{"id":"070c1e4f-f91a-43ef-b09a-af19ee05bd3b","arxiv_id":"2606.21313","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"J(G) for inner forms G of p-adic GL_n is defined over \bar Q_l, admits explicit formulas via reductive centralizers, shares Hochschild homology with C_c^infty(G), and supports rudimentary Hecke compatibility for sheaves on Bun_n.","lead":"The paper proves that Braverman-Kazhdan's asymptotic Hecke algebra J(G) for inner forms of p-adic GL_n is defined over the algebraic closure of Q_l and that representations extend to J(G)-modules in this setting. This enables compatibility statements with Hecke operators and categorical local Langlands, plus explicit formulas and homology comparisons.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"Reader correctly flags the reliance on types plus the Suzuki generalization as the load-bearing step; the paper addresses the latter by supplying its own proof, so the concern does not rise to a load-bearing objection.","tokens_in":1756,"tokens_out":240,"duration_ms":14195,"concrete_test":"Compare the statement and proof of the generalized Suzuki theorem (as given in the paper) with the original Suzuki result for GL_n; confirm that the modifications for inner forms preserve the \bar Q_l-coefficient property used to define J(G).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that J(G) and the extendability property are defined over \bar Q_l rests on Bushnell-Kutzko/Sécherre-Stevens types (known to exist for inner forms of GL_n) together with the paper's supplied generalization of Suzuki's theorem. Because the manuscript explicitly provides a proof of that generalization, the argument is internally self-contained on the key step; no hidden assumption about field of definition or choice of types is left unaddressed in the stated method.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper studies the Braverman-Kazhdan asymptotic Hecke algebra J(G) for inner forms G of p-adic GL_n. It proves that both J(G) and the property that a G-representation extends to a J(G)-module are defined over \bar Q_l, enabling their use in the categorical local Langlands correspondence. It establishes a basic compatibility with Hecke operators to interpret stalks of sheaves on Bun_n (including the Whittaker sheaf) via J(GL_n)-modules, gives explicit formulas for many elements of J(G) in terms of reductive centralizers of L-parameters, shows that J(G) has the same Hochschild homology as C_c^∞(G) with the Kazhdan-Lusztig bijection appearing in the isomorphism, and proceeds by means of Bushnell-Kutzko and Sécherre-Stevens types while supplying a proof of a generalization of Suzuki's theorem for GL_n.","tokens_in":1856,"tokens_out":744,"duration_ms":16586,"significance":"If the central claims hold, the results provide a concrete bridge between the asymptotic Hecke algebra construction and the categorical local Langlands program for inner forms of GL_n. The field-of-definition statement is load-bearing for the categorical applications, the explicit formulas and homology computation supply usable tools, and the supplied proof of the generalized Suzuki theorem makes the argument self-contained on a key technical step. These contributions are proportionate to the scope of the manuscript and strengthen the toolkit for studying representations of inner forms in a geometric context.","major_comments":[{"comment":"The central claim that J(G) is defined over \bar Q_l rests on the existence of Bushnell-Kutzko/Sécherre-Stevens types (known for inner forms) together with the paper's generalization of Suzuki's theorem. Because the manuscript supplies an explicit proof of that generalization, the argument is internally self-contained; however, the precise statement of the generalized theorem (including the exact hypotheses on the types and the field of definition) should be isolated as a numbered theorem with a self-contained proof section so that the dependence is transparent.","section":"Introduction / § on generalized Suzuki theorem"},{"comment":"The rudimentary compatibility with Hecke operators is used to discuss stalks of sheaves on Bun_n corresponding to trivial vector bundles on the L-parameter stack. The precise functoriality statement (which Hecke operators are involved and how they act on the J(G)-module structure) needs to be stated as a proposition with a clear reference to the relevant diagram or exact sequence, as this step directly supports the interpretation of the Whittaker sheaf.","section":"Section on compatibility with Hecke operators"}],"minor_comments":[{"comment":"Notation for the asymptotic Hecke algebra is introduced as Σ(G) in the title but rendered as J(G) throughout the abstract and body; adopt a single consistent symbol and update the title accordingly.","section":"Title and abstract"},{"comment":"The phrase 'rudimentary form of compatibility' is informal; replace with a precise description of the limited range of Hecke operators for which compatibility is proved.","section":"Abstract"},{"comment":"References to the original Braverman-Kazhdan construction and to the categorical local Langlands correspondence should include the most recent relevant citations (e.g., works post-2020 on Bun_n and L-parameters) to situate the contribution.","section":"Introduction"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the positive assessment and constructive suggestions. We address each major comment below.","responses":[{"response":"We agree that extracting the generalized Suzuki theorem into a numbered statement with its own self-contained proof section will improve transparency regarding the field-of-definition claim. We will make this structural change in the revised manuscript.","revision_made":"yes","referee_comment":"[Introduction / § on generalized Suzuki theorem] The central claim that J(G) is defined over  bar Q_l rests on the existence of Bushnell-Kutzko/Sécherre-Stevens types (known for inner forms) together with the paper's generalization of Suzuki's theorem. Because the manuscript supplies an explicit proof of that generalization, the argument is internally self-contained; however, the precise statement of the generalized theorem (including the exact hypotheses on the types and the field of definition) should be isolated as a numbered theorem with a self-contained proof section so that the dependence is transparent."},{"response":"We will formulate the precise functoriality statement as a numbered proposition, with explicit reference to the relevant diagram or exact sequence, in the section discussing compatibility with Hecke operators.","revision_made":"yes","referee_comment":"[Section on compatibility with Hecke operators] The rudimentary compatibility with Hecke operators is used to discuss stalks of sheaves on Bun_n corresponding to trivial vector bundles on the L-parameter stack. The precise functoriality statement (which Hecke operators are involved and how they act on the J(G)-module structure) needs to be stated as a proposition with a clear reference to the relevant diagram or exact sequence, as this step directly supports the interpretation of the Whittaker sheaf."}],"tokens_in":1609,"tokens_out":373,"duration_ms":19937,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main point is that Dawydiak carries the Braverman-Kazhdan asymptotic Hecke algebra J(G) over to inner forms of p-adic GL_n and proves it is defined over bar Q_l, so the extendability property for representations makes sense in categorical local Langlands. He also gives explicit formulas for many elements of J(G) via reductive centralizers of L-parameters, shows the Hochschild homology of J(G) equals that of C_c^infty(G), and notes that the Kazhdan-Lusztig bijection appears in the isomorphism. A rudimentary compatibility with Hecke operators is stated, which lets one talk about stalks of certain sheaves on Bun_n in terms of J(GL_n)-modules.\n\nThe paper does the useful work of supplying its own proof of the generalized Suzuki theorem rather than leaving that step as an assumption. The route through Bushnell-Kutzko and Secherre-Stevens types is the standard one for these groups, and those types are known to exist for inner forms, so the foundation is not invented here.\n\nThe soft spots are modest. The compatibility statement is described as rudimentary, so it is not a full functoriality result. Without the full proofs in hand it is hard to judge the length or technical difficulty of the type-theoretic adaptations for inner forms, but the abstract gives no sign of circularity or unfalsifiable steps. The argument rests on external type theory plus the supplied generalization, which keeps the circularity burden low.\n\nThis is for people already working inside p-adic representation theory and the categorical Langlands program for GL_n and its forms. A reader who needs the bar Q_l definition or the explicit centralizer formulas will get concrete value. It is an incremental but self-contained step that fills a documented gap.\n\nI would send it to peer review. The claims are specific, the key generalization is proved in the paper, and the methods are standard enough that referees can check them directly.","headline":"This paper extends the asymptotic Hecke algebra to inner forms of GL_n, shows it is defined over bar Q_l, supplies explicit centralizer formulas, and includes a proof of the needed generalized Suzuki theorem.","tokens_in":2368,"tokens_out":486,"would_cite":false,"duration_ms":17101,"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 asymptotic Hecke algebra J(G) for inner forms of p-adic GL_n is defined over the algebraic closure of Q_l","keywords":["asymptotic Hecke algebra","inner forms of GL_n","categorical local Langlands","Bushnell-Kutzko types","Sécherre-Stevens types","Hochschild homology","Kazhdan-Lusztig bijection","Suzuki theorem"],"falsifier":"Finding an inner form $G$ of $\\mathrm{GL}_n$ where $J(G)$ cannot be defined over $\\overline{Q}_l$ or where the extension property does not align with the categorical local Langlands correspondence would falsify the main claim.","tokens_in":2651,"feed_emoji":"","tokens_out":659,"duration_ms":18890,"temperature":0.7,"texified_at":"2026-08-05T21:10:39.843166+00:00","pith_summary":"The paper shows that for inner forms $G$ of p-adic $\\mathrm{GL}_n$ the Braverman-Kazhdan asymptotic Hecke algebra $J(G)$ is defined over the algebraic closure of the l-adic numbers. The same holds for the property that a representation of $G$ extends to a module over $J(G)$. This makes both objects available for use in the categorical local Langlands correspondence. The proof uses Bushnell-Kutzko and Sécherre-Stevens types and supplies a generalization of Suzuki's theorem.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":5472,"prompt_tokens":447,"completion_tokens":5025,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":447,"completion_tokens_details":{"reasoning_tokens":4631}},"feed_headline":"Asymptotic Hecke algebra for GL_n inner forms defined over bar Q_l","feed_subtitle":"This puts the algebra and its module extension property into the setting of the categorical local Langlands correspondence.","key_machinery":"Bushnell-Kutzko and Sécherre-Stevens types for inner forms of $\\mathrm{GL}_n$ with a generalization of Suzuki's theorem","core_discovery":"$J(G)$ and the extension property for $G$-representations to $J(G)$-modules are defined over $\\overline{Q}_l$ for inner forms $G$ of p-adic $\\mathrm{GL}_n$. This is established by working with types and a generalized version of Suzuki's theorem. The algebra also admits explicit formulas for its elements in terms of reductive centralizers of L-parameters and shares Hochschild homology with the algebra of compactly supported smooth functions on $G$.","pith_inferences":["If the definition over \\overline{Q}_l holds, it may allow direct comparison of J(G)-modules with objects in the categorical correspondence for non-split groups.","The compatibility with Hecke operators could lead to new ways to compute invariants of representations using the asymptotic algebra.","The homology isomorphism suggests that J(G) and C_c^∞(G) are related by a map that preserves more structure than just homology."],"forward_implications":["Stalks of sheaves on Bun_n corresponding to trivial vector bundles on the L-parameter stack, including the Whittaker sheaf, can be discussed in terms of J(GL_n)-modules.","Explicit formulas for many functions in J(G) are given using the reductive centralizer of L-parameters.","J(G) has the same Hochschild homology as C_c^∞(G) and the Kazhdan-Lusztig bijection appears in the isomorphism."],"fun_headline_variants":["GL_n inner forms have J(G) defined over bar Q_l","Asymptotic Hecke algebra over bar Q_l for GL_n inner forms","Braverman-Kazhdan J(G) over bar Q_l for p-adic GL_n","J(G) module extensions over bar Q_l for GL_n forms","J(G) shares Hochschild homology with C_c^infty(G) for GL_n"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The constructions and proofs rely on the existence and good properties of Bushnell-Kutzko and Sécherre-Stevens types for the inner forms of $\\mathrm{GL}_n$ together with a generalization of Suzuki's theorem.","fun_headline_variants_meta":{"raw":{"variants":["GL_n inner forms have J(G) defined over bar Q_l","Asymptotic Hecke algebra over bar Q_l for GL_n inner forms","Braverman-Kazhdan J(G) over bar Q_l for p-adic GL_n","J(G) module extensions over bar Q_l for GL_n forms","J(G) shares Hochschild homology with C_c^infty(G) for GL_n"]},"model":"grok-4.3","cost_usd":0.008464,"raw_usage":{"total_tokens":3846,"prompt_tokens":706,"num_sources_used":0,"completion_tokens":101,"cost_in_usd_ticks":84637000,"prompt_tokens_details":{"text_tokens":706,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3039,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":706,"tokens_out":101,"duration_ms":19550,"temperature":1.0,"reasoning_tokens":3039,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T12:55:18.811871+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Finding an inner form $G$ of $\\mathrm{GL}_n$ where $J(G)$ cannot be defined over $\\overline{Q}_l$ or where the extension property does not align with the categorical local Langlands correspondence would falsify the main claim.","supporting_citations":[],"review_version":1}