{"id":"bf2578a9-65ed-4971-ad17-8732637990d1","arxiv_id":"2411.19846","paper_version":2,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A local Langlands correspondence and a categorical Hecke-algebra refinement are established for all non-singular depth-zero representations of reductive p-adic groups.","lead":"This paper proves a matching (local Langlands correspondence) between all non-singular depth-zero representations of a reductive group over a non-archimedean local field and their dual-side parameters. It reaches this through an equivalence of categories involving Hecke algebras, and shows the matching behaves well under standard operations like parabolic induction.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The categorical LLC hinges on Proposition 3.5's equivariant splitting, which is asserted by analogy to Cases I–V; an unsplit Galois-side case would invalidate Theorem 8.7 and hence Theorems 1–2.","rationale":"I read the paper as a chain of comparisons: the p-adic Hecke algebra of a non-singular depth-zero Bernstein block is compared with the Galois-side twisted affine Hecke algebra via Proposition 6.9, Proposition 8.5, and finally Theorem 8.7, with Proposition 8.8 providing the localized isomorphism that yields the categorical equivalence in Theorems 9.4 and 9.6. The decisive step is the comparison of 2-cocycles, and its proof depends on the equivariant splittings in Propositions 2.10 and 3.5. Proposition 2.10 is checked in detail through Cases I–V, but Proposition 3.5 is abbreviated to an analogy with those cases. Since the two sides are genuinely different (dual groups, component groups π0(S^+_φ), and η-pushouts), the analogy is not self-evident; this is an internal incompleteness rather than a demonstrated contradiction. I found no circularity, no fitted parameters, and no overclaiming: the finite-length restriction and the non-canonicity of enhancements are stated explicitly, and the self-flagged missing reference in Proposition 6.6 is in fact proved. The concern therefore does not move the verdict away from CONDITIONAL, but it identifies where a specialist check is most needed.","tokens_in":79954,"tokens_out":4873,"duration_ms":46002,"concrete_test":"Independently write out the Galois-side case check omitted in the proof of Proposition 3.5: for each root-system type occurring in W(L∨,T∨)^{WF} (at minimum types A_n with outer automorphisms, D_n with n≥4, and E6), and each nontrivial stabilizer W(L∨,T∨)^{WF}_{φ_T,η}, construct extension (3.26) and verify that its pushout along η admits a splitting equivariant for W(NG∨(L∨),T∨)^{WF}_{η,φ_T}. A good starting point is the D4 triality case, where the diagram automorphism group is largest; if that case only splits nonequivariantly, Theorem 8.7 fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central construction is the algebra isomorphism (1.8) of localized End_G(Π_s) with localized H(s∨,q_F^{1/2}), obtained in Proposition 8.8 only after Theorem 8.7 identifies the 2-cocycles ♮_{s∨} and ♮_τ. Theorem 8.7(a) proves this by matching the data for A_{η,~w} and B_{~w} through the W(...)-equivariant isomorphism B(ζ^0,ζ^⋊) from Lemma 4.4. Lemma 4.3 derives the equivariance of ζ^0 from Propositions 2.10 and 3.5. Proposition 2.10 is a long, explicit case check (Cases I–V), but Proposition 3.5's proof ends with 'the case-by-case check is entirely analogous to Cases I–V' after reducing (3.26) to the analogous dual-side extensions. That is the weakest load-bearing point: the Galois-side extensions involve different groups (W(NG∨(L∨),T∨)^{WF}_{η,φ_T}, component groups π0(S^+_φ), and pushouts along η), and an equivariant splitting is exactly what is needed for the 2-cocycle comparison. If even one unexamined root-system case fails to admit a splitting equivariant for the stated stabilizer, the equality ♮_{s∨}=♮_τ can fail, so Proposition 8.8, Theorem 9.4, Theorem 9.6 and Theorem 1 have no proof. The paper is transparent about the difficulty (it calls Theorem 8.7 the most difficult step), but the decisive check is not written out in the reviewed text.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper establishes a local Langlands correspondence for all non-singular depth-zero representations of a connected reductive p-adic group, viewed as a rigid inner twist, and a categorical refinement in terms of twisted affine Hecke algebras. The main results are Theorem 1 (a bijection between irreducible non-singular depth-zero representations and non-singular enhanced depth-zero L-parameters trivial on wild inertia, with compatibility properties) and Theorem 2 (a finite-length categorical equivalence between the group side and a direct sum of twisted affine Hecke algebras constructed from L-parameter geometry). The proof is a chain of reductions to Morris types, Kaletha's LLC for non-singular supercuspidals, the AMS3 Hecke algebra for enhanced L-parameters, and Solleveld's analysis of End_G(Pi_s). The new technical core is the comparison of the two Hecke algebras, especially the comparison of 2-cocycles in Theorem 8.7, which relies on equivariant splitting results for central extensions on both the p-adic and Galois sides.","tokens_in":80130,"tokens_out":2640,"duration_ms":26811,"significance":"If the main results are correct, this is a major advance: it gives the LLC for all non-singular depth-zero representations, going well beyond the supercuspidal case, and it provides a categorical refinement in the spirit of recent spectral-side equivalences. The paper is careful and transparent about the difficult steps, and it builds on substantial prior machinery rather than introducing ad hoc assumptions. The q-parameter comparison is made quite explicit via reduction to principal series of quasi-split groups (Propositions 6.9, 8.4), and the authors are honest about the non-canonical choices in the LLC and about the obstruction to extending the categorical equivalence beyond finite-length modules. The main weakness is that the most load-bearing technical step, the equivariant splitting on the Galois side (Proposition 3.5), is not written out but only asserted by analogy to the p-adic case-by-case verification; since Theorem 8.7 and hence Theorems 1 and 2 depend on it, this is a genuine gap in the present text.","major_comments":[{"comment":"The proof of Proposition 3.5 ends with the statement that the remaining case-by-case check is 'entirely analogous to Cases I–V' from Proposition 2.10. This is a load-bearing assertion: the equivariant splitting of the extensions E^{0,phi'_T}_eta is used in Lemma 4.3, then in Lemma 4.4, and ultimately in Theorem 8.7(a) to identify the 2-cocycles on the two sides. The Galois-side extensions involve different groups (W(N_{G^vee}(L^vee),T^vee)^{WF}_{eta,phi_T}, component groups pi_0(S^+_phi), and pushouts along eta), so the analogy is not formal. If even one root-system case fails to admit a splitting equivariant for the stated stabilizer, the equality of 2-cocycles in Theorem 8.7 can fail and the LLC in Theorem 1 has no proof. I request that the omitted case check be supplied, or at minimum that the reduction to the p-adic cases be made precise enough that the reader can verify each case without rerunning the argument.","section":"§3.2, Proposition 3.5"},{"comment":"Lemma 8.1 asserts that the construction of H(s^vee,q_F^{1/2}) from [AMS3] adapts to rigid inner twists and to the component groups pi_0(S^+_phi) used in this paper, with 'all the results in [AMS3, §3]' remaining valid. This is a one-paragraph sketch rather than a proof, and it is used critically: the whole Galois-side Hecke algebra, including the 2-cocycle comparison in Theorem 8.7, depends on it. The substitutions listed in (8.3) are plausible, but the subsequent statements about equivariant local systems and the canonical parametrization of modules by enhanced L-parameters require checking that the rigid-inner-twist version does not introduce new choices or break the arguments in [AMS3]. I recommend expanding this lemma into a detailed verification, or explicitly citing a published source where the rigid-inner-twist adaptation is carried out.","section":"§8.1, Lemma 8.1"},{"comment":"The comparison of the 2-cocycles is presented as the most difficult step, but the argument is compressed: Theorem 8.7(a) reduces the equality to the existence of the equivariant isomorphisms in Proposition 8.6, whose proof refers to Appendices A and B and to 'similar isomorphisms without subscripts G'. The appendix material is not included in the reviewed text, so the reader cannot verify the key equality ♮_{s^vee} = ♮_tau. Given that Proposition 8.8, Theorem 9.4, Theorem 9.6, and Theorem 1 all rest on this, the paper should either include the full appendix arguments or give a precise pointer to where in the published literature the analogous comparison is proved.","section":"§8.3, Theorem 8.7 and Proposition 8.8"},{"comment":"Theorem 1 is stated as a bijection with a canonical map pi -> phi_pi, but the construction in §4 and §10 involves several choices: coherent splittings ϵ, isomorphisms ζ_0 in Lemma 4.4, and the choice of representatives in the LLC from Kaletha. The paper notes that the enhancement map is not canonical without a Whittaker datum and that the LLC is not uniquely specified. This is not an error, but it should be stated more prominently in Theorem 1 and its proof: the bijection depends on those choices, and only the L-parameter map pi -> phi_pi is claimed canonical. The current formulation could mislead a reader into thinking the full bijection is canonical.","section":"§10, Theorem 1"}],"minor_comments":[{"comment":"The statement 'The p-adic Kazhdan–Lusztig conjecture holds for Rep^0(G)_{ns}' appears without explanation. It would be helpful to give a reference or a short explanation of what is meant here, since the phrase is not standard and the following text does not return to it.","section":"§1"},{"comment":"The case-by-case verification in Cases I–V is long and relies on explicit coordinates and classification data. A table summarizing the relevant groups, stabilizers, and the splitting used in each case would improve readability and make it easier to check the claimed equivariance.","section":"§2"},{"comment":"The notation M^vee is used both for Z_{G^vee}(P_F) and later for the group in [AMS3], which can cause confusion. Consider using a different letter, e.g. M_0^vee, for the depth-zero centralizer.","section":"§3.1"},{"comment":"The map in (4.11) is written as an induction map from (S^+_phi)_eta to S^+_phi, but in the surrounding text it is referred to as a canonical bijection by Clifford theory. Please clarify whether this is induction or its adjoint, and specify the normalization of the induction functor for possibly disconnected groups.","section":"§4"},{"comment":"The notation Can(U) for analytic functions on U is used without definition. It is understandable from context, but a one-line definition would help.","section":"§7"},{"comment":"The proof of Proposition 8.4 appeals to [Sol10, Lemma 5.2] for principal series of quasi-split groups. Since this is a key comparison of q-parameters, the dependence on [Sol10] should be made precise, especially regarding the normalization of q-parameters and the treatment of the q^* parameters.","section":"§8.2"}],"recommendation":"major_revision","confidential_remarks":"This is a strong and ambitious paper, but the central claim is not fully verifiable from the text as it stands. The weakest point identified by the stress test is indeed real: Proposition 3.5's proof by analogy is not sufficient for a proposition on which the whole 2-cocycle comparison rests. Similarly, Lemma 8.1 is a sketch of an adaptation of a substantial external construction. I would not recommend rejection, because the overall strategy is sound and the authors are transparent about the difficulty, and it is plausible that the missing details can be supplied. However, the paper should not be accepted until either the omitted case checks and adaptation proofs are written out, or the authors point to a precise reference that proves the rigid-inner-twist version of the AMS3 construction. The paper is worthy of publication in a top journal if these gaps are addressed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear X,\n\nWhat you should know: this paper proves an explicit local Langlands correspondence, and a categorical refinement via twisted affine Hecke algebras, for all non-singular depth-zero representations of reductive p-adic groups. That goes beyond the supercuspidal cases treated by DeBacker–Reeder and Kaletha, so the main claim is genuinely new. The architecture is a chain of reductions to established machinery (Kaletha’s LLC, Morris types, AMS3 Hecke algebras), with the new work concentrated in equivariant splitting lemmas, the q-parameter comparison, and the 2-cocycle comparison.\n\nThe paper deserves real credit for transparency. The abstract states the finite-length restriction. The introduction acknowledges that the enhancements are not canonical. Proposition 6.6 is flagged as a result the authors could not find a reference for. I found no circularity and no fitted parameters: the two Hecke algebras being compared are built independently, one from p-adic types and one from L-parameter geometry.\n\nThe cuspidal-level equivariance in Theorem 4.8 is a substantive strengthening of Kaletha’s correspondence, and the categorical equivalence in Theorem 9.6 is exactly the kind of statement specialists will want. The listed properties of the LLC—central characters, temperedness, discrete series, cuspidal support, parabolic induction, Langlands classification—are coherent with the stated construction.\n\nThe soft spots are real but localized. The load-bearing step is Theorem 8.7, the 2-cocycle comparison, and it rests on Proposition 3.5, whose proof ends by saying the case-by-case check is “entirely analogous” to Cases I–V without writing those cases. The stress-test note is right to isolate this. I cannot certify the unexamined root-system cases; if one of them fails to admit the required equivariant splitting, the equality of 2-cocycles, and with it Theorems 9.4, 9.6, and 1, would lose its proof. Lemma 8.1 is a second genuine gap: adapting the AMS3 construction to rigid inner twists and the component groups used here is done in a one-paragraph sketch with a substitution table. It is plausible, but it is not a full proof. These are not fatal in my view—the reductions are explicit and the gaps are precisely locatable—but they are exactly what a referee should push on. The finite-length-only categorical equivalence and the choice-dependent enhancements are stated limitations, not hidden defects.\n\nWho is this for: specialists in p-adic representation theory and the local Langlands program. It deserves a serious referee: the result is important, the paper is honest, and the risky steps are confined enough that a competent referee can check them. I would send it to review, with a request that the referee verify Proposition 3.5 or require the authors to expand it, and that they scrutinize Lemma 8.1.","headline":"A serious, transparent 96-page LLC paper with a real new result; the main risk is the unexpanded case check in Proposition 3.5 and the sketch-level Lemma 8.1, both worth referee scrutiny.","tokens_in":80871,"tokens_out":2168,"would_cite":true,"duration_ms":23579,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22E50","20C08","20G25"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves a bijective refined local Langlands correspondence and a matching category equivalence for all non-singular depth-zero representations of a p-adic reductive group.","keywords":["local Langlands correspondence","depth-zero representations","non-singular cuspidal support","affine Hecke algebras","enhanced L-parameters","p-adic reductive groups","Deligne–Lusztig packets","rigid inner twists"],"falsifier":"Find one root system type not covered by Cases I–V in Section 2.1 (or the analogous cases in Section 3.2) for which the extension in Proposition 2.10 or Proposition 3.5 fails to split equivariantly; that would invalidate Theorem 8.7 and hence the categorical equivalence and the bijection. A more arithmetic falsifier would be to exhibit a non-singular depth-zero Bernstein block where the 2-cocycle of the p-adic Hecke algebra is not cohomologous to the parameter-side cocycle under the constructed isomorphism.","tokens_in":79562,"feed_emoji":"🔁","tokens_out":8233,"duration_ms":66045,"temperature":0.7,"pith_summary":"The paper proves a refined local Langlands correspondence for every depth-zero representation of a connected reductive group over a non-archimedean local field whose cuspidal support is non-singular. The correspondence is a bijection between such representations and enhanced L-parameters that are trivial on wild inertia, and it is upgraded to an equivalence of categories: finite-length non-singular depth-zero representations are equivalent to finite-length modules over a direct sum of twisted affine Hecke algebras built from Langlands parameters. This matters because it gives a uniform, structural handle on a large class of p-adic representations, with the Hecke-algebra comparison carrying all the arithmetic content. The bijection is compatible with central characters, temperedness, essential square-integrability, cuspidal support, parabolic induction, the Langlands classification, and twists by depth-zero characters.","feed_headline":"Langlands bijection proven for non-singular depth-zero representations","feed_subtitle":"The match is categorical: finite-length representations correspond to modules over twisted affine Hecke algebras.","key_machinery":"The load-bearing machinery is a comparison of twisted affine Hecke algebras on the two sides of the correspondence, meaning crossed products of an affine Hecke algebra by a finite group with a 2-cocycle. On the p-adic side, the type-theoretic construction gives the Hecke algebra $H(G, P_f, \\sigma)$ as such a crossed product; on the Galois side, the enhanced-parameter Bernstein component carries an algebra $H(s^\\vee, q_F^{1/2})$ built from the geometry of the parameter variety. The proof establishes a canonical isomorphism of root systems $R_\\sigma \\cong R_{s^\\vee}$, equality of q-parameters (Propositions 6.9 and 8.4), and, in the most difficult step, equality of the 2-cocycles after localization (Theorem 8.7). The 2-cocycle comparison rests on equivariant splittings of families of central extensions on the p-adic side (Proposition 2.10) and on the Galois side (Proposition 3.5), verified case-by-case over root system types.","core_discovery":"The central claim is Theorem 1: there is a bijection between irreducible non-singular depth-zero G-representations and G-relevant enhanced depth-zero L-parameters trivial on wild inertia, with the map $\\pi \\mapsto \\phi_\\pi$ canonical, and with compatibility with the listed representation-theoretic invariants. Here enhanced L-parameters are L-parameters together with an irreducible representation of the relevant component group. Theorem 2 strengthens this to an equivalence of categories between finite-length non-singular depth-zero G-representations and finite-length modules over a direct sum of twisted affine Hecke algebras $H(s^\\vee, q_F^{1/2})$ attached to Bernstein components of enhanced L-parameters. The paper also proves a new equivariance result at the supercuspidal level (Theorem 4.8) and a Kazhdan–Lusztig conjecture statement for the block category. In the authors' approach the bijection is obtained by specializing the category equivalence, so the category equivalence is the primary discovery and the bijection is its shadow.","pith_inferences":["A natural next step is to push the same comparison through general types to arbitrary-depth non-singular representations; the paper notes that the relevant Hecke algebras from such types are known to be isomorphic to depth-zero ones, which makes that extension plausible.","The restriction to finite length in the categorical statement may be real, not just technical: the authors point to cuspidal-level 2-cocycles as obstructions to a Morita equivalence for infinite-length modules, so one should not expect the equivalence to extend literally.","If the 2-cocycle comparison is as robust as claimed, the same Hecke-algebra machinery could be applied to singular depth-zero supercuspidals, where L-packets mix supercuspidal and non-supercuspidal members, despite the paper not treating them."],"forward_implications":["All non-singular depth-zero irreducible representations acquire a canonical L-parameter, and each L-packet is finite and parametrized by the component-group enhancements.","Temperedness, essential square-integrability, cuspidal support, parabolic induction, and the Langlands classification are all respected by the correspondence.","The category equivalence means questions about such representations can be studied through modules over explicit Hecke algebras attached to parameters, giving an algebraic shadow of geometric Langlands-style equivalences in low depth.","The p-adic Kazhdan–Lusztig conjecture holds for the non-singular depth-zero block category, as a direct corollary of Theorem 2."],"supporting_citations":[{"why":"Constructs the first depth-zero regular supercuspidal L-packets, the starting point for the cuspidal LLC used here.","marker":"[DeRe]"},{"why":"Supplies the non-singular supercuspidal LLC and the coherent splittings and extensions of tori that the paper strengthens to equivariance.","marker":"[Kal3]"},{"why":"Gives the Hecke algebra presentation for depth-zero Bernstein blocks as a crossed product of an affine Hecke algebra with a twisted group algebra.","marker":"[Mor1]"},{"why":"Provides the type-theoretic cover and the Hecke algebra for the Bernstein block that the paper compares with the parameter-side algebra.","marker":"[Mor2]"},{"why":"Provides the Bernstein decomposition of enhanced L-parameters and the cuspidal support map on the Galois side.","marker":"[AMS1]"},{"why":"Constructs the twisted affine Hecke algebra $H(s^\\vee, q_F^{1/2})$ and the parametrization of its irreducible modules by enhanced L-parameters.","marker":"[AMS3]"},{"why":"Describes $\\mathrm{End}_G(\\Pi_s)$ and its localization to twisted graded Hecke algebras, carrying the categorical comparison on the p-adic side.","marker":"[Sol5]"},{"why":"Gives the principal-series Hecke algebra comparison used to identify q-parameters on both sides.","marker":"[Sol10]"}],"fun_headline_variants":["Categorical Langlands for non-singular depth-zero reps","Langlands correspondence for depth-zero via categorical equivalence","Hecke algebra equivalence yields Langlands for depth-zero reps","Categorical LLC for non-singular depth-zero representations","Langlands bijection as shadow of Hecke algebra equivalence"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole argument depends on the assertion that certain families of central extensions split equivariantly with respect to the relevant Weyl-group actions, a claim verified by a lengthy case check over root system types; a failure in any unexamined case would break the Hecke-algebra isomorphism and with it the correspondence.","fun_headline_variants_meta":{"raw":{"variants":["Categorical Langlands for non-singular depth-zero reps","Langlands correspondence for depth-zero via categorical equivalence","Hecke algebra equivalence yields Langlands for depth-zero reps","Categorical LLC for non-singular depth-zero representations","Langlands bijection as shadow of Hecke algebra equivalence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000489,"raw_usage":{"total_tokens":2365,"prompt_tokens":863,"completion_tokens":1502,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":479,"completion_tokens_details":{"reasoning_tokens":1422}},"tokens_in":479,"tokens_out":1502,"duration_ms":9274,"temperature":1.0,"reasoning_tokens":1422,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T05:46:44.270654+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find one root system type not covered by Cases I–V in Section 2.1 (or the analogous cases in Section 3.2) for which the extension in Proposition 2.10 or Proposition 3.5 fails to split equivariantly; that would invalidate Theorem 8.7 and hence the categorical equivalence and the bijection. A more arithmetic falsifier would be to exhibit a non-singular depth-zero Bernstein block where the 2-cocycle of the p-adic Hecke algebra is not cohomologous to the parameter-side cocycle under the constructed isomorphism.","supporting_citations":[],"review_version":1}