{"id":"d13a31e4-3fde-41f1-a9e9-27b41172518b","arxiv_id":"2607.10010","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.5,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Weights uniquely determine simple modules for quasi-simply-connected affine Hecke algebras, and induction/restriction along isogenies preserve semisimplicity, Hermitian duals and unitarity.","lead":"Affine Hecke algebra simple modules are uniquely determined by their weights when the algebra is quasi-simply-connected, and characters determine Jordan–Hölder factors. The paper then shows induction and restriction along isogenies preserve semisimplicity, Hermitian duals, and unitarity. This organises representation theory of p-adic groups via Bernstein blocks and isogenies of root data.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The paper's strongest claim (linear independence of A-characters for quasi-simply-connected affine Hecke algebras, with the consequent uniqueness and isogeny-functoriality statements) is supported by a complete algebraic argument whose only non-routine combinatorial input is Lemma 7. That lemma is proved in a few lines by the classical gallery argument plus a short induction for adjoint B_n; both halves are standard and appear free of gaps. The parameter restrictions that make the Kato criterion apply are stated explicitly and used correctly. The later inductive extraction of Jordan–Hölder multiplicities and the graded-algebra treatment of induction/restriction along isogenies do not rely on any further unproved facts. Because the subject is pure algebra with fully written proofs and no experimental or numerical components, the reader's high-confidence ACCEPT verdict is appropriate; the stress-test finds no reason to adjust it.","tokens_in":16890,"tokens_out":581,"duration_ms":5761,"concrete_test":"Independently verify the adjoint B_n case of Lemma 7 for n=2 (or n=3) by enumerating all t in (C^x)^n with non-trivial stabiliser and checking that every w in Stab_W(t) factors into signed reflections that also fix t; if any counter-example appears the uniqueness theorems collapse, otherwise the combinatorial foundation is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest assumption (Lemma 7 on stabilisers generated by reflections for quasi-simply-connected root data, feeding Theorem 8) is the natural load-bearing combinatorial step, but the argument given is standard and appears correct. For simply-connected summands the proof reduces to the usual affine-Weyl gallery argument that any element of the affine stabiliser is a product of affine reflections fixing the lift of t; projecting yields ordinary reflections. For the remaining adjoint B_n case the short induction on signed permutations works: if the first coordinate is unique then the S_n factor fixes it and one inducts; otherwise a signed reflection in the stabiliser reduces to that situation. The parameter conditions (1)–(2) then force the Kato c_alpha inverse to vanish, so Ind_A^t is irreducible and tempered+antitempered modules with the same central character coincide. The subsequent induction on parabolic rank that yields linear independence of A-characters (Theorem 12) and uniqueness of simples by weights (Corollary 10) therefore rests on solid ground. The isogeny results (graded structure, preservation of semisimplicity/Hermitian duals/unitarity) are independent of this lemma and are cleanly proved via the explicit X_2/X_1-grading. No internal inconsistency or hidden gap that would undermine the central claims is visible.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper studies the extent to which A-weights and A-characters determine simple modules for affine Hecke algebras H(R) with parameters satisfying the usual non-vanishing conditions. For the newly defined class of quasi-simply-connected root data it proves that tempered+antitempered simples are uniquely determined by central character (Theorem 8), that A-characters of all simples are linearly independent (Theorem 12), and therefore that weights uniquely determine simples (Corollary 10) and characters determine Jordan–Hölder multiplicities of finite-length modules (Theorem 14). For general isogeny classes the same statements hold up to explicit outer automorphisms coming from the isogeny. The second half develops the graded algebra structure of a central morphism H1\to H2 (Proposition 21), shows that induction and restriction along isogenies preserve semisimplicity (Theorem 20), and constructs an explicit graded Hermitian pairing that realises Ind(V†)≅(Ind V)† and multiplies signatures by the index, hence preserves unitarity (Theorems 34–35).","tokens_in":17180,"tokens_out":759,"duration_ms":6588,"significance":"The uniqueness and linear-independence statements complete and unify earlier results of Evens–Mirković (simply-connected equal-parameter case) and Barbasch–Ciubotaru (graded Hecke algebras), while the independent proof of Antor–Okada is acknowledged. The isogeny package supplies a clean graded-algebra description of induction/restriction that preserves the Hermitian and unitary structures; this is markedly different from the parabolic case treated by Opdam–Solleveld and is useful for reducing questions about arbitrary isogeny classes to the quasi-simply-connected setting. The arguments are self-contained once the standard Langlands classification and Kato’s criterion are granted, and the explicit graded pairing is a concrete computational tool.","major_comments":[],"minor_comments":[{"comment":"Definition 6 of “quasi-simply-connected” is clear, but a one-sentence remark that the only non-simply-connected primitive summands that can appear are adjoint Bn with unequal short-root parameters would help the reader see why the class is natural.","section":null},{"comment":"In the proof of Lemma 7 the gallery argument for the simply-connected case is standard; a brief pointer to Humphreys §1.5 (already cited) or to the corresponding statement in Lusztig would make the reference complete.","section":null},{"comment":"The linear-independence argument of Theorem 12 is inductive on parabolic rank and ultimately reduces to the tempered+antitempered case via the Iwahori–Matsumoto involution; a short sentence at the beginning of the proof outlining this strategy would improve readability.","section":null},{"comment":"Section 4.1 (rank-1 example) is useful but could be shortened; the explicit matrices for the principal series are standard and the essential point is the behaviour of the two one-dimensional modules under the isogeny.","section":null},{"comment":"A few typographical inconsistencies appear: “Jordan-H¨ older” (missing umlaut), “antitempered” versus “anti-tempered”, and occasional missing spaces after punctuation. These are easily corrected in production.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is short, cleanly written and already cites the concurrent independent work of Antor–Okada. I see no novelty or citation issues. The result is solid enough for a standard research journal in representation theory; an accept recommendation is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper does two clean things. First, it shows that for quasi-simply-connected affine Hecke algebras (simply-connected summands or adjoint B with unequal short-root parameters), A-characters of simples are linearly independent, so weights determine simples and characters determine Jordan–Hölder multiplicities of finite-length modules. That generalises Evens–Mirković (simply-connected equal parameters) and the graded Barbasch–Ciubotaru result; Antor–Okada got the character independence independently by a different route. Second, it treats induction/restriction along central morphisms and isogenies: the target is a graded algebra over the source, and the functors preserve semisimplicity, Hermitian duals and unitarity via an explicit graded pairing that is different from the Opdam–Solleveld parabolic case.\n\nThe proofs are fully written and algebraic. The load-bearing combinatorial step (Lemma 7: stabilisers generated by reflections) is short and standard: affine-Weyl gallery for simply-connected summands, signed-permutation induction for adjoint B_n. Parameter conditions then force Kato’s criterion, so tempered+antitempered modules with the same central character coincide; the rest of the character-independence argument is induction on parabolic rank via Langlands classification and weight separation in standard modules (Proposition 2). The isogeny half is independent of that lemma and is clean: free finite bimodule, explicit X2/X1-grading, signature multiplies by the index. Rank-1 PGL2/SL2 example checks out.\n\nSoft spots are minor. “Quasi-simply-connected” is a convenient packaging of the cases where the argument works; the B_n induction is short but correct. No circularity with the author’s own prior work; external inputs (Solleveld, Kato, Evens) are standard. Citation pattern is appropriate.\n\nThis is for people who work with affine Hecke algebras or Bernstein blocks of p-adic groups and need uniqueness or isogeny functoriality. It deserves a serious referee. I would accept it for peer review and would cite the isogeny/Hermitian statements if I needed them.","headline":"Solid algebraic extension of weight-uniqueness for quasi-simply-connected affine Hecke algebras, plus clean isogeny functoriality for Hermitian duals and unitarity.","tokens_in":17768,"tokens_out":537,"would_cite":true,"duration_ms":4780,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20C08","22E50"],"pacs":[],"model":"grok-4.5","headline":"Weights uniquely determine simple modules for quasi-simply-connected affine Hecke algebras, and isogenies preserve unitarity.","keywords":["affine Hecke algebras","weights","characters","Langlands classification","isogenies","unitarity","Hermitian forms","quasi-simply-connected"],"falsifier":"Exhibit a quasi-simply-connected affine Hecke algebra (especially adjoint type B with unequal short-root parameters) and two non-isomorphic tempered+antitempered simple modules that share the same central character, or two non-isomorphic simples with identical sets of A-weights.","tokens_in":17769,"feed_emoji":"⚖️","tokens_out":891,"duration_ms":7122,"temperature":0.7,"pith_summary":"Affine Hecke algebras control the representation theory of p-adic groups via Bernstein blocks, and their simple modules carry a collection of weights coming from a large commutative subalgebra. This paper proves that when the root datum is quasi-simply-connected those weights already determine each simple module, and that the formal characters of simple modules are linearly independent; consequently any finite-length module can be reconstructed from its character by a finite algorithm once the tempered modules for proper parabolics are known. The same uniqueness is used to study induction and restriction along central morphisms (isogenies) of root data: the larger algebra becomes a graded module over the smaller one, induction and restriction preserve semisimplicity, and both operations send Hermitian forms to Hermitian forms while preserving unitarity. The results therefore give a precise dictionary between representations of isogenous groups that keeps track of the analytic properties needed for the Langlands correspondence.","feed_headline":"Weights alone pin down simple modules for Hecke algebras","feed_subtitle":"Isogenies preserve unitarity, giving a clean dictionary between isogenous groups","key_machinery":"The Langlands classification of simple modules as unique irreducible quotients of standard modules M(P, π, t), combined with the fact that stabilisers of weights are generated by reflections (Lemma 7) for quasi-simply-connected root data; this forces tempered+antitempered modules with the same central character to be isomorphic and yields linear independence of characters by induction on parabolic rank.","core_discovery":"For a quasi-simply-connected affine Hecke algebra the A-characters of simple modules are linearly independent over C, so the set of weights uniquely determines each simple module and characters determine Jordan–Hölder multiplicities of finite-length modules. Along isogenies, induction and restriction preserve semisimplicity, Hermitian duals and unitarity.","pith_inferences":["The graded structure that makes the Hermitian form on induced modules work may give a model for other non-parabolic induction functors that still preserve unitarity.","Once tempered modules are tabulated for classical types, the algorithm of Theorem 14 becomes a practical machine for computing Jordan–Hölder series of Iwahori-fixed representations of the corresponding p-adic groups.","The linear-independence statement for characters should survive passage to the completed Hecke algebra used in the full local Langlands correspondence."],"forward_implications":["Any finite-length module over a quasi-simply-connected affine Hecke algebra can be reconstructed from its character once tempered modules for proper parabolics are known.","Induction and restriction along isogenies give a graded correspondence that preserves unitarity and Hermitian duals between the representation theories of isogenous groups.","Conjugacy-class packets of simple modules for non-quasi-simply-connected algebras are still uniquely determined by their weights.","The same character-independence holds for the associated graded affine Hecke algebras."],"fun_headline_variants":["Weights alone fix simple modules of affine Hecke algebras","Affine Hecke simples uniquely fixed by their weight sets","Isogenies preserve unitarity of affine Hecke modules","Weights determine simples and Jordan-Hölder multiplicities","Induction along isogenies preserves Hecke hermiticity"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The claim that every weight stabiliser is generated by reflections when the root datum is quasi-simply-connected, proved by a short induction for adjoint type B; if that generation fails for some parameters the uniqueness theorems collapse.","fun_headline_variants_meta":{"raw":{"variants":["Weights alone fix simple modules of affine Hecke algebras","Affine Hecke simples uniquely fixed by their weight sets","Isogenies preserve unitarity of affine Hecke modules","Weights determine simples and Jordan-Hölder multiplicities","Induction along isogenies preserves Hecke hermiticity"]},"model":"grok-4.5","effort":"low","cost_usd":0.006874,"raw_usage":{"total_tokens":1525,"prompt_tokens":569,"num_sources_used":0,"completion_tokens":61,"cost_in_usd_ticks":68740000,"prompt_tokens_details":{"text_tokens":569,"audio_tokens":0,"image_tokens":0,"cached_tokens":0},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":895,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":569,"tokens_out":61,"duration_ms":6390,"temperature":1.0,"reasoning_tokens":895,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T01:03:08.163633+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a quasi-simply-connected affine Hecke algebra (especially adjoint type B with unequal short-root parameters) and two non-isomorphic tempered+antitempered simple modules that share the same central character, or two non-isomorphic simples with identical sets of A-weights.","supporting_citations":[],"review_version":1}