{"id":"4d086311-c27f-463b-82f3-585ad6d776ef","arxiv_id":"2607.01598","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Translates Gomez-Gourevitch-Sahi criterion into Zelevinsky data for degenerate Whittaker vanishing and shows vanishing implies Prasad's pole inequalities but converse fails via counterexamples.","lead":"The paper translates an orbit-theoretic criterion for vanishing of twisted Jacquet modules of GL_{2n} representations into Langlands-Zelevinsky data. It proves one direction of Prasad's conjecture on adjoint L-function poles but disproves the converse with explicit counterexamples for GL_4 and GL_6.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest assumption is precisely the step the paper claims to carry out; the explicit counter-examples and the GL_6 calculation provide independent verification that the translation is being applied correctly inside the stated range. No further load-bearing gap is visible from the argument structure.","tokens_in":1846,"tokens_out":280,"duration_ms":21655,"concrete_test":"For n=2 (GL_4), enumerate all irreducible representations whose Zelevinsky data m^t has longest segment exactly length 2; compute the corresponding degenerate Whittaker dimension directly via the known classification or via the GGS wave-front criterion and verify that none vanish.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is the explicit translation of the GGS orbit-theoretic vanishing criterion into the statement that π_{N,ψ}=0 ⇔ m^t contains a segment of length ≥ n+1. The manuscript performs this translation for irreducible admissible representations of GL_{2n}(F) and supplies explicit checks (counter-examples to the Prasad L-function converse on GL_4 and GL_6) that confirm the directionality of the claimed equivalence. No internal inconsistency appears in the reduction steps or in the handling of the Zelevinsky multisegment data.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper translates the Gomez--Gourevitch--Sahi orbit-theoretic criterion for vanishing of the twisted Jacquet module π_{N,ψ} on GL_{2n}(F) into Langlands--Zelevinsky data: for an irreducible admissible π = L(m), π_{N,ψ} vanishes if and only if the dual multisegment m^t contains a segment of length at least n+1. It proves that this vanishing implies the adjoint L-function pole inequalities conjectured by Prasad, but constructs explicit counterexamples showing the converse fails, including one on GL_4(F) and three of the 32 generalized Steinberg constituents on GL_6(F).","tokens_in":1945,"tokens_out":530,"duration_ms":21744,"significance":"The combinatorial criterion in terms of Zelevinsky segments supplies an explicit, computable test for degenerate Whittaker models that builds directly on the cited GGS theorem and standard facts about L-functions. The counterexamples on low-rank groups clarify the precise relationship to Prasad's conjecture and include concrete calculations that can be independently verified. The work avoids circularity and provides falsifiable predictions via the explicit multisegment data.","major_comments":[{"comment":"The translation of the GGS wavefront-set condition into the segment-length criterion (stated in the abstract and proved in the main theorem) assumes the correspondence holds without additional restrictions for all irreducible admissible representations; the manuscript should explicitly verify this for non-tempered cases or cite the precise step where the orbit dimension maps to segment length ≥ n+1.","section":"Main translation theorem"},{"comment":"For the GL_4(F) counterexample, the explicit multisegment m and the computation confirming that L(s, π × π^∨) has the predicted pole while π_{N,ψ} does not vanish should be expanded to include the precise Zelevinsky data and the L-function order calculation.","section":"Counterexample section (GL_4)"}],"minor_comments":[{"comment":"Notation for the Zelevinsky dual m^t and the segment length should be introduced with a brief reminder in the introduction for readers unfamiliar with the classification.","section":"Introduction"},{"comment":"The count of 32 constituents on GL_6(F) and the identification of the three violating ones would benefit from a short table or explicit list of the multisegments.","section":"GL_6 calculation"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading of the manuscript and the recommendation of minor revision. We address each major comment below.","responses":[{"response":"The GGS theorem is stated for all irreducible admissible representations, and the Zelevinsky classification provides a bijection between multisegments and irreducible representations that holds uniformly, including for non-tempered representations. The mapping from wavefront set (via orbit dimension) to the existence of a segment of length at least n+1 in m^t follows directly from the standard combinatorial description of the wavefront set in terms of Zelevinsky data. We will add a clarifying remark after the statement of the main theorem, citing the relevant properties of the Zelevinsky classification that ensure the correspondence applies without further restrictions.","revision_made":"yes","referee_comment":"The translation of the GGS wavefront-set condition into the segment-length criterion (stated in the abstract and proved in the main theorem) assumes the correspondence holds without additional restrictions for all irreducible admissible representations; the manuscript should explicitly verify this for non-tempered cases or cite the precise step where the orbit dimension maps to segment length ≥ n+1."},{"response":"We agree that providing the explicit multisegment data and the detailed order computation for the adjoint L-function will strengthen the presentation of the counterexample. We will expand the relevant section (currently containing the GL_4 example) to include the precise Zelevinsky multisegment m together with the explicit calculation of the pole order of L(s, π × π^∨).","revision_made":"yes","referee_comment":"For the GL_4(F) counterexample, the explicit multisegment m and the computation confirming that L(s, π × π^∨) has the predicted pole while π_{N,ψ} does not vanish should be expanded to include the precise Zelevinsky data and the L-function order calculation."}],"tokens_in":1531,"tokens_out":416,"duration_ms":21527,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main things to know are the explicit translation of the Gomez-Gourevitch-Sahi vanishing criterion into Zelevinsky language and the concrete counterexamples to the converse direction of Prasad's conjecture.\n\nThe paper states that for π = L(m) on GL_{2n}(F), the twisted Jacquet module vanishes exactly when the dual multisegment m^t contains a segment of length at least n+1. It proves the vanishing direction implies the expected adjoint L-function pole conditions, then gives counterexamples on GL4 where the L-function condition holds but the module does not vanish. For the 32 generalized Steinberg constituents of a certain principal series on GL6, exactly three violate the converse.\n\nThe translation step and the explicit checks are the useful parts. They turn an orbit-theoretic statement into something directly usable with Langlands-Zelevinsky parameters and show the directionality of the implication with small-rank calculations that can be inspected.\n\nThe work rests on the cited GGS theorem, which is standard, and the translation appears to go through without extra restrictions on the field or the representation. The counterexamples are limited to n=2 and n=3, which is enough to disprove a general converse but leaves the general picture open. No internal inconsistencies show up in the stated claims.\n\nThis is for people working on Whittaker models and L-function conjectures for classical groups. A reader who needs a usable criterion in Zelevinsky data or wants to see where Prasad-type predictions break will find it directly applicable.\n\nIt deserves a serious referee. The statements are precise and the counterexamples are concrete enough to check.","headline":"The paper translates the GGS orbit criterion into a clean segment-length condition on Zelevinsky data and supplies explicit counterexamples showing the converse of Prasad's L-function conjecture fails on GL4 and GL6.","tokens_in":2434,"tokens_out":416,"would_cite":false,"duration_ms":18229,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22E50"],"pacs":[],"model":"grok-4.3","headline":"For representations of GL_{2n}, the twisted Jacquet module vanishes if and only if the Zelevinsky dual contains a segment of length at least n+1.","keywords":["degenerate Whittaker models","Zelevinsky classification","twisted Jacquet modules","GL(2n)","Langlands classification","adjoint L-functions","wave-front sets"],"falsifier":"Finding an irreducible representation of GL_{2n} where the Zelevinsky dual has no segment of length n+1 or more but the twisted Jacquet module still vanishes, or the opposite case.","tokens_in":2716,"feed_emoji":"","tokens_out":638,"duration_ms":43460,"temperature":0.7,"pith_summary":"The paper translates an orbit-theoretic criterion for the vanishing of twisted Jacquet modules into the Langlands-Zelevinsky classification for irreducible representations of GL_{2n} over non-Archimedean local fields. It shows that for π given by multisegment m, the module π_{N,ψ} is zero exactly when the dual m^t has a segment longer than n. This gives a concrete combinatorial test and partially resolves a conjecture of Prasad relating the vanishing to poles of the adjoint L-function, while providing counterexamples to the full conjecture.","feed_headline":"Whittaker model vanishes iff Zelevinsky dual has segment of length n+1","feed_subtitle":"Criterion for GL_{2n} gives explicit test in Langlands data and tests Prasad conjecture on L-functions","key_machinery":"The Zelevinsky dual m^t of the multisegment m parametrizing the representation, whose maximal segment length controls the vanishing of the twisted Jacquet module.","core_discovery":"The central claim is that if π = L(m) is an irreducible admissible representation of GL_{2n}(F), then the twisted Jacquet module π_{N,ψ} vanishes if and only if the Zelevinsky dual m^t contains a segment of length at least n+1. This equivalence follows from translating the wave-front set condition of Gomez-Gourevitch-Sahi into the language of multisegments.","pith_inferences":["The combinatorial condition may allow direct computation of vanishing without computing wave-front sets for other groups.","Counterexamples to Prasad's conjecture suggest that L-function poles are necessary but not sufficient for non-vanishing in some cases."],"forward_implications":["Vanishing implies the pole conditions on the adjoint L-function L(s, π × π^∨) predicted by Prasad.","For principal series containing the trivial representation, the vanishing can be determined explicitly for each generalized Steinberg constituent.","The criterion holds for all irreducible admissible representations of GL_{2n}(F)."],"fun_headline_variants":["GL_{2n} Whittaker vanishes iff Zelevinsky dual has n+1 segment","Degenerate Whittaker criterion for GL_{2n} in Zelevinsky data","π_{N,ψ} vanishes iff m^t has segment length >=n+1","Wavefront sets translated to explicit vanishing test in GL_{2n}","Langlands data determines degenerate Whittaker vanishing for GL_{2n}"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The orbit-theoretic vanishing criterion can be translated directly into the Langlands-Zelevinsky classification without additional restrictions on the representation or the field.","fun_headline_variants_meta":{"raw":{"variants":["GL_{2n} Whittaker vanishes iff Zelevinsky dual has n+1 segment","Degenerate Whittaker criterion for GL_{2n} in Zelevinsky data","π_{N,ψ} vanishes iff m^t has segment length >=n+1","Wavefront sets translated to explicit vanishing test in GL_{2n}","Langlands data determines degenerate Whittaker vanishing for GL_{2n}"]},"model":"grok-4.3","cost_usd":0.00634,"raw_usage":{"total_tokens":2960,"prompt_tokens":794,"num_sources_used":0,"completion_tokens":98,"cost_in_usd_ticks":63403000,"prompt_tokens_details":{"text_tokens":794,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2068,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":794,"tokens_out":98,"duration_ms":23955,"temperature":1.0,"reasoning_tokens":2068,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-03T03:40:38.397294+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Finding an irreducible representation of GL_{2n} where the Zelevinsky dual has no segment of length n+1 or more but the twisted Jacquet module still vanishes, or the opposite case.","supporting_citations":[],"review_version":1}