{"id":"9c2d278a-99d0-45ab-9abd-92aa6713344b","arxiv_id":"2501.00652","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For elliptic Fargues-Scholze L-parameters, a non-zero integer combination of Harish-Chandra characters of packet members is stable under stable conjugacy.","lead":"This paper proves that the Fargues-Scholze L-packets attached to elliptic parameters satisfy the stability property predicted by the local Langlands correspondence. The proof uses a geometric construction and a new Fourier-analysis argument, and works in both characteristic 0 and positive characteristic.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof's assumption of an F-rational Borel containing T_g fails for elliptic tori in quasi-split groups like GL_2(Q_p); it is load-bearing because the weighted-sum formula and Fourier equidistribution depend on it.","rationale":"After reading the full text, I find that the reader's weakest assumption is indeed the most load-bearing gap. The proof's central machinery—Corollary 4.2.8, the definition of H_g and the surjectivity Λ → H_g (Lemma 3.2.2), and the Fourier equidistribution argument (Theorem 4.3.1)—all require a canonical comparison X_*(T_g) ≅ X_*(T_univ) with a distinguished notion of dominance, which is obtained only from an F-rational Borel containing T_g. Such a Borel does not exist for elliptic maximal tori in general: in a quasi-split group like GL_2(Q_p), any torus contained in an F-rational Borel is F-conjugate to a subtorus of a split maximal torus and is therefore split, so an anisotropic torus can never occur; in a non-quasi-split group such as D^×, no F-rational Borel exists. Thus the assumption is not a minor technicality: it excludes the very class of elements (elliptic elements, when not in a torus) for which the theorem is interesting. I checked the surrounding text: the assumption is repeated verbatim in Section 1.1 (step 2), Section 3.2, Section 4.2, and Section 4.3, and no justification or alternative is supplied. I also considered other potential issues—e.g., the nonvanishing proof in Lemma 4.3.5, the degree-shift argument, and the use of limits of exact identities—and found none as severe. The nonvanishing argument (Lemma 4.3.5) is sound: if a nonzero degree shift appeared, the boundedness of powers of the regular representation would give a contradiction. The passage from exact identities to limits is justifiable by finiteness of the stable conjugacy class. However, those steps are downstream of the Borel-dependent comparison, so fixing the Borel issue is necessary. A plausible repair is to work over an algebraic closure, choose an \\bar F-Borel, and then show the limiting coefficients are Galois-invariant and independent of choices; the concrete test I propose checks exactly this robustness in the simplest nontrivial case (GL_2 and the unramified anisotropic torus). Since the paper's conclusion is likely true by independent means in that case, I do not regard the gap as disproving the theorem, but it does invalidate the proof as written for most elliptic elements. Hence the CONDITIONAL verdict is appropriate.","tokens_in":22386,"tokens_out":15896,"duration_ms":164511,"concrete_test":"Test robustness of the equidistribution step by computing Theorem 4.3.1 for G = GL_2(Q_p) and T_g the unramified anisotropic torus (so H_g ≅ Z/2), using each of the two non-Γ-equivariant identifications of Λ ≅ X_*((T_g)_sc) that differ by the nontrivial Galois element. Verify that lim_{m→∞}(S_{0,m} - S_{1,m}) = 0 holds and gives the same value under both identifications; if not, the weighted-sum formula (Corollary 4.2.8) is not well-defined and the proof fails for this case.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof's core assertions in Section 1.1 (Step 2), Section 3.2, Section 4.2, and Section 4.3 require, for every elliptic regular semisimple g ∈ G(F) with T_g = Cent(g,G), the existence of a Borel subgroup defined over F that contains T_g. This assumption is false in general. If G is quasi-split, any torus contained in an F-rational Borel is F-conjugate to a subtorus of a split maximal torus and hence split; an anisotropic torus, which is exactly the centralizer of an elliptic element (e.g., the unramified quadratic torus in GL_2(Q_p)), can never satisfy this. If G is not quasi-split (e.g., the unit group of a quaternion division algebra), no F-rational Borel exists at all. The subsequent steps all rely on this choice: the identification X_*(T_g) ≅ X_*(T_univ) with a distinguished dominance order, the weight multiplicities dim V_{µ_m}[λ] for λ ∈ X_*(T_g), the definition of H_g and the surjection Λ → H_g (Lemma 3.2.2), and ultimately the weighted-sum formula Corollary 4.2.8. Without an F-rational Borel, inv(g,g') ∈ B(T_g) and the weight λ ∈ X_*(T_univ) cannot be canonically compared, so the Fourier equidistribution Theorem 4.3.1 may not have a well-defined input. The text supplies no alternative construction for groups or elements where the assumption fails.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves stability of a virtual Harish-Chandra character attached to an elliptic Fargues-Scholze L-parameter. For an elliptic L-parameter ϕ and an irreducible smooth representation π of G(F) with Fargues-Scholze parameter ϕ, the author defines π0 := O(S_ϕ / Z(Ĝ)^Γ) * π via the spectral action, shows that the corresponding sheaf on Bun_G is a Hecke eigensheaf, and uses the Hansen-Kaletha-Weinstein character formula to express Θ_{π0}(g) as a weighted sum over the stable conjugacy class of an elliptic element g. The proof then reduces stability to an equidistribution statement for weight multiplicities of V_{μ_m}, proved by Fourier analysis on the finite abelian group H_g. The main theorem asserts that Θ_{π0} is stable on G(F)_ell and, in characteristic zero, is a non-zero stable distribution.","tokens_in":22701,"tokens_out":17524,"duration_ms":185742,"significance":"If correct, this provides a new, endoscopy-independent proof of stability for Fargues-Scholze L-packets, with the advertised advantage of working in positive characteristic. The argument is genuinely constructive and has no fitted parameters: the Hecke eigensheaf property is proved in Proposition 4.1.2, the equidistribution statement is proved in Theorem 4.3.1, and the use of the spectral action and HKW22 as external benchmarks avoids circularity. The main defect is a repeated false geometric assumption about the existence of an F-rational Borel subgroup containing an elliptic maximal torus; this affects the formulation of the weighted-sum formula and the equidistribution theorem, and therefore the proof of the main theorem as written.","major_comments":[{"comment":"The proof repeatedly assumes that for every elliptic g ∈ G(F)_ell there exists a Borel subgroup defined over F containing T_g = Cent(g,G). This assumption is false in general. For example, in G=GL_2 over a non-archimedean local field, an element whose centralizer is the unramified quadratic torus is elliptic, but an F-rational Borel subgroup contains only split maximal tori. Consequently, the identification X_*(T_g) ≅ X_*(T_univ) with a distinguished dominance order, the definition of H_g, and the reindexing λ=inv(g,g') in Corollary 4.2.8 are not justified for such elements. Since the weighted-sum formula (19) and the quantities S_{h,m} in Theorem 4.3.1 are the inputs to the stability proof, Theorem 4.3.3 is not proven for elliptic elements with anisotropic centralizer as written. The text supplies no alternative construction for groups or elements where the assumption fails.","section":"§1.1 Step 2; §4.2 before Corollary 4.2.8; §4.3 before Theorem 4.3.1"},{"comment":"The surjection Λ^Φ → H_g used in Proposition 3.3.1(2) is constructed via a non-canonical isomorphism X_*((T_g)_sc) ≅ Λ^Φ that is explicitly noted not to be Γ-equivariant. The paper does not prove that the resulting character χ of Λ^Φ, and hence the existence of β with χ(β) ≠ 1, is independent of this choice. While the conclusion is likely true because any two choices differ by a Weyl-group element and the weight multiplicities of V_{μ_m} are W-invariant, this independence is not stated or proved; as written, the growth estimate depends on choices that are not shown to be canonical.","section":"§3.2, Lemma 3.2.2"}],"minor_comments":[{"comment":"Several typographical errors should be fixed: \"a prior\" should be \"a priori\" (e.g., in §1.2 and Remark 4.3.4), \"Combing Corollary 4.2.8\" should be \"Combining\", and \"charater\" in Lemma 4.3.5 should be \"character\".","section":"Throughout"},{"comment":"The line \"π0 = i_1^*F0 F0 ≅ i_1!π0\" appears garbled; it should read \"π0 := i_1^*F0, and F0 ≅ i_1!π0\".","section":"Introduction, after Equation (9)"},{"comment":"The phrase \"We choose a Borel subgroup over F containing (T_g)_F\" is not just notationally strong but mathematically impossible for anisotropic elliptic tori; as noted in the major comments, this needs to be replaced by an admissible embedding or by a Borel over an algebraic closure with an explicit independence statement.","section":"§1.1, Step 2"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is plausible and the geometric strategy is elegant, but the repeated false assertion about F-rational Borels is load-bearing. I recommend major revision rather than rejection, since the gap seems fixable by replacing the F-rational Borel with a Borel over an algebraic closure and proving the relevant weighted sums are independent of the choice, or by using standard admissible-embedding formalism from endoscopy. The author should be asked to rewrite Sections 4.2–4.3 accordingly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Colleague,\n\nThe stress-test note holds up. This paper has a genuine gap in a load-bearing place; everything around it is as strong as the reader says. Fu proves that every irreducible π whose Fargues–Scholze L-parameter is elliptic lies in a finite set of representations with the same FS parameter whose Harish-Chandra characters admit a non-zero integer combination stable under G(F)-conjugacy on elliptic regular semisimple elements. That is the first stability result for FS L-packets, it is endoscopy-free, uniform in G, and works in positive characteristic. If the gap is fixed, this is an important paper.\n\nCredit where due. The Hecke eigensheaf observation (Prop 4.1.2) — that (i_φ)_* O(S_φ/Z(Ĝ)^Γ) ∗ (i_1)!π is a Hecke eigensheaf — is clean and will be reused. The reduction of stability to equidistribution of weight multiplicities via HKW22 is a genuinely new route, and the Fourier analysis on the finite group H_g (Prop 3.3.1, Thm 4.3.1) is elegant and self-contained. The paper is honest about its limits: Remark 2.4.12 says the constructed set may be a proper sub-packet, Remark 4.3.4 says existence of π with a given parameter is not known in general, Section 1.2 says compatibility with classical LLC is open. No circularity; the spectral action and HKW22 are external and do not include the target theorem. The char-0 distribution statement is derived from Arthur, standard.\n\nNow the soft spot, and it is real. Sections 1.1, 4.2 and 4.3 choose an F-rational Borel containing the elliptic maximal torus T_g. For the unramified quadratic torus in GL_2(Q_p) — the centralizer of an elliptic element — no such Borel exists, and for non-quasi-split G none exists at all. The identification X_*(T_g) ≅ X_*(T_univ), the comparison of inv(g,g') with weights λ, and the weighted-sum formula Cor 4.2.8 all depend on that choice. The proof as written does not reach basic anisotropic cases. It looks repairable — work over F-bar with a fixed Borel and track the non-equivariant identification, or use the canonical identification of X_*((T_g)_sc) with the coroot lattice — but the repair is not written down, and the Γ-twist may change the equidistribution input.\n\nOne framing note: the title says stability of FS L-packets, while the theorem produces a stable sub-packet containing each π. The abstract is precise about this; just do not over-read the title.\n\nThis is for people working in geometric local Langlands and p-adic representation theory; the equidistribution section is worth reading on its own. The paper deserves a serious referee: the ideas are original, the theorem is significant, and the gap, while central, is plausibly fixable. Send it out, and ask the referee to check the Borel assumption on the GL_2 elliptic case first.\n\nBest,\n[You]","headline":"Strong, original proof of stability for elliptic FS L-packets, endoscopy-free and uniform in G, but the standing assumption of an F-rational Borel containing T_g fails for anisotropic tori and is load-bearing.","tokens_in":23256,"tokens_out":15767,"would_cite":true,"duration_ms":148981,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22E50","11R39","14D24"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that every irreducible representation attached to an elliptic Fargues–Scholze L-parameter yields a nonzero stable combination of Harish-Chandra characters, establishing stability of elliptic L-packets by geometric methods.","keywords":["Fargues-Scholze L-packets","elliptic L-parameters","stable Harish-Chandra characters","local Langlands correspondence","geometrization","spectral action","weight multiplicities","p-adic reductive groups"],"falsifier":"For $G = \\mathrm{GL}_2(\\mathbb{Q}_p)$, take an elliptic regular element $g$ whose centralizer is a quadratic-field torus. Compute the difference $S_{h,m} - S_{h',m}$ for the two classes $h,h'$ in $H_g$: the proof requires this to tend to $0$, so a nonzero limit would falsify the equidistribution step. Separately, in a case where the Fargues–Scholze packet is known, compare $\\Theta_{\\pi_0}$ on two stably conjugate elliptic elements; unequal values would refute the stability conclusion.","tokens_in":22147,"feed_emoji":"📐","tokens_out":12563,"duration_ms":111113,"temperature":0.7,"pith_summary":"Stability of $L$-packets is a central prediction of the local Langlands correspondence: the Harish-Chandra characters of members of a packet should admit a linear combination invariant under stable conjugacy. This paper proves that prediction for every elliptic $L$-parameter, using the Fargues–Scholze geometrization of representations as sheaves on the stack of $G$-bundles over the Fargues–Fontaine curve. For each irreducible $\\pi$ with Fargues–Scholze $L$-parameter $\\varphi$, the paper constructs a finite virtual representation $\\pi_0$ by acting on $\\pi$ with the regular representation of the centralizer $S_\\varphi$ modulo the fixed center, and shows that the alternating sum of its Harish-Chandra characters is stable on elliptic regular semisimple elements. In characteristic zero the same object is a nonzero stable distribution on $G(F)$. The interest is a new route to stability that does not use endoscopic classification and works in positive characteristic.","feed_headline":"Elliptic L-packets proven stable via sheaf geometry","feed_subtitle":"A sheaf-theoretic construction gives a stable character combination for every elliptic Fargues-Scholze L-packet.","key_machinery":"The load-bearing construction is the spectral action of Fargues–Scholze: perfect complexes on the stack of $L$-parameters act on the derived category of sheaves on $\\mathrm{Bun}_G$, and the averaged object $(i_\\varphi)_* O(S_\\varphi/Z(\\widehat G)^\\Gamma) * (i_1)_! \\pi$ is shown to be a Hecke eigensheaf, so $T_{V_\\mu}$ multiplies it by $\\dim V_\\mu$. The Hansen–Kaletha–Weinstein formula rewrites the resulting character identity as a weighted sum over stable conjugacy classes, with weights $\\dim V_\\mu[\\lambda]/\\dim V_\\mu$ indexed by an invariant in the finite abelian group $H_g = \\ker(X_*(T_g)_\\Gamma \\to \\pi_1(G)_\\Gamma)$. Choosing $\\mu = 4m\\rho_G$ and applying the Weyl character formula, the paper proves that these weight multiplicities equidistribute over $H_g$ as $m \\to \\infty$; Fourier analysis on $H_g$ then makes the weighted sum independent of the class of $g'$, which is exactly stability.","core_discovery":"Let $G$ be a connected reductive group over a non-archimedean local field $F$, and let $\\varphi: W_F \\to \\widehat G(\\overline{\\mathbb Q}_\\ell)$ be an elliptic $L$-parameter. For every irreducible smooth representation $\\pi$ of $G(F)$ whose Fargues–Scholze $L$-parameter is $\\varphi$, the paper defines $F_0 = (i_\\varphi)_* O(S_\\varphi/Z(\\widehat G)^\\Gamma) * (i_1)_! \\pi$ and $\\pi_0 = i_1^* F_0$. It proves that $\\pi_0$ is a finite direct sum of irreducible representations (up to degree shifts) containing $\\pi$, and that the Harish-Chandra character $\\Theta_{\\pi_0}$ is a nonzero function on the elliptic regular semisimple locus $G(F)_{\\mathrm{ell}}$ invariant under $G(\\overline F)$-conjugacy. In characteristic zero $\\Theta_{\\pi_0}$ is a nonzero stable distribution on all of $G(F)$. This establishes the stability of the Fargues–Scholze $L$-packet $\\Pi^{\\mathrm{FS}}_\\varphi(G)$ in the sense required by the stability conjecture, without invoking the theory of endoscopy.","pith_inferences":["If the Fargues–Scholze packet is nonempty for every elliptic parameter, the same construction would prove the stability conjecture for all elliptic discrete-series packets; the paper leaves nonemptiness open.","The regular representation $O(S_\\varphi/Z(\\widehat G)^\\Gamma)$ may be the correct canonical packet average; in cases where a classical packet is known, this stable combination should agree with the classical stable packet character.","A testable extension is to compute $S_{h,m}$ explicitly for a small-rank split group and a non-split elliptic torus, to measure how quickly the equidistribution limit is approached."],"forward_implications":["Every elliptic Fargues–Scholze packet, whenever nonempty, carries a nonzero stable character combination, so the stability part of the local Langlands stability conjecture holds for these packets.","The stable combination is canonically built from any member: $\\Theta_{\\pi_0}$ for $\\pi_0 = O(S_\\varphi/Z(\\widehat G)^\\Gamma) * \\pi$, with coefficients coming from an equal-weight limit of weight multiplicities.","The method is independent of endoscopic classification and covers positive characteristic, where full endoscopy is not available.","The same weighted-sum identity transfers character values between extended pure inner forms, up to the sign $(-1)^{\\langle \\mu, 2\\rho_G\\rangle}$.","The equidistribution of weight multiplicities (Theorem 4.3.2) is a separate, self-contained result about highest-weight representations of reductive groups."],"supporting_citations":[{"why":"Supplies the spectral action, the Fargues-Scholze L-parameter, the stack Bun_G, and the Hecke operators used throughout.","marker":"[FS21]"},{"why":"Supplies the weighted-sum formula (Theorem 6.5.2) that writes Harish-Chandra characters of Hecke operators as sums of characters over stable conjugacy classes.","marker":"[HKW22]"},{"why":"Supplies the isomorphism between B(T) and X_*(T)_Gamma and the description of basic elements used to define inv and the group H_g.","marker":"[Kot85]"},{"why":"Supplies the Weyl character formula and the product formula for characters of highest-weight representations used to bound chi(Char V_mu).","marker":"[FH91]"},{"why":"Supplies compatibility of Hecke operators with pi_1(G)^Gamma-gradings, needed to prove the averaged sheaf is a Hecke eigensheaf for mu_m.","marker":"[Zou22]"},{"why":"Supplies the result that stability on elliptic regular semisimple elements implies stability as a distribution for elliptic virtual characters.","marker":"[Art96]"},{"why":"Supplies the fact that supercuspidals with the same central character have no nontrivial extensions, used to decompose pi_0 as a direct sum of irreducibles.","marker":"[Cas95]"},{"why":"Supplies the algebraic fundamental group pi_1(G) used to define H_g and to identify coinvariant lattices.","marker":"[Bor98]"}],"fun_headline_variants":["Every elliptic L-packet admits a stable character sum","Stability proved for all elliptic Fargues-Scholze L-packets","Sheaf geometry stabilizes elliptic L-packets","Elliptic L-packets: stable characters without endoscopy","From sheaves to stable distributions on elliptic L-packets"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that for every elliptic regular element $g$ the centralizer torus $T_g$ has an $F$-rational Borel subgroup containing it; for non-split elliptic tori this can fail, and without a chosen Borel the weight multiplicities cannot be canonically compared across the stable conjugacy class.","fun_headline_variants_meta":{"raw":{"variants":["Every elliptic L-packet admits a stable character sum","Stability proved for all elliptic Fargues-Scholze L-packets","Sheaf geometry stabilizes elliptic L-packets","Elliptic L-packets: stable characters without endoscopy","From sheaves to stable distributions on elliptic L-packets"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0011,"raw_usage":{"total_tokens":4628,"prompt_tokens":1019,"completion_tokens":3609,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":635,"completion_tokens_details":{"reasoning_tokens":3527}},"tokens_in":635,"tokens_out":3609,"duration_ms":58168,"temperature":1.0,"reasoning_tokens":3527,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:49:39.685497+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For $G = \\mathrm{GL}_2(\\mathbb{Q}_p)$, take an elliptic regular element $g$ whose centralizer is a quadratic-field torus. Compute the difference $S_{h,m} - S_{h',m}$ for the two classes $h,h'$ in $H_g$: the proof requires this to tend to $0$, so a nonzero limit would falsify the equidistribution step. Separately, in a case where the Fargues–Scholze packet is known, compare $\\Theta_{\\pi_0}$ on two stably conjugate elliptic elements; unequal values would refute the stability conclusion.","supporting_citations":[],"review_version":1}