{"id":"6d0b874c-490b-4267-a9f8-8f0de49bf121","arxiv_id":"2505.16040","paper_version":3,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Depth-zero Hecke algebra parameters equal unipotent Hecke algebra parameters, proving a version of Lusztig's conjecture under tameness.","lead":"This paper proves that the parameters of affine Hecke algebras attached to depth-zero types for p-adic groups match the parameters of Hecke algebras attached to unipotent types. This makes the parameters of arbitrary Bernstein blocks explicitly computable and confirms a conjecture of Lusztig under tameness assumptions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The paper's central claim ultimately rests on [GM20, Cor 4.7.6] and [GM20, Cor 2.6.6] to transfer q-parameter equality from finite fields; if these theorems do not have the exact content used, Theorem 4.4.1 collapses, but no internal flaw is apparent.","rationale":"The reader's weakest_assumption identifies the same load-bearing point: the finite-field q-parameter equality in Proposition 3.2.3 depends on the external compatibility results [GM20, Cor 4.7.6] and [GM20, Cor 2.6.6]. My stress-test reviewed the entire proof chain. The reductions in Section 3.3 carefully handle regular embeddings, product decompositions, and restriction of scalars, and the applications of Proposition 2.1 in Section 4.4 are plausible because the relevant quotients (component groups of parahoric stabilizers) are finite abelian groups. The construction of G_θ and the proof of Theorem 4.4.1 are internally consistent, and the argument that H_K-rel = H_Kθ-rel via a root in the parahoric quotient is sound. No internal contradiction or unstated assumption beyond the quoted external theorems was found. Therefore the only meaningful risk is a misquotation or a subtle mismatch in the hypotheses of [GM20], which is a published and widely used reference. Since the paper honestly cites these results and the reductions aim to meet their assumptions, I do not see a basis for changing the reader's ACCEPT verdict. The concrete test proposed would settle the residual uncertainty by verifying the exact statements of the two corollaries.","tokens_in":32239,"tokens_out":28942,"duration_ms":237498,"concrete_test":"Check the exact statements of [GM20, Corollary 4.7.6] and [GM20, Corollary 2.6.6]. For Cor 4.7.6, verify that it applies after the reduction to f-simple adjoint groups (not only absolutely simple groups) and that the Jordan decomposition used is compatible with the one from [DM90, Theorem 7.1] after the reductions in Section 3.3. For Cor 2.6.6, verify that it directly yields dim(ρ_1)/dim(ρ_2) = dim(u*_1)/dim(u*_2) for the two constituents of a parabolically induced representation of length two; if it instead only gives degree formulas up to a common factor, confirm that the factor is identical for both constituents, so the ratio equality still holds.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The main theorem reduces to the equality of q-parameters in Proposition 3.2.3: q(R^G_M(ρ_M)) = q(R^{G*_s}_{M*_s}(u*_M)). The proof of this proposition invokes [GM20, Cor 4.7.6], that Jordan decompositions commute with parabolic induction when Z(G) is connected and G_ad is simple, and [GM20, Cor 2.6.6], that dimension ratios of constituents are preserved by the Jordan decomposition. The reductions in Section 3.3 appear sound: regular embeddings and the product decomposition use Corollary 2.2 with abelian or trivial quotients, so the hypotheses of Proposition 2.1 are satisfied; the absolutely simple case via restriction of scalars is justified using [Tay19, Cor 8.8]. Thus the only possible failure point is the precise content of the two quoted corollaries. If [GM20, Cor 4.7.6] requires G_ad to be absolutely simple rather than merely f-simple, or if [GM20, Cor 2.6.6] only preserves individual dimensions up to a factor that might differ between the two constituents of a length-two induced representation, then equation (4.4.1c) would not follow and the Weyl group and parameter equalities in Theorem 4.4.1 would be ungrounded. This is a genuine external dependency, not an internal inconsistency.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proves that for a depth-zero type (K, ρ) of a connected reductive p-adic group G, the affine Hecke algebra H(W(ρ_M)_{aff}, q) appearing in the structure theorem of [AFMO24a, Thm 5.3.6] is isomorphic to the affine Hecke algebra attached to a unipotent type for a group G_θ that splits over an unramified extension (Theorem 4.4.1). The key finite-field ingredient is Proposition 3.2.3, which equates the q-parameter of the parabolic induction of a cuspidal representation with that of a unipotent cuspidal representation via Jordan decomposition. Combining this with [AFMO24b] yields Theorem 4.5.1 and, under the usual tameness assumptions, a version of Lusztig's conjecture on parameters of Hecke algebras (Theorem 4.6.2). The proof is detailed, with explicit reduction steps in Sections 2 and 3.3.","tokens_in":32549,"tokens_out":24401,"duration_ms":197743,"significance":"If correct, the main result gives an explicit reduction of the parameters of arbitrary Bernstein Hecke algebras to the unipotent case, where they are already described by Lusztig. This is a substantial step beyond previous results of Roche and others. The proof is coherent; the main fragility is the reliance on [GM20, Cor 4.7.6 and 2.6.6] for the finite-field q-parameter equality. However, the paper's reduction in Section 3.3 explicitly reduces to the connected-center, absolutely-simple case, which is the hypothesis these results require, so this is an external dependency rather than an internal inconsistency. The paper also honestly acknowledges the non-canonical choices in the disconnected-center case (Remark 3.2.2).","major_comments":[],"minor_comments":[{"comment":"The title page reads \"forp-adic groups\"; it should read \"for p-adic groups\".","section":"Title page"},{"comment":"The text \"irreducible constitutes\" should be \"irreducible constituents\".","section":"Section 2, proof of Proposition 2.1"},{"comment":"Please provide the precise statements (with theorem numbers or page references) of [GM20, Corollary 4.7.6] and [GM20, Corollary 2.6.6] as used in the proof of Proposition 3.2.3, including the hypotheses on the group; the reduction in Section 3.3 is designed to meet them, but the reader cannot verify this without the exact statements.","section":"Section 3.4"},{"comment":"The notation is inconsistent: the representation is introduced as u_x, but the theorem states q_{θ,s} = q(ind_{K_{θ,h}}^{K_{θ,x}}(ρ_{θ,x})); please unify the notation.","section":"Section 4.3, Theorem 4.3.7"},{"comment":"The grammar \"the equivalence relations ∼ is defined\" should be \"the equivalence relation ∼ is defined\".","section":"Section 3.2, paragraph before Proposition 3.2.3"},{"comment":"The phrase \"a prioridepends\" should be \"a priori depends\".","section":"Section 3.3, Remark 3.3.1"},{"comment":"The notation \"H K-rel\" appears without subscripts; it should be H_{K-rel} for consistency with the surrounding text.","section":"Section 4.4, proof of Theorem 4.4.1"}],"recommendation":"minor_revision","confidential_remarks":"The paper makes a significant contribution and the central argument is convincing. The only substantive concern is that the proof of Proposition 3.2.3 hinges on the exact content of two corollaries from [GM20] that are not quoted; asking the author to state them precisely will make the paper self-contained enough for verification. The manuscript fits the scope of the journal well."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this paper delivers a genuine reduction. It proves that the affine Hecke algebra attached to any depth-zero type is isomorphic to one attached to a unipotent type (Thm 4.4.1), and under the usual tame hypotheses this gives a version of Lusztig's parameter conjecture. That is a substantial step, not a re-announcement of known results. The new content is the q-parameter comparison over finite fields (Prop 3.2.3, Cor 3.5.6) and the construction of the group G_theta in Section 4.3. The parameters are computed as dimension ratios of induced representations, defined independently of the conclusion, so there is no circularity.\n\nThe paper is honest about its debts. The q-parameter equality rests on [GM20, Cor 4.7.6] (Jordan decomposition commutes with parabolic induction) and [GM20, Cor 2.6.6] (dimension ratios preserved). If those two corollaries do not have exactly the content used, Thm 4.4.1 loses its foundation. The stress-test note asks precisely this question; I could not find a mismatch in how they are cited, but a referee needs to pull both corollaries and check the hypotheses against the reduction arguments in Section 3.3. Those reductions look sound to me: Prop 2.1 is a clean dimension-counting argument, and the regular embedding plus product decomposition steps fill in properly. The paper also depends on the author's own [AFMO24a,b], which are not yet peer-reviewed; the comparison in Section 4.2 is only as solid as those preprints.\n\nThe soft spots are in proportion: mostly external, not internal. The proof of Lemma 4.2.4(2) is a little compressed when it extends [AFMO24a, Lemma 3.8.9] to all H by saying the same proof works for general H; that should be expanded or checked. The construction of G_theta is intricate, especially the Galois descent of the inner twist, but the written steps line up. The author explicitly acknowledges the non-canonical choice of Jordan decomposition for disconnected centers, and mentions Eteve's categorical alternative; that is the right level of caution.\n\nWho should read it: anyone computing Hecke algebra parameters for p-adic groups, and anyone comparing type-theoretic Hecke algebras with Solleveld's. A serious referee should be assigned. I would accept it after a referee checks the two GM20 corollaries and the AFMO24 dependency. My own verdict is close to the reader's: ACCEPT, moderate confidence.","headline":"A careful reduction of q-parameters to unipotent ones; worth refereeing, with the main caveat being its reliance on two external Jordan-decomposition compatibility results.","tokens_in":33087,"tokens_out":3030,"would_cite":true,"duration_ms":25548,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22E50","20C08","20C33"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every depth-zero type's affine Hecke algebra is isomorphic to a unipotent type's.","keywords":["p-adic groups","Hecke algebras","types","unipotent representations","Bernstein blocks","affine Hecke algebras","q-parameters","depth-zero types"],"falsifier":"Compute directly the dimension ratio of the two constituents of the induced representation for a non-unipotent cuspidal representation of a maximal Levi subgroup of $\\mathrm{GL}_3$ over the field of two elements; if it differs from the unipotent-side ratio, the finite-field equality on which Theorem 4.4.1 relies fails.","tokens_in":32042,"feed_emoji":"🧮","tokens_out":9973,"duration_ms":84983,"temperature":0.7,"pith_summary":"This paper establishes that the affine Hecke algebra attached to any depth-zero type of a connected reductive p-adic group is isomorphic to the affine Hecke algebra attached to a unipotent type for an associated connected reductive group that splits over an unramified extension. Because Hecke algebras attached to types govern Bernstein blocks, this reduces the calculation of the previously unknown parameters of these algebras—dimension ratios of induced representations—to the explicitly known unipotent case. The proof compares a numerical invariant, the q-parameter of a length-two representation, across finite-field parabolic inductions and then lifts the equality to p-adic intertwiners. A conjecture on parameters for arbitrary Bernstein blocks follows under the assumption that the group splits over a tamely ramified extension and the residue characteristic does not divide the order of the absolute Weyl group.","feed_headline":"Every depth-zero type's Hecke parameters match unipotent cases","feed_subtitle":"The unknown dimension ratios now equal explicitly known unipotent-type parameters, so they are computable.","key_machinery":"The load-bearing object is the q-parameter: for a finite-dimensional representation of length two, the ratio $\\dim(\\pi_1)/\\dim(\\pi_2)$ of its constituents. Affine Hecke algebra parameters $q_s$ are exactly such ratios, via the standard identification of q-parameters with dimension ratios of parabolically induced finite reductive representations. The argument shows these ratios are invariant under two moves compatible with the construction of types: passing through a group homomorphism with abelian cokernel (Proposition 2.1), and passing from a cuspidal representation to the unipotent representation attached to it by the Jordan decomposition over a finite field (Proposition 3.2.3). To transfer the equality to p-adic groups, the paper constructs a connected reductive group $G_\\theta$ from the $\\theta$-orthogonal affine root system of the original datum, normalizing the affine roots by positive scalars so that they form a genuine affine root system, and proves the parahoric quotients of $G$ and $G_\\theta$ are adjointly isomorphic up to duals; this makes the unipotent comparison valid even when $G$ is ramified.","core_discovery":"The central claim is Theorem 4.4.1: for a depth-zero type $(K, \\rho)$ of a connected reductive p-adic group $G$, the sets of $K$-relevant affine hyperplanes and the affine Weyl groups they generate coincide with the corresponding objects for a unipotent type $(K_\\theta, u)$ of an associated group $G_\\theta$ that splits over an unramified extension. The parameters agree on simple reflections, so the affine Hecke algebras $H_C(W(\\rho_M)_{\\mathrm{aff}}, q)$ and $H_C(W(u_{\\rho_M})_{\\mathrm{aff}}, q_\\theta)$ are isomorphic. Since earlier work had already identified the full Hecke algebra attached to any tame type with a depth-zero Hecke algebra, the same isomorphism holds for the affine part of Hecke algebras attached to tame types. Under the assumptions that $G$ splits over a tamely ramified extension and the residue characteristic does not divide the order of the absolute Weyl group, this proves the conjecture that the parameters of the Hecke algebra attached to an arbitrary Bernstein block coincide with those of a unipotent Bernstein block.","pith_inferences":["The paper proves the isomorphism only for the affine Hecke algebra factor; if the twisted group algebra factors were also shown to be isomorphic, the entire depth-zero Hecke algebra would be Morita equivalent to a unipotent one.","The construction of $G_\\theta$ by normalizing affine roots suggests that, for ramified groups, $G_\\theta$ is the correct dual-side object for a Bernstein block; identifying it in explicit examples would test whether the comparison extends to the full block.","Because the q-parameter comparison is essentially a finite-field computation, the same route could yield other invariants of depth-zero Bernstein blocks—for instance formal degrees—once the full Hecke algebra isomorphism is in hand.","The scaling factors used in the normalization enter only through the construction of $G_\\theta$; computing them in one ramified example would show whether the unipotent comparison is genuinely needed there."],"forward_implications":["The q-parameters of any depth-zero affine Hecke algebra can now be read off from the explicit unipotent parameter lists rather than computed hyperplane by hyperplane.","The same equality holds for the affine Hecke algebra factors attached to tame types, so the only unknown part of those Hecke algebras is the twisted group algebra factor.","For groups splitting over a tamely ramified extension with residue characteristic not dividing the absolute Weyl group order, every Bernstein block has a Hecke algebra whose parameters equal those of a unipotent block.","The isomorphism preserves the standard anti-involutions, so the involutive structure of the affine Hecke algebra is also captured by the unipotent model.","The earlier principal-series identification of such Hecke algebras with Iwahori–Hecke algebras appears as the special case where the depth-zero datum is induced from a torus."],"supporting_citations":[{"why":"Supplies the explicit semidirect-product description of the depth-zero Hecke algebra whose affine part is the object under study.","marker":"[AFMO24a, Theorem 5.3.6]"},{"why":"Reduces Hecke algebras for tame types to depth-zero Hecke algebras, extending the isomorphism to those types.","marker":"[AFMO24b, Theorem 4.4.1]"},{"why":"States that Jordan decompositions commute with parabolic induction in the connected-center simple-adjoint case, the key compatibility for Proposition 3.2.3.","marker":"[GM20, Cor 4.7.6]"},{"why":"Supplies the preservation of constituent dimension ratios under the Jordan decomposition, used to equate q-parameters.","marker":"[GM20, Cor 2.6.6]"},{"why":"Identifies affine Hecke algebra parameters with dimension ratios of parabolically induced finite reductive representations.","marker":"[HL80, Theorem 3.18 (ii)]"},{"why":"Provides the refined Jordan decomposition bijection used to define the unipotent representation $u^*_M$.","marker":"[DM90, Theorem 7.1]"},{"why":"Gives the explicit classification of unipotent p-adic representations whose parameters are the target of the comparison.","marker":"[Lus95]"},{"why":"Prior description of depth-zero Hecke algebras as semidirect products, generalized by the paper's earlier work and extended here.","marker":"[Mor93]"}],"fun_headline_variants":["Depth-zero Hecke parameters equal unipotent ones","Tame-type Hecke parameters match unipotent cases","Hecke parameters for p-adic groups now computable","Lusztig's conjecture on Hecke parameters proven","Affine Hecke algebras align for depth-zero and unipotent"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The calculation assumes that the dimension ratio of an induced representation survives unchanged when the representation is replaced by its unipotent counterpart through the Jordan decomposition in the relevant finite reductive groups.","fun_headline_variants_meta":{"raw":{"variants":["Depth-zero Hecke parameters equal unipotent ones","Tame-type Hecke parameters match unipotent cases","Hecke parameters for p-adic groups now computable","Lusztig's conjecture on Hecke parameters proven","Affine Hecke algebras align for depth-zero and unipotent"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000761,"raw_usage":{"total_tokens":3490,"prompt_tokens":1166,"completion_tokens":2324,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":782,"completion_tokens_details":{"reasoning_tokens":2246}},"tokens_in":782,"tokens_out":2324,"duration_ms":16310,"temperature":1.0,"reasoning_tokens":2246,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T15:07:55.608663+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute directly the dimension ratio of the two constituents of the induced representation for a non-unipotent cuspidal representation of a maximal Levi subgroup of $\\mathrm{GL}_3$ over the field of two elements; if it differs from the unipotent-side ratio, the finite-field equality on which Theorem 4.4.1 relies fails.","supporting_citations":[],"review_version":1}