ω) such that (π^ω,H) ∼= Ind_{G_1(ω)}^{G(ω)}(ρ,H_1). ... Thus by [GGN25, Theorem 4.2], we obtain that D is diagonalizable with the above formula."

The projective formal-degree diagonalization is obtained by applying the untwisted theorem [GGN25, Theorem 4.2] to the central extension G(ω)=T⋊_{(1,ω)}G and then translating back. The paper proves the translation (Mackey correspondence and weight identification), but the substantive structural input—existence of the inducing representation and the diagonal form D = ∑_{δ} d_{π1}δ 1_{d_{π1}δ}(D)—is imported wholesale from the same authors' preprint. This is a self-citation chain, though not an identity-by-construction.

full rationale

The paper is not circular in the strictest sense: the twisted results are not assumed as inputs, no fitted parameter is renamed as a prediction, and the central-extension machinery (Propositions 2.4, 2.5, and Lemma 4.2) is actually proved, giving independent content to the twist. However, the derivation chain is dominated by self-citation to [GGN25] (same author group, unpublished preprint) and [GGLN25]. Every major theorem—Proposition 2.1, Theorem 3.1, Theorem 3.2, Theorem 3.7, Theorem 3.8, Theorem 4.3, Theorem 4.5, and especially Theorem 4.6—reduces, via the central extension, to a corresponding theorem in [GGN25]. The capstone Theorem 4.6 is not proved at all; its proof is explicitly deferred to [GGN25, Theorem 5.12] and to the already-cited Theorem 4.3. The transfer facts needed for the final formula are not written. This is a load-bearing self-citation chain, not an equivalence by construction. Score 4 reflects: substantial independent content in the transfer/dimension calculations, but a central claim whose support is a same-author preprint with an omitted proof.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters or invented entities appear; the central objects (twisted group von Neumann algebras, formal degree operators, Murray-von Neumann dimensions) are standard. The main axiom burden is external: the paper rests on the companion preprints [GGN25] and [GGLN25] by the same authors, especially the untwisted Atiyah-Schmid formula [GGN25, Theorem 5.12].

assumptions (5)

domain assumption Twisted group von Neumann algebra machinery: existence of twisted Plancherel weight phi^omega_G, identification L^2(L^omega(G), phi^omega_G) = L^2(G), modular operator given by pointwise multiplication by Delta_G, and density of lambda^omega_G(L^1_omega(G)) (Section 1.1, after [Sut80]).
Invoked throughout, e.g. Proposition 2.1 and Theorem 3.2, as the base theory for the twisted setting.
domain assumption Mackey's central extension correspondence: omega-projective representations of G correspond to representations of T x_(1,omega) G with the Weil topology; square integrability and factoriality are preserved (Section 2 and Proposition 3.5, [Mac58], [Mac57]).
Basis for lifting all untwisted results to the projective setting.
domain assumption Almost unimodular structure theory from the companion preprint [GGN25], including: equivalence with almost periodicity of the Plancherel weight, basic construction as crossed product (Theorem 5.1), reduction of square integrable representations to ker Delta_G (Theorems 4.1, 4.2, 4.5, 4.6), dimension
This is the foundational input; the paper proves the twisted versions by reduction to [GGN25].
domain assumption Murray-von Neumann dimension theory for strictly semifinite weights from [GGLN25], including existence of standard intertwiners and dimension independence of the choice of intertwiner (Section 4).
Used to set up the dimensions in Theorem 4.3 and Theorem 4.6.
standard math Duflo-Moore and Moore theory of formal degree operators and semi-invariant weights for (projective) representations of non-unimodular groups (Section 3, [DM76], [Moo77]).
Quoted for the definition and key orthogonality properties of formal degrees.

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Pith. "Pith review of Projective representations of almost unimodular groups." pith.science (2026). https://pith.science/paper/EVRT57JF

@misc{pith2026250909065,
  author       = {Pith},
  title        = {Pith review of: Projective representations of almost unimodular groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EVRT57JF}},
  note         = {Machine review of arXiv:2509.09065}
}

abstract

Given an almost unimodular $G$, so that the Plancherel weight $\varphi_G$ on the group von Neumann algebra $L(G)$ is almost periodic, we show that the basic construction for the inclusion $L(G)^{\varphi_G} \leq L(G)$ is isomorphic to a twisted group von Neumann algebra of $G \times \Delta_G(G)\hat{\ }$ with a continuous 2-cocycle, where $\Delta_G$ is the modular function. We show that when $G$ is second countable and admits a Borel 2-cocycle, $G$ is almost unimodular if and only if the central extension $\mathbb{T} \rtimes_{(1,\omega)} G$ is almost unimodular. Using this result and the connection between $\omega$-projective representations of $G$ and the representations of $\mathbb{T} \rtimes_{(1,\omega)} G$, we show that the formal degrees of irreducible and factorial square integrable projective representations behaved similarly to their representations counterparts and obtain the Atiyah--Schmid formula in the setting of second countable almost unimodular groups with a 2-cocycle twist and a finite covolume subgroup, which uses the Murray--von Neumann dimension for certain Hilbert space modules over the twisted group von Neumann algebra with its twisted Plancherel weight.

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

Paolo Aniello, Square integrable projective representations and square integrable representations modulo a relatively central subgroup, Int. J. Geom. Methods Mod. Phys. 3 (2006), no. 2, 233--267. 2213819

2006

[2] [2]

Michael Atiyah and Wilfried Schmid, A geometric construction of the discrete series for semisimple L ie groups , Invent. Math. 42 (1977), 1--62. 463358

1977

[3] [3]

Alain Connes, \' E tats presque p\'eriodiques sur une alg\`ebre de von N eumann , C. R. Acad. Sci. Paris S\'er. A-B 274 (1972), A1402--A1405. 0295092

1972

[4] [4]

Functional Analysis 16 (1974), 415--445

, Almost periodic states and factors of type III _ 1 , J. Functional Analysis 16 (1974), 415--445. 0358374 (50 \#10840)

1974

[5] [5]

Duflo and Calvin C

M. Duflo and Calvin C. Moore, On the regular representation of a nonunimodular locally compact group., J. Functional Analysis (1976), no. no. 2,, 209--243. 393335

1976

[6] [6]

C. M. Edwards, C -algebras of central group extensions. I , Ann. Inst. H. Poincar\'e Sect. A (N.S.) 10 (1969), 229--246. 248284

1969

[7] [7]

C. M. Edwards and J. T. Lewis, Twisted group algebras. I , Comm. Math. Phys. 13 (1969), 119--130. 253064

1969

[8] [8]

Ulrik Enstad, The density theorem for projective representations via twisted group von N eumann algebras , J. Math. Anal. Appl. 511 (2022), no. 2, Paper No. 126072, 25. 4385190

2022