{"id":"65d7bcae-6579-4312-8277-28ab3e20dbd4","arxiv_id":"2506.04449","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For large q, positive-depth Deligne-Lusztig induction matches the Yu-Kaletha-FKS algebraic construction for all Howe-unramified elliptic pairs, via a characterization theorem and a Green function comparison.","lead":"This paper shows that, for large finite residue fields, the geometric construction called positive-depth Deligne-Lusztig induction produces exactly the supercuspidal representations built algebraically by Yu, Kaletha, and Fintzen-Kaletha-Spice, with the correct Langlands parametrization. It also develops positive-depth Green functions and proves a positive-depth Springer hypothesis, explaining geometrically why orbital integrals appear in supercuspidal character formulas.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The comparison theorem hangs on the unproved scalar-product formula [Cha24, Thm 6.2]; without it R^G_r_{T_r}(theta) need not be +/-irreducible, so Theorem 5.6 and its corollaries lack an object to compare.","rationale":"The reader correctly identified the reliance on [Cha24, Theorem 6.2] as the weakest assumption. I agree that this is the single most load-bearing point: without the scalar-product formula, the irreducibility of R^G_r_{T_r}(theta) up to sign fails, and the entire comparison theorem collapses. The paper is otherwise careful, transparent about its q-assumptions, and provides independent checks of the near-necessity of (*) in Section 10. However, because the central theorem is logically conditional on an unproved preprint result by one of the authors, the most honest verdict is CONDITIONAL: accept the paper's internal reasoning, but mark the main theorem as pending independent verification of [Cha24, Theorem 6.2] (and, for Section 9, [BC24, Theorem 10.9]). This is not a rejection: the mathematics presented here is coherent, and the authors flag the dependency, but the reader's ACCEPT with moderate confidence is one step stronger than the evidence supports.","tokens_in":56863,"tokens_out":12130,"duration_ms":119805,"concrete_test":"Independently re-derive the special case of [Cha24, Theorem 6.2] that Theorem 5.6 actually uses: for a regular unramified elliptic Howe-factorizable pair, prove directly from Definition 5.1 and the Grothendieck-Lefschetz fixed-point formula that <R^G_r_{T_r}(theta), R^G_r_{T_r}(theta)> = 1, and check which hypotheses are needed. If the computation requires extra conditions not stated in the present paper (e.g. split-genericity, a bound on p, or a particular choice of Borel), then the first sentence of Theorem 5.6 is unsupported as written.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The proof of Theorem 5.6 starts with 'By Theorem 5.2, either R^G_r_{T_r}(theta) or -R^G_r_{T_r}(theta) is an irreducible representation of G_{x,0}'. Theorem 5.2 is quoted verbatim from [Cha24, Theorem 6.2], a preprint that is not reproved here. If that scalar-product/Mackey formula fails, or carries an unstated hypothesis, then Theorem 3.2 only gives uniqueness, and there is no irreducible object to identify with the FKS construction. This is not merely a cosmetic dependency: Proposition 6.9 also uses Theorem 5.2 to cancel the leading term in the Green-function orthogonality, so Theorem 8.2, Theorem 8.3, and Corollary 8.4 inherit the same fragility. The large-q condition is not the soft spot: Section 10 tests it explicitly and shows it is near sharp. The genuinely load-bearing unverified input is the quoted [Cha24] theorem; for Section 9, [BC24, Theorem 10.9] is a second preprint input, but the central comparison already rests on [Cha24]. This is a verification gap, not an internal contradiction.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a positive-depth analogue of Deligne--Lusztig induction for unramified elliptic pairs (T, θ) in a p-adic reductive group, under a largeness assumption on the residue field size q. The central result is Theorem 5.6: for regular θ and q satisfying Henniart's inequality (∗), the Fintzen--Kaletha--Spice twisted Yu representation τ^{FKS}_Ψ is isomorphic to (−1)^{r(G0)−r(T)+r(T,θ)} R^G_r_{T_r}(θ), so that compact induction of the latter is the expected supercuspidal representation. This is obtained from a ``litmus test'' characterization theorem (Theorem 3.2) proved by a Cauchy--Schwarz estimate, together with character formulas for the algebraic side (Proposition 4.3) and for positive-depth Deligne--Lusztig induction (Proposition 5.5). The paper then defines Green functions for both the geometric and algebraic constructions, proves an orthogonality relation (Proposition 6.9), and uses it to extend the comparison from regular θ to arbitrary θ (Theorem 8.3, Corollary 8.4). A further application is a positive-depth Springer hypothesis in the 0-toral setting (Theorem 9.3) and a geometric derivation of a supercuspidal character formula. The final section presents a detailed small-q analysis for G2, showing that failures of the characterization are intimately tied to unipotent representations.","tokens_in":56972,"tokens_out":14377,"duration_ms":124028,"significance":"If the central comparison is correct, the paper gives a geometric realization of Kaletha's Howe-unramified regular supercuspidal L-packets, with the ε_ram twist appearing automatically rather than as an external correction. The characterization theorem (Theorem 3.2) is a new and potentially widely applicable tool, and the paper explicitly demonstrates its sharpness through the G2 small-q analysis. The Green-function orthogonality (Proposition 6.9) and the character formula for FKS--Yu virtual representations (Theorem 7.2) are independent structural contributions. The paper also contains concrete machine-verifiable character table data for G2(F3) and G2(F5), which is a commendable feature. However, the main comparison and the Section 9 applications rest on unrefereed preprint results ([Cha24, Theorem 6.2] and [BC24, Theorem 10.9]), and one central theorem (Theorem 8.2) is stated with a notational collision that makes it formally tautological. These issues are load-bearing rather than cosmetic, so the current version needs revision.","major_comments":[{"comment":"The proof of Theorem 5.6 begins with 'By Theorem 5.2, either R^G_r_T_r(θ) or −R^G_r_T_r(θ) is an irreducible representation of G_{x,0}', and Theorem 5.2 is quoted verbatim from [Cha24, Theorem 6.2], a preprint by the first author that is not reproved in this paper. This is load-bearing: it is the only input guaranteeing that ±R^G_r_T_r(θ) is irreducible, which Theorem 3.2 then needs to identify with the FKS construction. The same theorem is also used in the proof of Proposition 6.9 and hence in Theorem 8.2, Theorem 8.3, and Corollary 8.4. If [Cha24, Theorem 6.2] carries an unstated hypothesis or is not yet available in refereed form, the main comparison of the paper is unsupported. Please include a proof of the needed case (or of a statement sufficient for unramified elliptic Howe-factorizable pairs) or explicitly reformulate the main theorems as conditional on [Cha24, Theorem 6.2] and state its current status.","section":"Section 5.1, Theorem 5.2"},{"comment":"The statement 'QGr_Tr(θ+) = (−1)^{r(G0)−r(T)+r(T,θ)} · QGr_Tr(θ+)' is formally a tautology because the symbol QGr_Tr is used for the geometric Green function (Definition 6.4) and the FKS–Yu Green function (Definition 7.1) without distinction. As written, the theorem asserts X = c·X, which can only hold when c = 1, but the sign is generally nontrivial. The proof makes it clear that the intended assertion is an equality between the two different Green functions. Please introduce distinct notations, for example Q^{geom} and Q^{FKS}, and restate Theorem 8.2, Theorem 8.3, and Corollary 8.4 accordingly.","section":"Section 8, Theorem 8.2"},{"comment":"The proof asserts that θ′ := θ·φ^{-1}_{-1}·φ′_{-1} is a regular character whenever φ′_{-1} is a regular depth-zero character of T. This is not automatic: for a nontrivial w ∈ W_G(T), the condition (θ·χ)^w = θ·χ is equivalent to χ^w χ^{-1} = θ^w θ^{-1}, so the set of χ making θ·χ non-regular is a finite union of cosets of proper subgroups of the character group. The existence of a suitable φ′_{-1} requires an argument using the largeness of q beyond the assumptions already stated in Section 2.1. Please add a short counting argument or a reference.","section":"Section 8, proof of Theorem 8.2"},{"comment":"The proof of the positive-depth Springer hypothesis uses [BC24, Theorem 10.9] to identify the function associated with pInd^{G_r}_{T_r}(L_θ) with the character of R^G_r_T_r(θ). This is a second load-bearing dependency on an unpublished preprint. Since the results of Section 9 — in particular Corollary 9.9 — depend on this identification, please either prove the needed statement, provide a precise reference to a publicly available version with theorem numbers, or clearly mark the results of Section 9 as conditional on [BC24].","section":"Section 9, Theorem 9.3"}],"minor_comments":[{"comment":"The notation |R^G_{j,r}_{T_{j,r}}(jθ)| is used to denote the sign-adjusted irreducible component of a virtual representation; this notation is only defined implicitly in the proof of Theorem 5.6. Please define it before first use in Theorem 5.8.","section":"Section 5.3, Theorem 5.8"},{"comment":"There is a numerical typo: the dimension of R^G_T(θ) for G2(F5) is computed as (q−1)^2(q+1)^2(q^2+q+1) = 17856, but the text then refers to 'three irreducible representations whose dimension is 17586'. Please correct the typo and verify the character labels.","section":"Section 10.1.4"},{"comment":"Table 4 is difficult to read because the rows and columns are not visually aligned; the values for different conjugacy classes run together. Consider formatting the character table as a proper matrix with clear column separations.","section":"Section 10, Table 4"},{"comment":"The notation |R^G_r_{G'_r}(α ⊗ φ)| in Conjecture 5.12 is used before being defined; please clarify that it denotes the unique irreducible component up to sign of the virtual representation.","section":"Section 5.5, Conjecture 5.12"},{"comment":"The phrase 'for some sign constant c ∈ {±1}' should be clarified: c is allowed to depend on the representation π, and the conclusion of the theorem forces the two constants to coincide. Stating this explicitly would avoid confusion.","section":"Section 3, Theorem 3.2"}],"recommendation":"major_revision","confidential_remarks":"The paper is technically rich and the authors are to be commended for the detailed small-q analysis and for the effort to make the analytic characterization theorem self-contained. However, the central comparison theorem depends on [Cha24, Theorem 6.2], an unpublished preprint by the first author, and Section 9 depends on [BC24, Theorem 10.9]. Given the field's standards, I would advise the editor that final acceptance should be conditioned on either the inclusion of proofs of these external statements or their publication/acceptance in refereed venues. The notational collision in Theorem 8.2 is a separate but important issue that must be fixed before the paper can be evaluated cleanly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear X,\n\nThis is the real comparison: for large q, positive-depth Deligne–Lusztig induction R^G_{T_r}(θ) matches the FKS-twisted Yu construction on Howe-unramified elliptic pairs, with the ε_ram twist appearing automatically. Theorem 5.6 is the main event, and the route through Theorem 3.2 is a genuinely nice move: a short Cauchy–Schwarz estimate with an explicit largeness bound, then independently computed character formulas on very regular elements force the isomorphism. The Green-function sections 6–8 are new and do real work: they package character formulas so that arbitrary θ follows from regular θ exactly along Lusztig's Green-function strategy. Section 10's small-q analysis is a strength—it tests the largeness condition and finds counterexamples, all unipotent, which supports rather than undermines the setup.\n\nThe soft spot is exactly where the stress-test points. Theorem 5.6 begins by invoking [Cha24, Thm 6.2] to get irreducibility of R^G_{T_r}(θ). That is a preprint quoted verbatim, not reproved. The irreducibility is load-bearing: without it Theorem 3.2 has only uniqueness and there is no object to identify with the FKS side. The Green-function orthogonality (Prop 6.9) also uses the same formula, so Theorems 8.2, 8.3, and Corollary 8.4 inherit the condition. This is a verification gap, not a sign of error; the authors are explicit that the paper waited on [Cha24]. But a referee cannot certify the paper from this text alone. The second preprint input [BC24] only enters Section 9 and is less central.\n\nI do not share the worry about the large-q condition itself. Section 10 checks it carefully and shows it is near sharp; that is honest behavior. The 0-toral comparison had a weaker q-condition but stronger hypotheses; here the trade-off is stated plainly in Remark 5.7.\n\nBottom line: the paper deserves serious refereeing. The central claim is coherent, the internal proof of Theorem 3.2 is complete, and the Green-function structure is a real advance. The referee's main job is to verify the [Cha24] dependency or ask the authors to state the comparison as conditional on it. I would bring it to reading group and would cite it, with that caveat.","headline":"A serious comparison theorem for positive-depth Deligne–Lusztig induction, conditional on Chan's scalar-product preprint; it deserves refereeing, but the referee must verify that dependency.","tokens_in":57693,"tokens_out":1805,"would_cite":true,"duration_ms":19659,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22E50","20G40","11F70"],"pacs":[],"model":"deepseek-v4-flash","headline":"For sufficiently large residue fields, positive-depth Deligne–Lusztig induction is the geometric realization of the Howe-unramified regular supercuspidal L-packets, and its Green functions extend the match to all characters.","keywords":["p-adic reductive groups","positive-depth Deligne–Lusztig induction","Green functions","supercuspidal representations","L-packets","Howe factorization","parahoric subgroups","local Langlands correspondence"],"falsifier":"Check the scalar-product formula by computing $\\langle R^{G_r}_{T_r}(\\theta), R^{G_r}_{T'_r}(\\theta')\\rangle$ for an unramified elliptic Howe-factorizable pair and comparing it with the number of Weyl-group elements sending $\\theta$ to $\\theta'$; any mismatch invalidates the irreducibility step. Alternatively, in the excluded small-$q$ cases $G_2$ over $\\mathbb{F}_3$ or $\\mathbb{F}_5$ with the Coxeter torus, search for a non-unipotent irreducible representation whose values on all regular semisimple elements equal $\\pm\\Theta_{R^G_T(\\theta)}$ without being isomorphic to $\\pm R^G_T(\\theta)$, which would refute the characterization theorem outside the largeness range.","tokens_in":56495,"feed_emoji":"🧩","tokens_out":11974,"duration_ms":109469,"temperature":0.7,"pith_summary":"This paper aims to show that positive-depth Deligne–Lusztig induction is not merely a formal analogue of the algebraic construction of supercuspidal representations but a genuine geometric realization of it. For residue fields large enough, the compact induction of the geometrically defined virtual representation $R^{G_r}_{T_r}(\\theta)$ is exactly the irreducible supercuspidal representation obtained from the algebraic construction, with the sign twist $\\varepsilon_{\\mathrm{ram}}$ appearing automatically rather than inserted by hand. The paper then extends the match from characters with trivial Weyl stabilizer to arbitrary characters by proving that the Green functions of geometric and algebraic origin coincide. If the results are correct, one geometric machine produces the correct supercuspidal L-packets and simultaneously explains, through a positive-depth Springer hypothesis, why orbital integrals appear in supercuspidal character formulas.","feed_headline":"Geometric induction builds regular supercuspidal L-packets","feed_subtitle":"For large residue fields the geometric construction matches the algebraic one, sign twist included.","key_machinery":"The machinery consists of three objects working together. First is the positive-depth Deligne–Lusztig induction functor $R^{G_r}_{T_r}(\\theta)$, defined by the $\\theta$-isotypic cohomology of the variety $X_{T_r\\subset G_r}$; its character at an unramified very regular element is the sum $\\sum_{w\\in W_{G_{x,0}}(T_\\gamma,T)} \\theta^w(\\gamma)$. Second is the “litmus test” uniqueness theorem: under the largeness inequality $(*)$, at most one irreducible parahoric representation can have that character shape, a fact proved by bounding the non-very-regular contribution through Cauchy–Schwarz. Third are the positive-depth Green functions $Q^{G_r}_{T_r}(\\theta_+)$—character values at unipotent elements—whose orthogonality and comparison formulas carry the result from regular $\\theta$ to all $\\theta$ and, via Fourier transform of coadjoint-orbit delta functions, to the Springer hypothesis.","core_discovery":"On the paper's own terms, the central discovery is the isomorphism of $T G_{x,0}$-representations $\\tau^{\\mathrm{FKS}}_{\\Psi} \\cong (-1)^{r(G_0)-r(T)+r(T,\\theta)} R^{G_r}_{T_r}(\\theta)$ for every regular $\\theta$ attached to an unramified elliptic pair, under odd non-bad $p$, the stated divisibility conditions, and the largeness inequality $(*)$. Because the right-hand side is the geometric object and the left-hand side is the twisted algebraic object, compact induction yields $\\pi^{\\mathrm{FKS}}_{\\Psi}$ as the irreducible supercuspidal representation corresponding to the pair $(T,\\theta\\cdot\\varepsilon_{\\mathrm{ram}})$, so positive-depth Deligne–Lusztig induction realizes the Howe-unramified regular supercuspidal L-packet. The comparison for arbitrary $\\theta$ follows by matching the unipotent restrictions of the two families, giving $\\circ\\tau^{\\mathrm{FKS}}_{(T,\\theta)} \\cong (-1)^{r(G_0)-r(T)+r(T,\\theta)} R^{G_r}_{T_r}(\\theta)$ for all unramified elliptic pairs.","pith_inferences":["Because the uniqueness theorem needs no Howe factorization and no restriction on $p$, the geometric side of the comparison is available in settings where the algebraic construction does not yet exist; the first testable payoff would be a comparison for non-Howe-factorizable characters or for $p=2$ once such representations are constructed.","The paper's small-$q$ analysis shows that failures of the characterization in $G_2$ over $\\mathbb{F}_3$ and $\\mathbb{F}_5$ all come from unipotent representations; this suggests a general principle, made precise for depth zero in the paper, that non-unipotent representations are pinned by regular-semisimple character values even when the largeness inequality fails.","The orthogonality relations for positive-depth Green functions are a reusable structure: they should feed directly into endoscopic character identities for positive-depth supercuspidal representations, in the same way classical Green functions do for depth zero.","The 0-toral Springer hypothesis is a proof of concept; once the trace-of-Frobenius computation for character sheaves is available for non-0-toral $\\theta$, the same argument should prove the full positive-depth Springer hypothesis and rederive the general supercuspidal character formula geometrically."],"forward_implications":["For regular $\\theta$ and large $q$, $\\mathrm{c-Ind}\\big(R^{G_r}_{T_r}(\\theta)\\big)$ is the irreducible supercuspidal representation $\\pi^{\\mathrm{FKS}}_{\\Psi}$, so the geometric functor realizes the Howe-unramified regular supercuspidal L-packets; in particular the assignment is compatible with the local Langlands correspondence.","For arbitrary $\\theta$, the algebraic virtual representation $\\circ\\tau^{\\mathrm{FKS}}_{(T,\\theta)}$ and the geometric representation $(-1)^{r(G_0)-r(T)+r(T,\\theta)}R^{G_r}_{T_r}(\\theta)$ are isomorphic, so the regular case determines the whole Howe-unramified family.","Every Howe-unramified supercuspidal type appears in some $R^{G_r}_{T_r}(\\theta)$, and Howe-unramified Kim–Yu types occur in the cohomology of the positive-depth Deligne–Lusztig varieties.","For 0-toral Howe-unramified regular pairs, the positive-depth Green function is the Fourier transform of the delta function on a coadjoint orbit, giving a geometric proof of the orbital-integral character formula.","The same comparison shows that positive-depth Deligne–Lusztig induction preserves stability: it maps stable conjugacy classes of unramified elliptic regular pairs to stable distributions."],"supporting_citations":[{"why":"Supplies the scalar-product/Mackey formula used to prove that $R^{G_r}_{T_r}(\\theta)$ or its negative is irreducible, the step that lets Theorem 3.2 apply.","marker":"[Cha24, Theorem 6.2]"},{"why":"Constructs the regular supercuspidal representations and L-packets via Howe factorization; this is the object the geometric side is compared with.","marker":"[Kal19]"},{"why":"Defines the twisted Yu construction and its sign character, giving $\\pi^{\\mathrm{FKS}}_{\\Psi}$ and the stability and endoscopy statements used in Section 5.3.","marker":"[FKS23]"},{"why":"Earlier comparison in the 0-toral setting, including the character $\\varepsilon_{\\mathrm{ram}}$ and the very-regular character formula, which this paper generalizes.","marker":"[CO25]"},{"why":"Original algebraic construction of tame supercuspidal representations that the twisted version and the reparametrization build on.","marker":"[Yu01]"},{"why":"Defines the positive-depth Deligne–Lusztig varieties and induction functor for parahoric subgroups that the paper's geometric side uses.","marker":"[CI21]"},{"why":"Provides the classical Deligne–Lusztig character formula, Green functions, and orthogonality relations that Sections 6 and 9 adapt to positive depth.","marker":"[DL76]"},{"why":"Introduced the analytic Cauchy–Schwarz characterization argument whose largeness inequality (*) is the quantitative hypothesis of Theorem 3.2.","marker":"[Hen92]"},{"why":"Constructs positive-depth character sheaves on parahoric subgroups and the trace formula used to prove the positive-depth Springer hypothesis.","marker":"[BC24]"},{"why":"Provides the depth-zero harmonic-analysis method and orbital-integral character formula that Section 9 lifts to positive-depth 0-toral representations.","marker":"[DR09]"}],"fun_headline_variants":["Positive-depth DL induction realizes supercuspidal L-packets","Sign-twisted isomorphism realizes L-packets geometrically","Green functions match algebra and geometry in deep induction","Positive-depth Springer hypothesis proven geometrically","Geometric realization for regular unramified pairs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the quoted scalar-product/Mackey formula for positive-depth Deligne–Lusztig induction, taken from another preprint and not reproved in this paper; if that formula fails in an unexamined case, the irreducibility step on which the main comparison theorems rest collapses.","fun_headline_variants_meta":{"raw":{"variants":["Positive-depth DL induction realizes supercuspidal L-packets","Sign-twisted isomorphism realizes L-packets geometrically","Green functions match algebra and geometry in deep induction","Positive-depth Springer hypothesis proven geometrically","Geometric realization for regular unramified pairs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000817,"raw_usage":{"total_tokens":3586,"prompt_tokens":961,"completion_tokens":2625,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":577,"completion_tokens_details":{"reasoning_tokens":2552}},"tokens_in":577,"tokens_out":2625,"duration_ms":18047,"temperature":1.0,"reasoning_tokens":2552,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T10:44:35.764101+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check the scalar-product formula by computing $\\langle R^{G_r}_{T_r}(\\theta), R^{G_r}_{T'_r}(\\theta')\\rangle$ for an unramified elliptic Howe-factorizable pair and comparing it with the number of Weyl-group elements sending $\\theta$ to $\\theta'$; any mismatch invalidates the irreducibility step. Alternatively, in the excluded small-$q$ cases $G_2$ over $\\mathbb{F}_3$ or $\\mathbb{F}_5$ with the Coxeter torus, search for a non-unipotent irreducible representation whose values on all regular semisimple elements equal $\\pm\\Theta_{R^G_T(\\theta)}$ without being isomorphic to $\\pm R^G_T(\\theta)$, which would refute the characterization theorem outside the largeness range.","supporting_citations":[],"review_version":1}