{"id":"3e983a33-eaf2-43ed-9e4c-d7fb717af363","arxiv_id":"2501.16339","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Equality of twisted local gamma factors up to GL_l identifies generic supercuspidal representations of quasi-split non-split SO_{2l} up to the outer automorphism, proved directly via partial Bessel functions.","lead":"A direct proof is given of the local converse theorem for quasi-split non-split even special orthogonal groups, using Howe vectors and partial Bessel functions. The main theorem determines irreducible generic supercuspidal representations from their local gamma factors, up to an outer automorphism.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Concern: key propositions cited to unpublished [15] (esp. Prop 6.9 and Lemma 6.3(1)) are load-bearing and unverified in the text; if either fails, Theorem 6.11 and hence Theorem 1.2 collapse.","rationale":"The reader's weakest_assumption identifies the same cluster: the proof leans on Proposition 4.5, Lemma 6.3(1), and Proposition 6.9 from the unpublished [15], plus the omitted proof of Corollary 4.11. My stress-test focuses on Proposition 6.9 and Lemma 6.3(1) because they are used directly in the pivotal Theorem 6.11 and in the earlier reductions that isolate B_m(g,f_{\\tilde w_{l-1}}). The central claim is Theorem 1.2, and its proof cannot be checked without these statements. This is not a manufactured objection: the paper's own text marks these results as '[15, Proposition ...]' and 'In prepartion', and the reader's conditional verdict is appropriate. I see no reason to move the verdict away from CONDITIONAL; the concern reinforces the need for the missing proofs or an available version of [15]. I also note the generic case has independent support from Haan-Kim-Kwon [14], so the mathematical statement is plausible; the issue is internal verifiability of this particular proof.","tokens_in":34928,"tokens_out":3999,"duration_ms":41296,"concrete_test":"Obtain [15] or independently verify Proposition 6.9 and Lemma 6.3(1). A concrete finite check for l = 2 and l = 3: write explicit matrices for t = diag(t_1,...,t_{l-1}, [[a, b rho],[b, a]], t_{l-1}^{-1},...,t_1^{-1}) with a != 1 and b != 0, and for w = t_{l-1}(w') \\tilde w_{l-1}; apply the embedding (6.5) into SO_{2l+1}, decompose the result according to Q_l w_l V_l, and check the stated criterion 'w in B_{l-1}(SO_{2l}) and a != 1' plus the displayed formula for A*. For Lemma 6.3(1), evaluate the defining partial-Bessel integral at t_{l-1}(a) and check that equality follows from equality of gamma factors or from the central-character condition without additional hidden assumptions. If either check fails or reveals extra hypotheses, Theorem 6.11 needs repair before Theorem 1.2 is established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing concern is the paper's dependence on the author's unpublished manuscript [15], listed as 'In prepartion'. Three statements are imported from it without proof: Proposition 4.5 (Bessel-support partition), Lemma 6.3(1) (vanishing/equality on the Levi cell), and Proposition 6.9 (torus-times-Weyl elements in Q_l w_l V_l). Of these, Proposition 6.9 is especially critical. In Theorem 6.11 it is the only mechanism that restricts the torus integration to T_l = {t : a != ±1}, produces the matrix A used to realize the zeta integral over GL_l, and sets up the two-to-one map tw <-> ctc w. If Proposition 6.9 fails for some w or some t with a != 1, the claimed identity B_m(g,f_{\\tilde w_{l-1}}) + B^c_m(g,f_{\\tilde w_{l-1}}) = 0 is unsupported, and the final appeal to uniqueness of Whittaker models does not go through. Lemma 6.3(1) is used earlier to equate W^f_m(t_{l-1}(a)) and W^{f'}_m(t_{l-1}(a)) in Propositions 6.5 and 6.7 and again in Theorem 6.11; the text only says 'This follows from the proof of [15, Proposition 4.8]'. Additionally, the proof of Corollary 4.11, which supplies the initial decomposition into B_m(g,f_{\\tilde w_n}), is omitted, though it is plausibly an adaptation of [44, Corollary 4.7]. These are not disagreements with external consensus; they are unverifiable internal dependencies. The generic case has independent support from Haan-Kim-Kwon [14], but the supercuspidal proof in this paper is not self-contained where it matters most.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a local converse theorem for irreducible generic supercuspidal representations of quasi-split non-split even special orthogonal groups over non-Archimedean local fields: if two such representations have the same central character and the same family of local gamma factors twisted by all irreducible generic representations of GL_n for n ≤ l, then they are isomorphic up to the outer automorphism. The proof uses Howe vectors and partial Bessel functions, following the strategy of Zhang, Jo, and Hazeltine–Liu for other classical groups, with a new treatment of the nonsplit torus and the outer automorphism. The paper also states, without proof, an analogous generic-case theorem and a weak rigidity theorem for automorphic representations.","tokens_in":35391,"tokens_out":2753,"duration_ms":27388,"significance":"If the central argument is correct, the paper gives a direct, Arthur-independent proof of the supercuspidal local converse theorem for quasi-split non-split SO_{2l}, a case where the previously available proofs used theta correspondence or Langlands functoriality. The use of Howe vectors and partial Bessel functions is conceptually valuable and yields, as a byproduct, intrinsic proofs of the equality of gamma factors for π and π^c. The paper is honest about its dependencies, but several of those dependencies are unpublished statements from the author's own manuscript [15], and one corollary central to the induction is stated without proof. The generic-case theorem is independently known from Haan–Kim–Kwon and from Arthur's work, so the main novelty rests on Theorem 1.2, whose proof is not self-contained at load-bearing points.","major_comments":[{"comment":"The proof of Corollary 4.11 is omitted ('The proof is an adaptation of [44, Corollary 4.7] which we omit'). This corollary supplies the initial decomposition Equation (4.2), from which every later vanishing statement in Theorem 6.1 and Theorems 6.7 and 6.11 proceeds. Since the Bruhat-cell structure and Bessel support in the quasi-split non-split case are exactly what the paper develops, a proof or a complete reference must be provided; an appeal to an unpublished manuscript is not sufficient for this load-bearing step.","section":"§4.4, Corollary 4.11"},{"comment":"Lemma 6.3(1) is used in Propositions 6.5 and 6.7 and again in Theorem 6.11 to identify the partial Bessel functions of π and π′ on the Levi cell t_{l−1}(a). The text says only 'This follows from the proof of [15, Proposition 4.8]', where [15] is listed as 'In prepartion'. If this vanishing/equality statement fails, the comparison of the non-intertwined zeta integrals in those propositions collapses. The proof needs to be included in this paper or the statement must be verifiable in a publicly accessible reference.","section":"§6.1, Lemma 6.3(1)"},{"comment":"Proposition 6.9 is quoted from [15, Proposition 7.2] and is the only mechanism in Theorem 6.11 that restricts the torus integration to T_l = {t : a ≠ 1}, produces the matrix A used to realize the zeta integral over GL_l, and sets up the two-to-one map tw ↔ ctcw. If the characterization is wrong for some w or some t with a ≠ 1, the identity B_m(g,f_{\\tilde w_{l−1}}) + B^c_m(g,f_{\\tilde w_{l−1}}) = 0 is unsupported. This is a load-bearing external dependency and should be proved in full in this paper or replaced by a checkable argument.","section":"§6.3, Proposition 6.9"},{"comment":"Theorem 1.3 (generic case) and Theorem 1.5 (weak rigidity) are stated without proof. The paper notes that Theorem 1.3 is independently due to Haan–Kim–Kwon and also follows from Arthur, but the manuscript presents the theorem as one of its main results and gives only 'similar arguments as in [24, §3.2]'. Since the generic case is not the paper's novel contribution, the authors should either include the proof or clearly state that these are recorded for completeness and that the paper's direct proof covers the supercuspidal case only.","section":"Theorems 1.3 and 1.5"}],"minor_comments":[{"comment":"The abstract and Theorem 1.2 state characteristic p ≠ 2, while Section 2 begins 'Let n,l ∈ N and F be a non-Archimedean local field of characteristic 0.' Please clarify the precise characteristic assumptions consistently across the paper, especially for the supercuspidal theorem versus the generic theorem.","section":"Introduction, §2"},{"comment":"Reference [15] is listed as 'In prepartion'; this should be 'In preparation'.","section":"References"},{"comment":"The proof of Theorem 6.11 refers to 'Equation (5.4)' in two places when evaluating the image of sections under the intertwining operator; the intended displayed equation appears to be (5.2). Please correct the cross-references.","section":"Theorem 6.11 proof"}],"recommendation":"major_revision","confidential_remarks":"The paper's central claim is plausible and the approach is a natural extension of existing Howe-vector methods, but the proof currently depends at several structurally central points on the author's unpublished manuscript [15], which is not publicly verifiable. Given the journal's standards, I think the appropriate route is major revision requiring the author to either prove those statements in this paper or to make [15] available in a verifiable form. The omitted proof of Corollary 4.11 should also be supplied. I would not reject on mathematical grounds, but the self-containedness requirement is important here because the whole point of the paper is a direct proof."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know this is a serious direct proof of the local converse theorem for quasi-split non-split SO_{2l}, using Howe vectors and partial Bessel functions adapted to a non-split torus. The theorem itself is not new: Haan-Kim-Kwon [14] proved it by theta correspondence, and it also follows from Arthur. What is new is the method, and that is worth taking seriously.\n\nThe paper does a lot well. The adaptation to the non-split torus is the real technical work, and the observation that all relevant Weyl elements are self-conjugate under the outer automorphism simplifies the argument relative to the split case. The supercuspidal case in characteristic p ≠ 2 is handled directly, which goes beyond the generic-case results in [14]. The proof is written in the established style of Zhang, Jo, and Hazeltine-Liu, and it is honest about what it omits.\n\nThe soft spots are real, and they are exactly where the stress-test lands. Proposition 6.9, imported from the unpublished [15], is load-bearing: it is the only mechanism that restricts the torus to a ≠ ±1, produces the matrix A for the GL_l zeta integral, and sets up the two-to-one map that makes the final uniqueness argument go through. Lemma 6.3(1) and Proposition 4.5 are also cited to [15], and Corollary 4.11 is stated without its proof. Theorems 1.3 and 1.5 are stated without proof, though 1.3 was independently established in [14]. None of this is a contradiction with the paper's own claims; it is simply not self-contained where it matters most.\n\nMy own read: the strategy is credible and the proof follows a pattern that has worked for other classical groups. The missing pieces are likely fillable. But as submitted, a referee cannot verify the central argument without access to [15]. That is a serious structural problem, not a cosmetic one.\n\nThis paper deserves a serious referee. I would send it out, with a clear instruction that the referee check the dependencies on [15] and that the author be asked to either include the omitted proofs or make [15] available and establish the cited statements. If the missing pieces check out, this will be the standard reference for the direct approach. As is, it is a promising manuscript, not a finished reference.\n\nBest.","headline":"A genuinely new direct Bessel-function proof strategy for quasi-split non-split SO_{2l}, but load-bearing results are cited to an unpublished manuscript; referee if the missing pieces can be supplied.","tokens_in":35851,"tokens_out":2746,"would_cite":false,"duration_ms":25819,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F70","22E50","11F85"],"pacs":[],"model":"deepseek-v4-flash","headline":"Local twisted gamma factors up to GL_l determine supercuspidal representations of quasi-split nonsplit SO_{2l} up to the outer automorphism.","keywords":["local converse theorem","quasi-split even special orthogonal groups","local gamma factors","Howe vectors","partial Bessel functions","supercuspidal representations","generic representations","outer automorphism"],"falsifier":"Take l = 2 (the quasi-split non-split group SO_4) and compute B_m(t w̃_1, f_{w̃_1}) and B_m^c(t w̃_1, f_{w̃_1}) for a non-c-fixed torus element t; if the equality of GL_2 gamma factors does not force the sum to vanish, Theorem 6.11 is false. Equivalently, one could search for two supercuspidal ψ-generic representations of quasi-split non-split SO_4(F) with equal gamma factors against all GL_1 and GL_2 twists that are neither isomorphic nor outer-conjugate.","tokens_in":34734,"feed_emoji":"🎯","tokens_out":8493,"duration_ms":78124,"temperature":0.7,"pith_summary":"This paper proves a local converse theorem for the quasi-split non-split even special orthogonal group SO_{2l} over a non-Archimedean local field of characteristic p ≠ 2. It shows that two irreducible generic supercuspidal representations with the same central character are isomorphic, or become isomorphic after applying the outer automorphism, whenever their twisted gamma factors agree for all twists by generic representations of GL_n with n ≤ l. The proof is direct, using Howe vectors and partial Bessel functions to extract information cell by cell from the Bruhat decomposition, rather than importing the result through Langlands functoriality. The paper also states a generic-case version over characteristic-zero fields and an automorphic weak-rigidity consequence, but those proofs are omitted.","feed_headline":"Local gamma factors pin down SO(2l) generic reps","feed_subtitle":"Equality against all GL(n) twists up to n=l forces supercuspidal reps to be isomorphic or outer-conjugate.","key_machinery":"The engine of the proof is the partial Bessel function B_m(g; f), defined by averaging a Whittaker function W^f over the Howe-vector subgroup U_m; it transforms under the generic character on upper-triangular unipotents and under an auxiliary character ψ_m on a compact subgroup H_m. The Bessel support of these functions is the set of Weyl elements supporting nonzero values, and it is partitioned into Bruhat cells B_n(SO_{2l}) for n = 1, ..., l−1. Equality of gamma factors for GL_k twists is shown to erase all cells with n ≤ k; the GL_{l−1} twist erases the surviving cell on c-fixed elements, and the GL_l twist shows that on the remaining elements B_m + B_m^c = 0. Because B_m^c is the partial Bessel function of π^c, uniqueness of Whittaker models forces π ≅ π′ or π ≅ π′^c.","core_discovery":"The paper's central claim is Theorem 1.2: over a non-Archimedean local field of characteristic p ≠ 2, if π and π′ are irreducible ψ-generic supercuspidal representations of quasi-split non-split SO_{2l}(F) with the same central character, and if γ(s, π × τ, ψ) = γ(s, π′ × τ, ψ) for every irreducible generic representation τ of GL_n(F) with n ≤ l, then π is isomorphic to π′ or to π′^c, where c is the outer automorphism. The proof compares partial Bessel functions attached to matrix coefficients of π and π′, shows that equality of gamma factors erases the Bruhat cells below rank l, and uses the GL_l twist to force the surviving cell terms to cancel after adding the outer-conjugate term. Uniqueness of Whittaker models then yields the dichotomy. The generic case and the automorphic weak-rigidity theorem are stated as consequences, with proofs referred to a future writeup.","pith_inferences":["[Editorial inference] The three unpublished Bessel-support statements are the natural target for independent verification; a self-contained proof of Proposition 4.5, Lemma 6.3(1), and Proposition 6.9 would remove the main external dependency of the argument.","[Editorial inference] The same partial-Bessel-function framework may extend to the positive-characteristic generic case, since Theorem 1.2 already works for p ≠ 2 and the missing step is the analogue of the supercuspidal-to-generic reduction.","[Editorial inference] Because the twist family leaves exactly the two-to-one ambiguity, adding a single invariant that changes sign under the outer automorphism on every pair π ≠ π^c would complete the classification; the paper notes that twisted exterior-square gamma factors are insufficient for SO_6, so the needed invariant must be something else."],"forward_implications":["For quasi-split non-split SO_{2l}, equality of gamma factors against GL_n twists for n ≤ l identifies a generic supercuspidal representation up to the two-to-one ambiguity π ↔ π^c.","The gamma factors of π and π^c coincide for all GL_n twists with n ≤ l, so this family of invariants cannot separate an outer-conjugate pair; any finer uniqueness statement needs an additional invariant.","Over characteristic-zero fields, the statement extends from supercuspidals to all irreducible generic representations, giving a local converse theorem at the full generic level.","The generic local theorem yields a weak rigidity statement for cuspidal automorphic representations: agreement of local components at almost all places forces agreement or outer-conjugacy at every place."],"supporting_citations":[{"why":"Supplies the Bessel-support partition, the vanishing on the Levi cell, and the torus–Weyl-element criterion that are load-bearing in Sections 6.1 and 6.3.","marker":"[15]"},{"why":"Gives the split SO_{2l} analogue of the argument, including the section construction and the smoothness argument for the key function.","marker":"[17]"},{"why":"Defines the Rankin–Selberg zeta integrals and twisted gamma factors for SO_{2l} × GL_n that the converse theorem uses.","marker":"[29]"},{"why":"Provides the Bruhat-cell decomposition theorem used to reduce differences of partial Bessel functions to the cells B_n(SO_{2l}).","marker":"[9]"},{"why":"Is the symplectic-group local converse theorem whose Bruhat-cell and Bessel-function techniques are adapted here, including Lemma 6.3.","marker":"[44]"},{"why":"Supplies the analytic criterion that turns vanishing of all GL_k zeta integrals on Whittaker functions into vanishing of the partial Bessel function on the Bruhat cell.","marker":"[6]"},{"why":"Provides the vanishing principle that partial Bessel functions vanish on Bruhat cells outside the Bessel support, which underlies the cell-by-cell reduction.","marker":"[8]"}],"fun_headline_variants":["Gamma factors up to n=l force SO(2l) supercuspidals to be outer-twins","Equality of twists up to n=l pins down SO(2l) generic supercuspidals","SO(2l) converse theorem: twists up to l settle supercuspidals","Local converse for SO(2l): gamma factors up to GL(l) decide","Twists up to l suffice: SO(2l) generic reps unique up to outer"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof leans on three structural claims about which Weyl cells can support the special functions and how torus elements land in them, all taken from the author's own manuscript in preparation; if any of those claims is wrong, the reduction to the final Bruhat cell and the closing GL_l computation collapse.","fun_headline_variants_meta":{"raw":{"variants":["Gamma factors up to n=l force SO(2l) supercuspidals to be outer-twins","Equality of twists up to n=l pins down SO(2l) generic supercuspidals","SO(2l) converse theorem: twists up to l settle supercuspidals","Local converse for SO(2l): gamma factors up to GL(l) decide","Twists up to l suffice: SO(2l) generic reps unique up to outer"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00068,"raw_usage":{"total_tokens":3008,"prompt_tokens":785,"completion_tokens":2223,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":401,"completion_tokens_details":{"reasoning_tokens":2105}},"tokens_in":401,"tokens_out":2223,"duration_ms":13507,"temperature":1.0,"reasoning_tokens":2105,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T20:37:38.637273+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take l = 2 (the quasi-split non-split group SO_4) and compute B_m(t w̃_1, f_{w̃_1}) and B_m^c(t w̃_1, f_{w̃_1}) for a non-c-fixed torus element t; if the equality of GL_2 gamma factors does not force the sum to vanish, Theorem 6.11 is false. Equivalently, one could search for two supercuspidal ψ-generic representations of quasi-split non-split SO_4(F) with equal gamma factors against all GL_1 and GL_2 twists that are neither isomorphic nor outer-conjugate.","supporting_citations":[{"cited_title":"5, 12, 13, 18, 26, 32 LOCAL CONVERSE THEOREM FOR QUASI-SPLIT SO 2l 33","cited_arxiv_id":null,"evidence_quote":"Supplies the Bessel-support partition, the vanishing on the Levi cell, and the torus–Weyl-element criterion that are load-bearing in Sections 6.1 and 6.3."},{"cited_title":"Theory, to appear","cited_arxiv_id":null,"evidence_quote":"Gives the split SO_{2l} analogue of the argument, including the section construction and the smoothness argument for the key function."},{"cited_title":"1, 2, 6, 7, 8, 9","cited_arxiv_id":null,"evidence_quote":"Defines the Rankin–Selberg zeta integrals and twisted gamma factors for SO_{2l} × GL_n that the converse theorem uses."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Bruhat-cell decomposition theorem used to reduce differences of partial Bessel functions to the cells B_n(SO_{2l})."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Is the symplectic-group local converse theorem whose Bruhat-cell and Bessel-function techniques are adapted here, including Lemma 6.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the analytic criterion that turns vanishing of all GL_k zeta integrals on Whittaker functions into vanishing of the partial Bessel function on the Bruhat cell."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the vanishing principle that partial Bessel functions vanish on Bruhat cells outside the Bessel support, which underlies the cell-by-cell reduction."}],"review_version":1}