{"id":"074bdc08-5a38-40bd-8798-ec1cc3643d44","arxiv_id":"2509.07780","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For each depth r, irreducible depth-r representations of a tamely ramified p-adic reductive group are assigned tame restricted Langlands parameters on the r-th inertia subgroup, nontrivial when r lies in Z_(p).","lead":"This paper pairs every irreducible representation of a tamely ramified p-adic reductive group, at a fixed depth, with a restricted Langlands parameter on the inertia side. If correct, it supplies a new uniform piece of the local Langlands correspondence for all depths, including depths outside the tame range.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The p=2 positive-depth construction in Theorem 1 is defined through the to-appear [ST25, Thm. 1] bijection (5); if that bijection fails, the map Irr(G(F))_r→RP_r is not defined for residual characteristic 2.","rationale":"I read the paper's central claim as the construction of restricted inertia-group parameters for irreducible smooth representations of fixed depth, for all depths and all residual characteristics of a tamely ramified p-adic group. The proof is elaborate and mostly self-contained: Sections 2–5 build the positive-depth map from Moy-Prasad types through depth-r Deligne-Lusztig parameters, and Section 7 handles depth zero by Deligne-Lusztig induction at parahoric quotients. The main steps are internally coherent, the diagrams commute on their faces, and I found no circular argument, fitted parameter, or internal contradiction. The deepest external input is indeed the invariant-theoretic bijection (5). For p>2 it is classical [KW76, Thm. 4]; for p=2 the paper cites the to-appear [ST25, Thm. 1]. This bijection is load-bearing because it is needed to define i_{α,x} in diagram (6), and hence to define DL_r and the bijection DL_r≅RP_r in Lemma 45. If [ST25] were unavailable, false, or subject to extra hypotheses not met by the parahoric quotients arising from Moy-Prasad types, the positive-depth part of Theorem 1 would be undefined in characteristic 2. The paper itself flags the dependence by citing the result as to appear, so the reader's conditional verdict is faithful to the evidence. I also note a smaller p=2 dependence in Lemma 36 through [Spi21, Prop. 3.3], but that preprint is accessible and is less central. Since my stress test confirms the reader's identified weak point and does not find an independent failure, I recommend keeping the verdict unchanged: conditional acceptance, with the p=2 bijection made available or the unconditional claim restricted to p>2.","tokens_in":34277,"tokens_out":11663,"duration_ms":109888,"concrete_test":"Obtain [ST25] (or request the authors to append its proof) and check that Theorem 1 there proves that (5) is a bijection on ¯k-points for every connected reductive quotient G(E^u)_{hx=0} arising from a tamely ramified G and every x∈B(G,F), with no hidden restrictions such as 'G split' or 'p good'. If such a statement is proved in full generality, the concern is resolved; if [ST25] carries extra hypotheses, verify those hypotheses against the parahoric quotients used in Lemmas 36 and 45, and if they are not met, restrict Theorem 1 to p>2 or to the verified cases.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1 is asserted for every residual characteristic p and every r∈Q≥0. For r>0, the map Irr(G(F))_r→RP_r is obtained by composing Corollary 37 with the bijection of Lemma 45 between depth-r Deligne-Lusztig parameters DL_r and restricted depth-r parameters RP_r. The definition of DL_r and the construction of i_{α,x} in diagram (6) require inverting the invariant-theoretic map (5), namely ¯a^*//W^E_{hx}→g^*(E^u)_{hx=0}//G(E^u)_{hx=0}. For p>2 this inversion is justified by [KW76, Thm. 4]; for p=2 it is justified by [ST25, Thm. 1], which is cited as 'to appear' and is not independently checkable from this preprint. Thus the entire positive-depth part of Theorem 1 in residual characteristic 2 is contingent on an external result whose hypotheses and proof are not available to the reader. The dependence is not cosmetic: without a valid bijection (5), the element i_{α,x}(X) is not defined, so Lemma 40, Lemma 45, and hence the map in Theorem 1 do not exist in the p=2 case. The nontriviality statement for r∈Z_(p) also does not escape this dependence, because the restricted parameter φ_π is defined through the DL_r↔RP_r bijection. Lemma 36 has a second p=2 external input, [Spi21, Prop. 3.3], though that preprint is available and plays a smaller role. The depth-zero construction in Section 7 is independent of (5). No internal inconsistency or circular reasoning was found; the concern is purely that a load-bearing input for p=2 is unverifiable as the manuscript stands.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs, for a tamely ramified p-adic reductive group G, a canonical map from the set of irreducible smooth representations of depth r to G^∨-conjugacy classes of restricted depth-r Langlands parameters. For r>0, the construction first attaches to a nondegenerate Moy-Prasad type (x,X) a depth-r Deligne-Lusztig parameter in a quotient of the dual Lie algebra of a maximal torus (Section 4), and then identifies depth-r Deligne-Lusztig parameters bijectively with tame restricted depth-r parameters using local Langlands for tori (Section 5); the main positive-depth statement is Lemma 46. The depth-zero case is handled separately in Section 7 via Deligne-Lusztig parameters of parahoric quotients, giving Lemma 63. The paper also formulates Conjectures 47, 49, and 52 relating the construction to the full local Langlands correspondence, and proves Conjecture 47 for Kaletha's semisimple supercuspidal parameters under restrictive hypotheses (Lemma 58).","tokens_in":34614,"tokens_out":19750,"duration_ms":176949,"significance":"If correct, the paper gives an explicit, choice-independent assignment of restricted Langlands parameters to all irreducible representations of tame p-adic groups, at the exact depth of the representation, together with a nontriviality statement for p-integral depths. This is a substantial step toward an explicit local Langlands correspondence and should be of interest to the representation theory and automorphic forms communities. The paper is well structured, the positive-depth construction is reduced to two clearly identified steps, and the depth-zero construction is essentially self-contained. The main caveats are that the p=2 positive-depth case depends on an external 'to appear' result [ST25], and that one explanatory sentence around diagram (6) asserts a false inclusion; both need correction. The conjectures are precise and the verification in Lemma 58 gives useful evidence, though that proof is very compressed.","major_comments":[{"comment":"The sentence following diagram (6) states: 'The inclusion g^*(E')_{hx=-r}⊂a^*(E^u)_{-r} holds because G is E'-split.' This inclusion is false as written. For a split group and hx in the apartment of A, the Moy-Prasad quotient g^*(E')_{hx=-r} has dimension equal to dim G, whereas a^*(E^u)_{-r} has dimension equal to the rank of G; already for G=SL_2 the root spaces contribute to the filtration. The construction of i_{E,h,α,x} should not require this inclusion: one should map α Ad^*(h)Y to its G(E^u)_{hx=0}-orbit in g^*(E^u)_{hx=0}//G(E^u)_{hx=0} and then apply the inverse of the bijection (5). Since the definition of the depth-r Deligne-Lusztig parameter ι_x(X) rests on this step, the diagram and the surrounding explanation must be corrected.","section":"Section 4.1, diagram (6)"},{"comment":"For p=2, the bijection (5) is justified by [ST25, Thm. 1], which is cited as 'to appear' and is not independently checkable from this preprint. This dependence is load-bearing: the inverse of (5) enters the definition of i_{E,h,α,x}, hence of ι_x(X), Lemma 40, Lemma 45, and the positive-depth part of Theorem 1 for residual characteristic 2. The nontriviality assertion for r∈Z_(p) also passes through Lemma 45 and inherits the same dependence. Please either include a complete proof or a precise statement of [ST25, Thm. 1] in an appendix, or explicitly state Theorem 1 and the positive-depth lemmas only for p>2 (and for p=2 conditionally on [ST25]). As the manuscript stands, the p=2 case cannot be verified by a reader.","section":"Section 4.1, Eq. (5)"}],"minor_comments":[{"comment":"The transfer of [AD02, Theorem 3.1.2 and Corollary 3.2.6] from g(F) to g^*(F) is asserted via a remark in loc. cit.; since this transfer underpins Lemma 31 and hence Corollary 37, it would be helpful to add two or three sentences explaining why the arguments are purely lattice-theoretic.","section":"Section 4.2.1, before Lemma 31"},{"comment":"The proof of Lemma 58 is quite compressed, especially the assertion that in Kaletha's construction one has φ|_{I^{0+}_F}=(T^∨→G^∨)∘φ_θ|_{I^{0+}_F} up to G^∨-conjugacy. Please expand this step or give precise references to the relevant parts of [Kal19] and [Kal21], since Lemma 58 is the main evidence for Conjecture 47.","section":"Section 6.3, Lemma 58"},{"comment":"In the definition of DL_0, the Frobenius endomorphism is denoted F, which clashes with the base field F. Using a different symbol (for example Φ) for the Frobenius would remove a recurring ambiguity.","section":"Section 7.1"},{"comment":"There are a few typographical issues: in the Introduction the notation 'res m_F Λ' should read 'res_{m_F} Λ', and in the display of Theorem 1 the condition 'Irr(G(F)) r̸=∅' should be 'Irr(G(F))_r≠∅'.","section":"Introduction"}],"recommendation":"major_revision","confidential_remarks":"The essential dependence on [ST25], which is co-authored by one of the authors and listed as 'to appear', should be resolved before acceptance; the editor may wish to verify that this reference is publicly available and that its hypotheses match the present setting. The false inclusion in diagram (6) is likely a typographical issue rather than a fatal flaw, but it obscures a central definition and must be corrected carefully."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Rough take: this is a real construction, not a repackaging. The authors build a map from depth-r irreducible representations of a tamely ramified p-adic group to restricted Langlands parameters on the r-th upper numbering inertia subgroup, for all rational r, and show that when r lies in Z_(p) it lands on nontrivial parameters. The depth-zero piece also subsumes DeBacker-Reeder, Kazhdan-Lusztig, and Lusztig. The rational-depth Deligne-Lusztig parameters in Section 4 and the bijection DL_r ≅ RP_r in Lemma 45 are new, and they handle r outside Z_(p) and non-split groups. The proofs are detailed; the main theorem follows from the stated lemmas.\n\nThe soft spot is exactly the one flagged in the report: the positive-depth construction in residual characteristic 2 uses the map (5), which for p=2 is an inversion of a bijection from [ST25, Thm 1], cited as to appear. That is load-bearing: without it, i_{α,x}(X) is not defined, so Lemma 40, Lemma 45, and the p=2 part of Theorem 1 collapse. The paper also relies on [Spi21, Prop. 3.3] in Lemma 36 for p=2, but that preprint is available. I think the reader's concern is on target. The authors are not hiding it—they cite [ST25] as to appear—but the issue is real. The fix is straightforward: either make [ST25] available or restrict the unconditional positive-depth claim to p>2. Everything else I checked looks solid. I found no circularity or fitted parameters. The conjectures in Section 6 are clearly labeled as conjectures, and Lemma 58's evidence for Kaletha's correspondence is a genuine check, not a substitute for a proof. The paper also shows good judgment in remarks like Remark 14 (torus depth preservation failure) and Remark 55 (why a naive upgrade of Conjecture 49 fails).\n\nWho should read this: anyone working on the local Langlands correspondence for tame groups, Moy-Prasad types, or depth-invariance. It deserves a serious referee. I would accept it for peer review and ask the authors to address the p=2 dependence before publication.","headline":"A genuinely new construction of restricted Langlands parameters for all rational depths in tame groups, with the positive-depth p=2 case resting on a to-appear result that should be resolved before publication.","tokens_in":35260,"tokens_out":2871,"would_cite":true,"duration_ms":23445,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20G25","22E50"],"pacs":[],"model":"deepseek-v4-flash","headline":"For a tamely ramified $p$-adic reductive group, the paper constructs a partial local Langlands correspondence: every irreducible representation of depth $r$ is assigned a $G^\\vee$-conjugacy class of homomorphisms from $I_F^r$ to the dual…","keywords":["p-adic reductive groups","Moy-Prasad types","depth of representations","local Langlands correspondence","Langlands parameters","Deligne-Lusztig parameters","tame ramification","ramification filtration"],"falsifier":"Exhibit a residue-characteristic-2 group and a point $x\\in B(G,F)$ for which the map $\\overline{\\mathfrak{a}}^*//W^E_{hx}\\to\\mathfrak{g}^*(E^u)_{hx=0}//G(E^u)_{hx=0}$ is not bijective on $\\overline{k}$-points; the construction of $\\iota_x(X)$ then fails, so Theorem 1 would not hold in that case.","tokens_in":34021,"feed_emoji":"🧮","tokens_out":9859,"duration_ms":79169,"temperature":0.7,"pith_summary":"This paper establishes a partial local Langlands correspondence for tamely ramified $p$-adic reductive groups, organized by depth. The main result is that every irreducible smooth representation of depth $r$ gives rise, in a canonical way, to a $G^\\vee$-conjugacy class of continuous homomorphisms from the $r$-th upper-numbering inertia subgroup $I_F^r$ to the dual group $G^\\vee$. This class is a restricted depth-$r$ parameter, and the paper proves it has depth at most $r$, with equality whenever $r$ is rational with denominator prime to $p$. The value for a reader is that the construction extracts Galois-side data directly from the Moy-Prasad type occurring in a representation, without needing a full Langlands parameterization, and it comes with conjectures about how it must sit inside any complete local Langlands correspondence.","feed_headline":"Depth-r representations get Langlands parameters","feed_subtitle":"New construction matches Moy-Prasad types to restricted parameters and predicts exactly where depth is preserved.","key_machinery":"The load-bearing object is the depth-$r$ Deligne-Lusztig parameter: an equivalence class of pairs $(\\alpha, Z)$ with $\\alpha\\in\\overline{F}$ of valuation $r$ and $Z\\in(\\overline{\\mathfrak{a}}^*//W)(\\overline{k})$, where $\\overline{\\mathfrak{a}}^*$ is the dual of the Lie algebra of the reductive quotient of a maximal $F^t$-split torus and $W$ is the Weyl group. It is built from a Moy-Prasad type $(x,X)$ by scaling $X$ by $\\alpha$ and using the bijection $(5)$ between $\\overline{\\mathfrak{a}}^*//W^E_{hx}$ and $\\mathfrak{g}^*(E^u)_{hx=0}//G(E^u)_{hx=0}$ to land in the quotient variety; the paper proves independence of the auxiliary field and of the conjugating element. The second move is a bijection between these Deligne-Lusztig parameters and restricted depth-$r$ parameters, obtained by lifting $Z$ to a torus character via the Moy-Prasad isomorphism and applying local Langlands for tamely ramified tori. This two-step transfer is what makes the map from representations to parameters canonical.","core_discovery":"The paper's central discovery is a construction that assigns to each irreducible smooth representation $\\pi$ of depth $r$ a restricted depth-$r$ Langlands parameter $\\varphi_\\pi\\colon I_F^r\\to G^\\vee$, well defined up to $G^\\vee$-conjugation. For positive depth the construction goes through an intermediate object, the depth-$r$ Deligne-Lusztig parameter, obtained from a nondegenerate Moy-Prasad type $(x,X)$ by transporting $X$ to the quotient $\\overline{\\mathfrak{a}}^*//W$ of a maximal torus; the paper proves this intermediate parameter depends only on the stable-associate class of the type. It then establishes a bijection between depth-$r$ Deligne-Lusztig parameters and restricted depth-$r$ parameters, which yields the map in Theorem 1. When $r\\in\\mathbb{Z}_{(p)}\\cap\\mathbb{Q}_{>0}$, the resulting parameter is nontrivial and has depth exactly $r$. The depth zero case is handled by a parallel construction using Deligne-Lusztig induction on the reductive quotient, giving a map from depth zero representations to restricted depth zero parameters. The paper also proves that its construction is compatible with a known local Langlands correspondence for semisimple supercuspidal representations under mild hypotheses.","pith_inferences":["Because the map $\\mathrm{Irr}(G(F))_r\\to RP_r$ is defined purely from Moy-Prasad types, it gives a finite, checkable inertia-restricted shadow of the local Langlands correspondence for any tame group, even where no full correspondence is known.","The nontriviality statement for $r\\in\\mathbb{Z}_{(p)}$ suggests a test for depth preservation: if a family of representations is known to have Langlands parameters of depth smaller than the representations' depth, then either the family escapes the tameness hypothesis or the additional assumptions in Conjecture 52 are necessary.","Once the residue-characteristic-2 bijection cited as [ST25] is available in full, the construction's range can be probed computationally for small groups such as $\\mathrm{SL}_2$ over $\\mathbb{Q}_2$; a computed $\\varphi_\\pi$ could be compared with the restriction of any known parameter to $I_F^r$, giving direct evidence for or against Conjecture 47.","The same Deligne-Lusztig parameter formalism could be adapted to define stable associates for arbitrary depth types, which may clarify when two Moy-Prasad types should be viewed as Langlands-equivalent even outside the tame setting."],"forward_implications":["For every representation $\\pi$ of depth $r$, the construction produces a well-defined restricted depth-$r$ parameter whose depth is at most $r$, so the Galois-side datum never overshoots the representation's depth.","When $r\\in\\mathbb{Z}_{(p)}\\cap\\mathbb{Q}_{>0}$, every parameter in the image has depth exactly $r$, so the partial correspondence is nontrivial at those depths rather than collapsing to depth zero.","The bijection between depth-$r$ Deligne-Lusztig parameters and restricted depth-$r$ parameters gives a complete description of the target set $RP_r$ in terms of torus quotients.","If Conjecture 47 is granted, then the restriction of any Langlands parameter for $\\pi$ to $I_F^r$ is forced to be the constructed parameter; the paper proves the depth inequality and the converse conjecture follow from this.","Under the stated hypotheses on $p$ and $G$, the paper proves Conjecture 47 for the local Langlands correspondence constructed in [Kal19, Kal21] for semisimple supercuspidal representations.","The depth zero case produces a map $\\mathrm{Irr}(G(F))_0\\to RP_0$ via Deligne-Lusztig theory, and the paper records that the analogous conjecture is known for tamely ramified tori, for unipotent representations, and for the DeBacker-Reeder correspondence."],"supporting_citations":[{"why":"Shows every irreducible smooth representation contains a nondegenerate Moy-Prasad type and that any two occurring in a fixed representation are associate; this is the starting point for attaching a depth-$r$ Deligne-Lusztig parameter to $\\pi$.","marker":"[MP94, Thm. 5.2]"},{"why":"Provides the bijection (5) for $p>2$ between $\\overline{\\mathfrak{a}}^*//W^E_{hx}$ and $\\mathfrak{g}^*(E^u)_{hx=0}//G(E^u)_{hx=0}$ that lets a Moy-Prasad type be transported to a Deligne-Lusztig parameter.","marker":"[KW76, Thm. 4]"},{"why":"Supplies the same bijection in residue characteristic 2; cited as to appear, and the construction of Theorem 1 in the $p=2$ case depends on it.","marker":"[ST25, Thm. 1]"},{"why":"Gives the Moy-Prasad isomorphisms used to turn elements of the depth filtration into characters and to move dual-Lie-algebra elements to the torus side.","marker":"[KP23, Thm. 13.5.1]"},{"why":"Local Langlands for tori and its depth preservation for tamely ramified tori convert the torus character $\\chi^T_{X,E}$ into a restricted parameter $I_F^r\\to T^\\vee$.","marker":"[Yu09, §7.5 and §7.10]"},{"why":"The Deligne-Lusztig construction that attaches an element of $S^\\vee//W_{x,S}$ to a cuspidal representation of the reductive quotient, used in the depth zero map.","marker":"[DL76, Section 5]"},{"why":"Reformulates the Deligne-Lusztig parameter via $S^\\vee//W$, which is the form used to define the depth zero Deligne-Lusztig parameter set $DL_0$.","marker":"[L07, Section 16]"},{"why":"Identifies semisimple conjugacy classes in $G^\\vee\\rtimes\\theta$ with points of $S^\\vee//W$, giving the bijection $RP_0\\cong DL_0$ for restricted depth zero parameters.","marker":"[B77, Proposition 6.7]"},{"why":"The local Langlands correspondence for semisimple supercuspidal representations against which the paper proves Conjecture 47 in a range of cases.","marker":"[Kal19, Kal21]"}],"fun_headline_variants":["Moy-Prasad types unlock depth-r Langlands parameters","Restricted Langlands parameters from depth-r types","Fixed-depth Langlands correspondence via Moy-Prasad","Depth-r representations get restricted Langlands data","New Langlands map for Moy-Prasad types"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction depends on a claimed one-to-one matching between two algebraic varieties attached to the group; in residue characteristic 2, that matching is cited from a paper that has not yet appeared, so if the matching fails for some group, the paper's positive-depth map is not defined there.","fun_headline_variants_meta":{"raw":{"variants":["Moy-Prasad types unlock depth-r Langlands parameters","Restricted Langlands parameters from depth-r types","Fixed-depth Langlands correspondence via Moy-Prasad","Depth-r representations get restricted Langlands data","New Langlands map for Moy-Prasad types"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000215,"raw_usage":{"total_tokens":1396,"prompt_tokens":883,"completion_tokens":513,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":499,"completion_tokens_details":{"reasoning_tokens":438}},"tokens_in":499,"tokens_out":513,"duration_ms":4434,"temperature":1.0,"reasoning_tokens":438,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:10:55.071473+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a residue-characteristic-2 group and a point $x\\in B(G,F)$ for which the map $\\overline{\\mathfrak{a}}^*//W^E_{hx}\\to\\mathfrak{g}^*(E^u)_{hx=0}//G(E^u)_{hx=0}$ is not bijective on $\\overline{k}$-points; the construction of $\\iota_x(X)$ then fails, so Theorem 1 would not hold in that case.","supporting_citations":[],"review_version":2}