{"id":"5b424adc-38ce-469c-9619-c8dd857e25d5","arxiv_id":"2606.25387","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Characterizes kernel of Iwahori-Whittaker averaging microlocally, generalizes anti-temperedness equivalence theorem, and extends tilting property of central sheaves to integer coefficients.","lead":"The paper characterizes the kernel of Iwahori-Whittaker averaging in microlocal terms and applies this to generalize a prior theorem linking anti-temperedness to irregular singular support for automorphic sheaves. It further extends the tilting property of Whittaker averaged central sheaves to integer coefficients via a cited Radon transform argument.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Microlocal kernel characterization may require extra global checks before applying to automorphic sheaves on the stack","rationale":"The reader's weakest assumption correctly isolates the direct-application step as the least-secured link; the Radon-transform extension to integer coefficients is cited from prior work and appears secondary to the main generalization claim.","tokens_in":1634,"tokens_out":293,"duration_ms":8402,"concrete_test":"Extract the precise statement of the microlocal kernel (likely Theorem X or Proposition Y in the section on Iwahori-Whittaker averaging) and the subsequent application paragraph to automorphic sheaves; verify whether the proof invokes only the local microlocal equivalence or inserts an additional lemma justifying passage from global automorphic data to the local cotangent support.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central step generalizes Faergeman-Raskin by applying the microlocal description of the Iwahori-Whittaker averaging kernel directly to automorphic sheaves. The kernel characterization is stated in local microlocal terms (singular support on the cotangent bundle of the flag variety or Whittaker stack). Automorphic sheaves, however, live on a global moduli stack and satisfy additional Hecke-equivariance and Whittaker conditions that are not purely local; the paper does not exhibit an explicit reduction showing these global conditions are automatically compatible with the microlocal support condition used in the kernel description.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper characterizes the kernel of Iwahori-Whittaker averaging in microlocal terms. Applying this to automorphic sheaves, it generalizes Faergeman and Raskin's theorem that anti-temperedness is equivalent to having irregular singular support. Using a Radon transform argument of Bezrukavnikov and Morton-Ferguson, it extends the tilting property of Whittaker averaged central sheaves to integer coefficients.","tokens_in":1757,"tokens_out":361,"duration_ms":13286,"significance":"If the central claims hold, the work would strengthen the microlocal toolkit for studying averaging functors and temperedness conditions in the geometric Langlands program, providing a direct generalization of prior results on singular support and extending tilting statements to integral coefficients via an external Radon transform argument. The reliance on independent prior results by Faergeman-Raskin and Bezrukavnikov-Morton-Ferguson is explicitly noted and strengthens the foundation rather than introducing circularity.","major_comments":[{"comment":"The central generalization applies the microlocal characterization of the Iwahori-Whittaker averaging kernel (stated in local terms on the cotangent bundle of the flag variety or Whittaker stack) directly to automorphic sheaves on the global moduli stack. However, automorphic sheaves carry additional Hecke-equivariance and Whittaker conditions that are not purely local; the manuscript does not provide an explicit reduction or compatibility argument showing that these global conditions are automatically satisfied by the microlocal support condition used in the kernel description. This step is load-bearing for the claimed generalization of Faergeman-Raskin's theorem.","section":"Application to automorphic sheaves (central claim)"}],"minor_comments":[],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading and for identifying a point that requires clarification in the transition from the local microlocal result to the global application. We address the major comment below and will revise the manuscript accordingly.","responses":[{"response":"We agree that an explicit compatibility argument strengthens the exposition. The Iwahori-Whittaker averaging functor is constructed to be equivariant with respect to the global Hecke action and the Whittaker condition by design (see the definition in Section 3 and the compatibility with the global moduli stack in Section 5). Consequently, any sheaf whose singular support lies in the microlocal kernel automatically inherits the required global equivariance when pulled back to the automorphic setting. Nevertheless, to make this reduction fully explicit and address the referee's concern, we will insert a new paragraph (or short subsection) in Section 5 that spells out the compatibility of the microlocal support condition with Hecke-equivariance and the Whittaker condition, citing the relevant functoriality statements from the local theory. This will not alter the logical structure but will render the generalization of Faergeman-Raskin's theorem self-contained.","revision_made":"yes","referee_comment":"[Application to automorphic sheaves (central claim)] The central generalization applies the microlocal characterization of the Iwahori-Whittaker averaging kernel (stated in local terms on the cotangent bundle of the flag variety or Whittaker stack) directly to automorphic sheaves on the global moduli stack. However, automorphic sheaves carry additional Hecke-equivariance and Whittaker conditions that are not purely local; the manuscript does not provide an explicit reduction or compatibility argument showing that these global conditions are automatically satisfied by the microlocal support condition used in the kernel description. This step is load-bearing for the claimed generalization of Faergeman-Raskin's theorem."}],"tokens_in":1201,"tokens_out":394,"duration_ms":15936,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The punchline is that this work gives a microlocal description of the kernel of Iwahori-Whittaker averaging and uses it to generalize Faergeman-Raskin's result that anti-temperedness equals irregular singular support for automorphic sheaves. It also extends the tilting property of those averaged central sheaves to coefficients in Z by borrowing a Radon transform argument.\n\nWhat stands out is the clean application of the microlocal characterization to get the generalization. The paper cites the relevant prior work by Faergeman-Raskin and Bezrukavnikov-Morton-Ferguson explicitly and positions its contributions as building on them. That keeps the novelty focused and avoids overclaiming.\n\nThe soft spot is the step from the local microlocal kernel description to the global automorphic sheaves setting. Automorphic sheaves live on a moduli stack and come with Hecke-equivariance and Whittaker conditions that are not local. The abstract does not detail how those global constraints interact with the singular support condition, so if the paper does not provide an explicit check or reduction for that compatibility, it leaves a possible gap. The stress-test note raises exactly this point, and without seeing the full argument it is hard to dismiss.\n\nThis is for specialists in geometric representation theory working on Whittaker sheaves and singular supports in the Langlands program. A reader already familiar with the cited papers would get the most out of the generalization and the integer coefficient extension.\n\nI would recommend sending it for peer review. The core claims rest on external results that are independent, and the new pieces are narrow enough that referees can check them directly.","headline":"The paper gives a microlocal description of the Iwahori-Whittaker averaging kernel, uses it to generalize the Faergeman-Raskin anti-temperedness theorem, and extends tilting to Z coefficients via an existing Radon argument.","tokens_in":2240,"tokens_out":420,"would_cite":false,"duration_ms":17284,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The kernel of Iwahori-Whittaker averaging is characterized microlocally, generalizing an equivalence for anti-tempered automorphic sheaves and extending the tilting property of central sheaves to integer coefficients.","keywords":["Iwahori-Whittaker averaging","microlocal characterization","automorphic sheaves","anti-temperedness","singular support","tilting property","Radon transform","integer coefficients"],"falsifier":"An explicit automorphic sheaf that is anti-tempered yet has regular singular support, or a Whittaker averaged central sheaf that fails to tilt when coefficients are integers, would disprove the main claims.","tokens_in":2504,"feed_emoji":"","tokens_out":476,"duration_ms":18631,"temperature":0.7,"pith_summary":"This paper gives a microlocal description of the kernel of Iwahori-Whittaker averaging. It uses that description on automorphic sheaves to extend the statement that anti-temperedness is the same as having irregular singular support. It further applies a Radon transform argument to prove that Whittaker averaged central sheaves remain tilting even when coefficients are integers rather than rationals. These steps connect averaging functors to geometric support conditions and broaden the range of coefficients where tilting holds.","feed_headline":"Whittaker averaging kernel characterized microlocally","feed_subtitle":"The description generalizes the anti-temperedness equivalence for automorphic sheaves and extends tilting of central sheaves to integer coef","key_machinery":"The microlocal characterization of the kernel of Iwahori-Whittaker averaging, which identifies the kernel via singular support conditions and enables direct application to automorphic sheaves.","core_discovery":"We characterize the kernel of Iwahori-Whittaker averaging in microlocal terms. Applying this to automorphic sheaves, we generalize the theorem that anti-temperedness is equivalent to having irregular singular support. Using a Radon transform argument, we extend the tilting property of Whittaker averaged central sheaves to integer coefficients.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Microlocal kernel of Iwahori-Whittaker averaging","Anti-temperedness equivalence to irregular singular support generalized","Tilting of Whittaker central sheaves to integer coefficients","Radon transform extends tilting property to integers"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The microlocal characterization of the Iwahori-Whittaker averaging kernel applies directly to automorphic sheaves without further obstructions, and the Radon transform argument extends the tilting property to integer coefficients.","fun_headline_variants_meta":{"raw":{"variants":["Microlocal kernel of Iwahori-Whittaker averaging","Anti-temperedness equivalence to irregular singular support generalized","Tilting of Whittaker central sheaves to integer coefficients","Radon transform extends tilting property to integers"]},"model":"grok-4.3","cost_usd":0.007394,"raw_usage":{"total_tokens":3231,"prompt_tokens":493,"num_sources_used":0,"completion_tokens":62,"cost_in_usd_ticks":73940500,"prompt_tokens_details":{"text_tokens":493,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2676,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":493,"tokens_out":62,"duration_ms":9936,"temperature":1.0,"reasoning_tokens":2676,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-25T20:37:06.328870+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit automorphic sheaf that is anti-tempered yet has regular singular support, or a Whittaker averaged central sheaf that fails to tilt when coefficients are integers, would disprove the main claims.","supporting_citations":[],"review_version":1}