{"id":"a7156c6a-a53d-4783-94e9-294028daf354","arxiv_id":"2601.00124","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For cotangent representations of reductive groups with a suitable torus, Borel-Moore homology of the zero fiber is torsion free and the K-theoretic restriction satisfies wheel-type ideal divisibility conditions.","lead":"This math paper proves that for many group actions, the homology of the zero-fiber of the moment map is torsion-free, so it embeds into a ring of invariants; it also describes divisibility conditions ('wheel conditions') satisfied by the K-theory image. A specialist audience interested in cohomological Hall algebras and their K-theoretic analogues would read it to see how quiver results extend to cotangent representations of reductive groups.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.4's torsion-freeness rests on Lemma 5.5, whose proof invokes [H25a, Cor 1.11] for purity of H^BM_G(μ_V^{-1}(0),Q) without verifying that μ_V^{-1}(0)/G is a symmetric quotient stack; if that purity fails, the Leray spectral sequence (5.3) may not degenerate and the central claim is unsuppor","rationale":"Reader's weakest_assumption identifies exactly the same step: Lemma 5.5's reliance on [H25a, Cor 1.11] and the purity-induced degeneration of (5.3). I agree. I considered alternative potential concerns—e.g., the stratification argument in Theorem 5.4's proof, the base-change step in K-theoretic Theorem 6.2, and the degree shifts in Lemma 5.6—but none is as consequential or as external to the paper as the purity input. The K-theory wheel-condition part is largely self-contained and its local computations with line-bundle resolutions are sound. The torsion-freeness theorem is presented as the main structural result, and its proof is conditional on a cited theorem whose scope is not demonstrated. Thus the concern is real, but it is not an internal contradiction; it is a gap in verification. The paper is honest about its conditional scope and even acknowledges a prior inaccuracy in this exact proof. Since the gap can be closed by checking the applicability of [H25a] or by a direct computation in a minimal example, the appropriate disposition remains CONDITIONAL as the Reader recommended. Verdict unchanged.","tokens_in":24073,"tokens_out":13397,"duration_ms":122587,"concrete_test":"Independently verify the hypotheses of [H25a, Cor 1.11] for μ^{-1}_V(0)/G with G=SL_2(C), V=Sym^2(C^2), T^s=(C^*)^2 as in Example 6.5; if the stack is not symmetric in the sense of [H25a], compute the E_2 page of (5.3) directly and look for a nonzero differential. If a nonzero d_2 exists, Lemma 5.5 fails and Theorem 5.4's torsion-freeness is false for this case; if the spectral sequence degenerates despite non-symmetry, the argument may be saved by a weaker purity input.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The pivotal step is Lemma 5.5, asserting that H^BM_{G×T^s}(μ^{-1}_V(0),Q) is free as a k_2-module. Its proof sets up the Leray spectral sequence for v_{λ,N}: Y_{λ,N}→S_{χ,N}. Equation (5.3) identifies E_2^{p,q} with H^p(S_{χ,N},Q)⊗H^{BM}_{-q+2c}(Y'_{λ,N},φ_{f'_{λ,N}}Q). The proof then claims this RHS is pure, citing [H25a, Cor 1.11] and [D22, Lemma 9.5], and concludes degeneration at E_2. The load-bearing gap is that [H25a] is a result on cohomological integrality for symmetric quotient stacks, while here G is an arbitrary reductive group and V an arbitrary representation; the stack μ^{-1}_V(0)/G is not shown to be symmetric. The paper supplies no verification that the hypotheses of [H25a, Cor 1.11] apply to the relevant cohomology groups, nor that purity of H^BM_G(μ^{-1}_V(0)) propagates to the fibers H^*(Y'_{λ,N},i^*Dφ_{f'}Q) appearing in (5.3). If [H25a] does not cover this class, Lemma 5.5 is unproven, and the subsequent localization argument in Theorem 5.4 cannot exclude S-torsion. The author's acknowledgment that Davison pointed out an inaccuracy in an earlier version of Theorem 5.4's proof underscores that this step is sensitive.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the cotangent stack T*(V/G) for a complex reductive group G and a finite-dimensional representation V. It constructs, following Kontsevich--Soibelman and Davison, a Hall induction map between Borel--Moore homology groups of the zero loci of moment maps associated to cocharacters, using equivariant dimensional reduction and critical vanishing cycles. The main homological result, Theorem 5.4, asserts that, under assumptions on an auxiliary torus T^s, the H_{G×T^s}-module H^BM_{G×T^s}(μ_V^{-1}(0),Q) is torsion free, equivalently that the restriction to the fixed point is an embedding. The proof relies on Lemma 5.5, whose proof cites a purity result of Hennecart and a theorem of Davison. In K-theory, Theorem 6.2 gives wheel-type divisibility conditions for the image of the restriction map to a fixed point, with applications to adjoint representations and to irreducible representations of SL_2(C).","tokens_in":24549,"tokens_out":9026,"duration_ms":87305,"significance":"If the proofs are completed, Theorem 5.4 would extend the Schiffmann--Vasserot and Davison embedding results for preprojective CoHA of quivers to arbitrary cotangent representations of reductive groups, giving a torsion-free equivariant-homology realization of Hall induction. Theorem 6.2 is a useful K-theoretic analog of wheel conditions outside the quiver setting, and the examples are informative. The paper is clearly written and exhibits the expected functorial structures. However, the central homological claim depends on cited purity statements whose hypotheses are not verified in the present setting, so the current proofs are conditional rather than complete.","major_comments":[{"comment":"The proof asserts that the right-hand side of (5.3) is pure, citing [H25a, Corollary 1.11] and [D22, Lemma 9.5], and concludes that the Leray spectral sequence degenerates at E2. But [H25a, Corollary 1.11] is a result for symmetric quotient stacks, and the manuscript does not verify that μ_V^{-1}(0)/G, or the fiber varieties Y'_{λ,N} appearing in (5.3), satisfy the hypotheses of that corollary for arbitrary reductive G and arbitrary representation V. It also does not explain how purity of H^BM_G(μ_V^{-1}(0),Q) propagates to the specific fiber cohomology groups in (5.3). Without this verification, the freeness of H^BM_{G×T^s}(μ_V^{-1}(0),Q) as a k_2-module is unsupported, and the subsequent localization argument in Theorem 5.4 cannot exclude S-torsion. This is load-bearing, and the author's own acknowledgement of a prior inaccuracy in the proof of Theorem 5.4 underscores the sensitivity o","section":"§5.1, Lemma 5.5 and Eq. (5.3)"},{"comment":"After Lemma 5.5, the proof reduces the torsion check to the locus bN = {(x,a) ∈ V×N : a.x = 0} by citing 'the argument in [SV22, Proposition 5.2]' for an isomorphism of localized pushforward maps. The statement and hypotheses of [SV22, Proposition 5.2] are not recorded, and the manuscript does not explain why that proposition applies to the present class of cotangent representations. This reduction is essential for the subsequent stratification by nilpotent orbits and for the final S-torsion-freeness, so the proof is incomplete at this point as written.","section":"§5.1, proof of Theorem 5.4 after Lemma 5.5"},{"comment":"The K-theoretic containment is derived from the chain j^* = i_0^* p^* = v_0^* i_V^* p^* = v_0^* p'_* i_V^! and the computation v_0^* p'_*[O_l] = 1 - χ_l^{-1}. The computation is plausible, but the manuscript does not justify that the square in the diagram is Cartesian as a scheme-theoretic fiber product: the condition that l⊕l' intersects μ_V^{-1}(0) in l∪l' does not by itself rule out embedded components or non-reduced structure. If the square is not Cartesian, the base-change identity f^* g_* = g'_* f^! cannot be applied. Theorem 6.2 should either state the scheme-theoretic hypothesis explicitly or prove the required fiber-product statement for the pairs of lines used in the examples.","section":"§6.2, proof of Theorem 6.2"}],"minor_comments":[{"comment":"The module structure of H^BM(X/G,Q) over H(BG,Q) is stated in a way that conflates the general construction with the smooth case. For singular X, such as μ_V^{-1}(0), the module structure used in Theorem 5.4 should be defined explicitly.","section":"§2.2, final paragraph"},{"comment":"Typos and small errors: 'embbedding' should be 'embedding'; 'a pont' in §2.3 should be 'a point'; the reference [RSYZ20] has an incomplete year '202'.","section":"§6.2"},{"comment":"The proof of associativity is only 'This is proven the same way as in [KS11]'. Given that the induction map is defined as a composition of five nontrivial steps, a precise reference or a few words explaining which compatibilities are used would improve the exposition.","section":"§4.4, Lemma 4.7"}],"recommendation":"major_revision","confidential_remarks":"The main risk is the reliance on [H25a, Corollary 1.11] for arbitrary reductive G and arbitrary V. The editor may wish to ask the author to verify the applicability of that purity result, or to supply an alternative proof of the degeneration used in Lemma 5.5. The recommendation is major revision rather than rejection because the claims may well be correct, but the current proofs have a central gap that must be addressed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a real extension of known quiver arguments to general reductive groups, and the wheel-condition part stands on its own. But Theorem 5.4 is not yet a theorem as written — the proof delegates the key purity step to [H25a, Cor 1.11], and the fit of that citation is not established.\n\nWhat's new: the Hall induction map for cotangent representations, the torsion-freeness statement, and the K-theoretic wheel conditions. The examples (adjoint representations, SL2/Sym^n) are concrete and useful. The exposition is honest about the conditional scope, and the acknowledgment of Davison's correction is a good sign, not a mark against the author.\n\nThe soft spot is exactly where your stress-test lands. Lemma 5.5 requires freeness over k2, and the proof invokes purity of H^BM_G(μ^{-1}_V(0)) from [H25a]. But [H25a] is about symmetric quotient stacks; here V is an arbitrary reductive representation and μ^{-1}(0)/G is not shown to be symmetric. The paper does not verify the hypotheses, nor does it show purity propagates to the fibers in the Leray spectral sequence. The earlier inaccuracy in this proof makes this more than a cosmetic gap. If purity fails, the spectral sequence need not degenerate and the torsion-freeness conclusion can't be reached. So the central result is plausible but conditional.\n\nThe wheel-condition theorem (6.2) is more self-contained: the base-change argument is transparent, and the examples are worked in enough detail to check. The notation is at times cramped, but the logic holds.\n\nBottom line: this paper is for specialists in CoHA and geometric representation theory. It deserves a serious referee, but the referee should ask for either a proof of the missing purity or a restatement of Theorem 5.4 that makes the dependence explicit. The wheel-condition part can be published as is.","headline":"Extends quiver CoHA results to cotangent representations, but the main torsion-freeness theorem should be read as conditional on unproved purity assumptions.","tokens_in":25006,"tokens_out":1974,"would_cite":false,"duration_ms":21190,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F08","14F43","16G20","20G05"],"pacs":[],"model":"deepseek-v4-flash","headline":"For cotangent representations of reductive groups, Hall induction lands in a torsion-free equivariant Borel–Moore homology module, so restriction to the fixed point is injective; in K-theory the image satisfies wheel-type divisibility condi","keywords":["Hall induction","cotangent representation","Borel-Moore homology","torsion freeness","wheel conditions","K-theoretic Hall algebra","vanishing cycles","moment map"],"falsifier":"Run the spectral sequence (5.3) on a cotangent representation whose zero fiber is not known to satisfy the cited purity condition; any non-zero differential on the E_2 page would produce an element annihilated by a non-zero equivariant class, i.e. torsion, directly contradicting Theorem 5.4.","tokens_in":1693,"feed_emoji":"🧩","tokens_out":2635,"duration_ms":107385,"temperature":0.7,"pith_summary":"This paper proves that for a cotangent representation of a complex reductive group, the Hall induction maps built from vanishing cycles and dimensional reduction land in a module over the equivariant cohomology of the acting group that is torsion free, provided an auxiliary torus with two specified weight directions is present. Equivalently, the restriction map from the Borel–Moore homology of the moment-map zero fiber to the equivariant cohomology of a fixed point is injective. A direct corollary is that this homology is concentrated in even degrees. In K-theory, the paper establishes a complementary statement: the image of the restriction map is contained in a Weyl-symmetric intersection of ideals generated by factors of the form 1 minus a line character, yielding a wheel condition that generalizes the known one-loop-quiver case. The upshot, if the theorem holds, is that Hall induction for arbitrary reductive-group representations admits the same kind of explicit symmetric-polynomial description that quiver cohomological Hall algebras enjoy.","feed_headline":"Cotangent Hall induction proven torsion-free","feed_subtitle":"Torus weights make the fixed-point restriction injective; K-theory gains wheel divisibility.","key_machinery":"The central object is the Hall induction map Ind^λ_ν, defined whenever λ⪯ν: it composes pullbacks, vanishing-cycle restriction and extension, and proper pushforward to map H^{BM}_{L_λ}(μ^{-1}_λ(0),Q)[d_λ+2l_λ] to H^{BM}_{L_ν}(μ^{-1}_ν(0),Q)[d_ν+2l_ν], with dimensional reduction translating vanishing-cycle cohomology of T^*V×g into Borel–Moore homology of the moment-map zero fiber. The torsion-freeness proof is carried by the auxiliary torus T^s, whose subtori C^*_1 and C^*_2 with weights (1,−1,0) and (1,0,−1) make the fixed locus a point and permit localization; the module is then shown free over the cohomology ring of C^*_2. In K-theory the operative mechanism is the base-change computation","core_discovery":"The central claim is Theorem 5.4: under three assumptions on an auxiliary torus T^s — it acts on T^*V×g preserving μ^{-1}_V(0), it commutes with G, and it contains subtori C^*_1, C^*_2 acting with weights (1,−1,0) and (1,0,−1) — the H_{G×T^s}-module H^{BM}_{G×T^s}(μ^{-1}_V(0),Q) is torsion free. This is equivalent to injectivity of the restriction map to the fixed point. The proof reduces torsion-freeness to freeness over the cohomology ring of C^*_2 via torus localization, and that freeness is obtained from a spectral sequence whose degeneration is supplied by a cited purity statement for the equivariant Borel–Moore homology of the zero fiber. Corollary 5.7, that the module is concentrated","pith_inferences":["The containment in Theorem 6.2 is stated as an inclusion; the worked examples suggest that for adjoint representations and SL_2 symmetric powers the intersection of ideals may describe the image exactly, which would give a complete shuffle-algebra presentation of the K-theoretic Hall algebra of cotangent representations.","The purity input in Lemma 5.5 is used only to force a spectral sequence to degenerate; if degeneration could be obtained from a weaker vanishing statement, Theorem 5.4 would extend beyond the cited purity setting, including singular quotient stacks.","For the additive character stacks of genus g, the wheel conditions impose divisibility on K-theory classes of G-local systems on Riemann surfaces; testing sharpness for small rank could connect these conditions to known generators of character-variety K-theory.","The even-degree concentration of the homology suggests a possible motivic refinement: if the Hall induction maps split on graded pieces, the structure might be upgraded to a categorical statement rather than a numeric one."],"forward_implications":["For every representation satisfying the torus hypotheses, Hall induction is an embedding into a torsion-free equivariant module, giving a concrete realization of the zero-fiber homology as a subspace of symmetric polynomials.","The Borel–Moore homology of μ^{-1}_V(0) is concentrated in even homological degrees, producing a parity collapse for the entire induction ladder.","In K-theory, the image of the restriction map obeys explicit wheel divisibility: any image class vanishes when both a line character and its paired dual-line character equal 1, for every Cartesian pair in the statement.","For the adjoint representation of GL_n the wheel conditions recover the one-loop-quiver preprojective conditions, and for SL_2 symmetric powers the theorem yields three explicit families of ideals.","The torsion-freeness statement permits computations of Hall induction on the fixed-point side via the localization isomorphism, rather than directly on the singular zero fiber."],"fun_headline_variants":["Torsion-free proof for cotangent Hall induction","Hall induction loses torsion in Borel-Moore homology","Fixed-point injectivity proven for cotangent Hall","K-theory wheel conditions verified for Hall induction"],"cache_read_input_tokens":26240,"weakest_assumption_plain":"The proof of the torsion-freeness theorem delegates the degeneration of its key spectral sequence to a cited purity theorem for equivariant Borel–Moore homology, and if that purity statement does not cover the present class of reductive-group representations, the freeness over the subtorus cohomology and hence the entire theorem may fail.","fun_headline_variants_meta":{"raw":{"variants":["Torsion-free proof for cotangent Hall induction","Hall induction loses torsion in Borel-Moore homology","Fixed-point injectivity proven for cotangent Hall","K-theory wheel conditions verified for Hall induction"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000165,"raw_usage":{"total_tokens":1018,"prompt_tokens":607,"completion_tokens":411,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":351,"completion_tokens_details":{"reasoning_tokens":359}},"tokens_in":351,"tokens_out":411,"duration_ms":4582,"temperature":1.0,"reasoning_tokens":359,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T13:08:16.680213+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the spectral sequence (5.3) on a cotangent representation whose zero fiber is not known to satisfy the cited purity condition; any non-zero differential on the E_2 page would produce an element annihilated by a non-zero equivariant class, i.e. torsion, directly contradicting Theorem 5.4.","supporting_citations":[],"review_version":1}