{"id":"4960895e-23e5-4a2b-95ef-b750cb3075bc","arxiv_id":"2509.09065","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Formal degree and Atiyah-Schmid dimension formulas hold for square integrable projective representations of second countable almost unimodular groups with a 2-cocycle twist.","lead":"This paper shows that the basic construction for an almost unimodular group is isomorphic to a twisted group von Neumann algebra, and it extends the Atiyah-Schmid dimension formula to square integrable projective representations of such groups with finite covolume subgroups. The results give a unified treatment of formal degrees for projective representations in the non-unimodular setting.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.6 is stated without proof: its conclusion rests entirely on the unverified companion [GGN25, Thm 5.12] plus a transfer argument that is not written out. If [GGN25] is wrong, the paper's central formula is unsupported.","rationale":"I read the paper as a projective/twisted generalization of the author's earlier almost-unimodular program. The genuinely new tools (Thm 2.2, Prop 2.4, Prop 2.5, Lemma 4.2, Thm 4.3) are largely proved in the text, and I found no internal contradiction in those proofs. The central advertised result, Thm 4.6, is however not proved in the manuscript; the proof is explicitly omitted and deferred to a computation 'similar' to Thm 4.3 plus [GGN25, Thm 5.12]. The load-bearing question is therefore whether [GGN25, Thm 5.12] is correct and whether the transfer through the central extension preserves the formal degree and covolume exactly as needed. The reader's weakest_assumption identified exactly this external dependency and the omitted proof. I agree. Because the argument that is present is coherent and the missing piece is explicitly identified by the author, the honest verdict is unchanged: conditional acceptance pending independent verification of the companion theorem and completion of the omitted derivation.","tokens_in":23202,"tokens_out":9892,"duration_ms":113121,"concrete_test":"Write out the omitted proof of Thm 4.6 in full and check four transfer points: (a) G(ω)=T⋊_{(1,ω)}G is second countable and almost unimodular (Prop 2.4); (b) the lifted representation π_{1,ω} of T⋊_{(1,ω)}kerΔ_G is irreducible square integrable and its formal degree equals d_{π1}; (c) the map θ^ω_Δ from Thm 4.3 is exactly eΩ_{G,1}∘θ_Δ∘eΩ^{-1}_{H,1}, with θ_Δ from [GGN25, Thm 5.8]; (d) combining [GGN25, Thm 5.12] with the dimension transfer of Thm 4.3 yields exactly d_{π1}(1/|Δ|∑δ)[μ_G:μ_H]. Separately, reduce to ω=1: Thm 4.6 must reproduce [GGN25, Thm 5.12] verbatim. If any step fails or requires an extra hypothesis, the theorem statement needs amendment.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After Theorem 4.6 the manuscript 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 the central result, and no derivation is supplied. The intended route is to lift G to G(ω)=T⋊_{(1,ω)}G, apply [GGN25, Thm 5.12] to H(ω)≤G(ω), and transfer the Murray–von Neumann dimension back through the corner isomorphisms of Prop 2.5/Lemma 4.2. The written parts of that route (Props 2.4, 2.5, Lemma 4.2, Thm 4.3) are plausible and internally coherent, but the final formula d_{π1}(1/|Δ|∑δ)[μ_G:μ_H] requires: (i) [GGN25, Thm 5.12] is correct; (ii) the lift π_{1,ω} of π1 has the same formal degree d_{π1}; (iii) the covolume transfers as [μ_{T×G}:μ_{T×H}]=[μ_G:μ_H]; (iv) the θ^ω_Δ map is the exact compression of the θ_Δ from [GGN25, Thm 5.8]. Point (i) is an unverified external premise: [GGN25] is an arXiv preprint by the same author, not machine-checked, and this paper does not reproduce any part of it. Points (ii)–(iv) are asserted rather than demonstrated in the omitted proof. Thus the central claim is conditional on a companion paper.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":23567,"tokens_out":9161,"duration_ms":92829,"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":[{"comment":"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.","section":"Theorem 4.6 (§4)"},{"comment":"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.","section":"Overall dependence on [GGN25]"}],"minor_comments":[{"comment":"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.","section":"§4 (notation)"},{"comment":"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'.","section":"Throughout"},{"comment":"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.","section":"Theorem 4.6 / Corollary 4.7"},{"comment":"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)'.","section":"Example 2.3"},{"comment":"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.","section":"Proposition 2.1"}],"recommendation":"major_revision","confidential_remarks":"The paper's main theorem, Theorem 4.6, is explicitly unproved and depends on the companion preprint arXiv:2504.08107. I would ask the editor to ensure that either the proof is supplied in the revision or [GGN25] has been accepted/published before publication; otherwise the manuscript's central claim remains conditional."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nWhere 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.\n\nThe 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.\n\nWho 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.\n\nRecommendation: 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.","headline":"A well-written projective analogue of the almost unimodular program; the main formula is asserted without proof and leans on an unpublished companion preprint.","tokens_in":24102,"tokens_out":2293,"would_cite":true,"duration_ms":24096,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22D25","46L10","22D10"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["projective representations","almost unimodular groups","Atiyah–Schmid formula","twisted group von Neumann algebras","Plancherel weight","formal degree","Murray–von Neumann dimension","2-cocycles"],"falsifier":"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.","tokens_in":23021,"feed_emoji":"🧮","tokens_out":6517,"duration_ms":66617,"temperature":0.7,"pith_summary":"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.","feed_headline":"Projective Atiyah–Schmid formula holds for almost unimodular groups","feed_subtitle":"Twisted 2-cocycles no longer break the dimension-counting formula for square-integrable representations.","key_machinery":"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","core_discovery":"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","pith_inferences":["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}."],"forward_implications":["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."],"fun_headline_variants":["Atiyah–Schmid formula proven for projective reps","Almost unimodular groups: projective Atiyah–Schmid","Twisted Atiyah–Schmid formula for projective reps","Projective Atiyah–Schmid: almost unimodular case"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Atiyah–Schmid formula proven for projective reps","Almost unimodular groups: projective Atiyah–Schmid","Twisted Atiyah–Schmid formula for projective reps","Projective Atiyah–Schmid: almost unimodular case"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000591,"raw_usage":{"total_tokens":2653,"prompt_tokens":833,"completion_tokens":1820,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":577,"completion_tokens_details":{"reasoning_tokens":1760}},"tokens_in":577,"tokens_out":1820,"duration_ms":14768,"temperature":1.0,"reasoning_tokens":1760,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T19:43:16.217285+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}