{"id":"68a85b61-e4f4-4c7a-acb2-ce9b9a34a6eb","arxiv_id":"2505.17953","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In good characteristic, the stalks of intersection cohomology complexes on Drinfeld's compactifications and Zastava schemes are described by the q-analogue of Kostant's partition function, independently of the coefficient field.","lead":"This paper proves that the local intersection cohomology of Drinfeld's compactifications and Zastava spaces has the same stalk dimensions in good modular characteristic as in characteristic zero. This removes a characteristic-zero restriction that had been a barrier for positive-characteristic geometric Langlands constructions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Main theorem depends on [2, Thm 8.2] (cited as Thm 5.8), whose modular costalk computation is not proved here; a failure there would invalidate Cor 5.11.","rationale":"I read the proof chain with the reader's weakest assumption in mind and found the same load-bearing point. The internal arguments are coherent: Theorem 5.6 reduces all stalk polynomials to the P_nu, Lemma 5.3 reduces the P_nu to curve-independence, and Corollary 5.11 computes them from Theorem 5.8. I found no circularity: the divisibility of P^{nu_i} by q used in Proposition 5.15 comes from the earlier support-degree bound for perverse sheaves, not from Corollary 5.11. The only step where the characteristic-independent q-analogue of Kostant's partition function enters is the imported companion result, so that is the single most load-bearing assumption. The authors state this dependence openly, and the companion is a parameter-free derivation of a different object, which the rubric treats as real evidence. The concern therefore does not overturn the reader's ACCEPT verdict, but it does justify the stated medium confidence and makes an independent check of [2, Thm 8.2] the natural next step.","tokens_in":43467,"tokens_out":10748,"duration_ms":99704,"concrete_test":"Independently verify [2, Theorem 8.2(2)] in the smallest nontrivial case G=SL_2 over F_p with p odd: compute the costalk ranks of Ga^infty_2 on S_{-n alpha} directly from the definition (5.6) and the explicit affine-Grassmannian stratification for n=1,2,3, and compare the generating function with P(n alpha, q^2). Agreement in these cases would confirm the imported formula; disagreement would refute Corollary 5.11 and hence the main theorem. A weaker but useful check is to audit the proof of [2, Thm 8.2] to confirm that the only modular input is the Mirkovic-Vilonen conjecture as established by [4,29], with no additional hidden hypothesis on the coefficient field.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central formula P_mu = q^{-1} P(mu,q^2) is assembled in Corollary 5.11 from the ranks of (i_{-mu})_! Ga^infty_2 supplied by Theorem 5.8(2), which is part of the same authors' companion preprint [2, Theorem 8.2]. That theorem is not reproduced, proved, or machine-checked in this paper; its proof is said to use the Mirkovic-Vilonen conjecture, known only in good characteristic (Remark 1.2(1)). Since Theorem 5.6 reduces every stalk polynomial for Zastava schemes and Drinfeld compactifications to the polynomials P_nu, and Corollary 5.11 is the only step that injects the characteristic-independent Kostant partition function, the field-independence claim is hostage to the correctness of [2, Thm 8.2] in the modular setting. This is a structural dependency rather than an internal inconsistency: the authors flag it explicitly, the companion is a parameter-free derivation of a different object, and the characteristic-zero ancestor is Gaitsgory's [17]. Nonetheless, because the main theorem is specifically about arbitrary good characteristic, the modular version of [2, Thm 8.2] must be independently confirmed before the central claim is fully settled.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proves that for a connected reductive group G with simply connected derived subgroup, over a coefficient field k of good characteristic, the dimensions of the cohomology of the stalks of the intersection cohomology complexes on Zastava schemes and on Drinfeld's compactification Bun̄_B are independent of k, and are given by the same formulas involving the q-analogue of Kostant's partition function as in characteristic zero. The proof reduces the problem to the computation of the stalks of the Gaitsgory sheaf Ga^∞_2 on the affine Grassmannian: Theorem 5.6 expresses all relevant stalk polynomials in terms of the polynomials P_µ, and Corollary 5.11 identifies P_µ with q^{-1}P(µ,q^2) using the costalk computation imported from the authors' companion paper [2, Thm 8.2]. The paper also contains detailed proofs of constructibility and curve-independence statements that were previously only sketched, and an appendix extending the results to principal ideal domains.","tokens_in":43654,"tokens_out":14765,"duration_ms":115646,"significance":"If correct, this settles the modular analogue of Braverman–Finkelberg–Gaitsgory–Mirković's characteristic-zero description, giving a uniform combinatorial answer for all good characteristics. The paper's main contribution is the reduction of the Zastava/Drinfeld stalk computation to the semiinfinite sheaf formalism, together with a full proof of constructibility via factorization and reduction to P^1; these fill gaps in the literature. The main caveat is structural rather than internal: the characteristic-independent formula for the costalks of Ga^∞_2 is imported from the companion preprint [2, Theorem 8.2], whose modular proof relies on the Mirković–Vilonen conjecture (now known in good characteristic). The authors flag this dependency explicitly in Remark 1.2(1), and the present paper's logic is conditional on that companion result. No circularity is apparent.","major_comments":[{"comment":"The central formula P_µ = q^{-1}P(µ,q^2) relies on the modular costalk computation of [2, Thm 8.2], which is not proved in this manuscript. This is a load-bearing external dependency: if the companion's theorem were incorrect or its hypotheses violated, the main field-independence claim would not follow. However, the authors state this clearly (Remark 1.2(1)), and the companion concerns a different object (semiinfinite sheaves on the affine Grassmannian) whose computation is independent of the target results. I therefore view this as an acceptable structural dependency rather than an internal gap, but the editor should ensure that the companion is reviewed in tandem.","section":"§5.2, Theorem 5.8 and Corollary 5.11"}],"minor_comments":[{"comment":"The phrase 'On the other have we have' contains a typo; it should read 'On the other hand, we have'.","section":"§5.4"},{"comment":"The notation Y^{≻0} is used in Lemma 5.3 and Corollary 5.11 but is not explicitly defined; it should be defined as Y^{⪰0}\\setminus\\{0\\}.","section":"§2.4 and Lemma 5.3"},{"comment":"The name 'Be˘ılinson' should be written as 'Beilinson'.","section":"§4.2, proof of Proposition 4.2"},{"comment":"The appeal to Braden's hyperbolic localization theorem (via [17, Lemma 2.2.4]) would benefit from a remark that the statement holds with arbitrary coefficients, since the reference [17] is written in characteristic 0.","section":"§5.6"}],"recommendation":"minor_revision","confidential_remarks":"The paper is part of a two-paper project with the companion [2]. The central result hinges on [2, Thm 8.2]; I recommend that the editor obtain a report on [2] as well, or require the authors to include the statement/proof in a way that can be checked. The present manuscript's own contributions (constructibility, factorization reduction, π_! identity) are carefully argued and appear correct."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know before reading. First, this is a genuinely new result: the good-characteristic modular analogue of the Braverman–Finkelberg–Gaitsgory–Mirković computation for Zastava schemes and Drinfeld compactifications, with independence of the coefficient field as the headline. Second, the final answer P_mu = q^{-1} P(mu,q^2) is imported from the authors' companion preprint [2, Thm 8.2]; that computation is not reproduced here. So the main theorem is exactly as strong as that companion.\n\nWhat is actually new: the paper runs Gaitsgory's semiinfinite machinery in reverse. Instead of deriving the Gaitsgory sheaf's costalks from the known characteristic-zero stalks on Bun_B, it proves the reverse implication. Theorem 5.9 (pi_! IC_Bun_U ≅ Ga^∞_2) is proved here, in Section 5.6, via Proposition 5.15, which uses their own constructibility results and some A^1-specific arguments from Feigin–Finkelberg–Kuznetsov–Mirković. Sections 2–4 are a careful review, but they also supply missing details for statements that are \"known to experts\": Proposition 4.2 (Zastava is affine over C^mu × Bun_T), Lemma 4.7 (curve independence), and Proposition 4.8 (local triviality). The constructibility theorem (Theorem 5.6) is proved in detail, filling a gap the authors explicitly flag in [10] and [12].\n\nThe soft spot is one load-bearing external input: [2, Thm 8.2] computes the modular costalks of the Gaitsgory sheaf, and that theorem rests on the Mirković–Vilonen conjecture in good characteristic. The authors flag this in Remark 1.2(1), and the dependency is structural, not circular—[2] is a separate affine-Grassmannian computation and does not assume the present results. Still, a referee should read the companion in parallel. If that costalk computation failed for some good-characteristic case, Corollary 5.11 and the field-independence claim would collapse. The risk is moderate, not fatal: the characteristic-zero ancestor is Gaitsgory's [17], so the answer is expected.\n\nThis paper is for specialists in geometric representation theory, especially those working on modular perverse sheaves and geometric Langlands. It deserves a serious referee, ideally someone who knows both the semiinfinite sheaf machinery and MV cycles. I would accept it if the referee verifies the companion's Theorem 8.2. My own verdict: moderate-confidence accept, with the companion as part of the package.","headline":"Solid modular extension of BFGM; the stated stalk formula is conditional on the companion's costalk computation, but the proof route is new and the paper earns referee time.","tokens_in":44255,"tokens_out":3247,"would_cite":true,"duration_ms":27942,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F43","14D24","22E67"],"pacs":[],"model":"deepseek-v4-flash","headline":"For connected reductive groups, the stalk dimensions of intersection cohomology on Drinfeld's compactifications do not depend on the coefficient field, as long as its characteristic is good for G.","keywords":["intersection cohomology","Drinfeld compactification","Zastava scheme","Gaitsgory sheaf","affine Grassmannian","q-analogue of Kostant partition function","modular coefficients","geometric Langlands"],"falsifier":"For $G = \\mathrm{SL}_3$ and $\\mu$ the sum of the two simple coroots, compute the cohomology of the stalks of $\\mathrm{IC}_{Z^\\mu}$ along the closed stratum over a field of characteristic 2, which is good for $\\mathrm{SL}_3$, and compare the ranks with the coefficients of $q^{-1}P(\\mu,q^2)$. The formula fixes the ranks completely, so any deviation would disprove the central claim.","tokens_in":43206,"feed_emoji":"📐","tokens_out":10829,"duration_ms":84785,"temperature":0.7,"pith_summary":"This paper proves that the local intersection cohomology of Zastava schemes and of Drinfeld's compactifications of the stack of B-torsors on a curve is governed by a coefficient-field-independent formula, as long as the coefficient field has good characteristic for G, for example any characteristic other than 2, 3, or 5. The governing quantity is a Laurent polynomial attached to a positive coroot combination $\\mu$: the stalk polynomial $P_\\mu$ equals $q^{-1}$ times Lusztig's $q$-analogue of Kostant's partition function evaluated at $q^2$. Since this formula contains no trace of the field $k$, the characteristic-0 description of these stalks obtained earlier extends to every field of good characteristic, and even to principal ideal domains in which bad primes are invertible. This matters because Drinfeld's compactifications are central objects in geometric Langlands, and positive-characteristic coefficients are needed for many modular and arithmetic applications.","feed_headline":"Stalk sizes on Drinfeld spaces do not depend on coefficient field","feed_subtitle":"Good-characteristic coefficients give the same stalk dimensions as characteristic 0, via one formula with Kostant's partition function.","key_machinery":"The argument is carried by a single isomorphism: $\\pi_!\\mathrm{IC}_{\\mathrm{Bun}_U}[\\cdots] \\cong \\mathrm{Ga}^\\infty_2$, where $\\mathrm{Ga}^\\infty_2$ is the Gaitsgory sheaf, a semi-infinite intersection cohomology object in a perverse $t$-structure on $I^\\infty_2$-equivariant sheaves on the affine Grassmannian. This isomorphism converts the unknown stalks of $\\mathrm{IC}$ on Zastava schemes and Drinfeld compactifications into costalks of $\\mathrm{Ga}^\\infty_2$, which are accessible through the companion computation summarized in Theorem 5.8. The final formula is expressed through Lusztig's $q$-analogue of Kostant's partition function, and the passage from one closed stratum to all strata uses the factorization property of Zastava schemes.","core_discovery":"The paper's central claim is that for a connected reductive group $G$ with simply connected derived subgroup, a smooth projective curve $C$, and any coefficient field $k$ of good characteristic, the generating functions for the dimensions of the cohomology of the stalks of the intersection cohomology complexes on the Zastava schemes $Z^\\mu$ and on Drinfeld's compactification $\\overline{\\mathrm{Bun}}_B$ are identical to those computed previously in characteristic 0. In particular, for any strictly positive coroot combination $\\mu$, the stalk polynomial along the closed stratum satisfies $P_\\mu = q^{-1}P(\\mu,q^2)$, where $P(\\mu,q)$ is the $q$-analogue of Kostant's partition function, and from this the full local description follows by factorization. The authors prove this by reversing the usual direction of Gaitsgory's argument: instead of deriving the stalks of the Gaitsgory sheaf from the known stalks of $\\mathrm{IC}$ on $\\overline{\\mathrm{Bun}}_B$, they establish the isomorphism $\\pi_!\\mathrm{IC}_{\\mathrm{Bun}_U} \\cong \\mathrm{Ga}^\\infty_2$ and then read off the Zastava stalks from the independently computed costalks of the Gaitsgory sheaf on the affine Grassmannian.","pith_inferences":["One consequence the authors leave implicit is that constructions in geometric Langlands that cite characteristic-0 stalk descriptions can now be re-examined over fields of positive good characteristic with the same numerical input.","The restriction to good characteristic is probably not a feature of the Zastava geometry itself: the paper's dependence on the companion theorem suggests that bad-characteristic failures, if any, would first appear in the costalks of the Gaitsgory sheaf on the affine Grassmannian, not in the isomorphism $\\pi_!\\mathrm{IC}_{\\mathrm{Bun}_U} \\cong \\mathrm{Ga}^\\infty_2$.","The same strategy should extend to the parabolic analogues of Drinfeld's compactifications, since the paper explicitly anticipates a variation of its techniques for those spaces; verifying the analogue of $P_\\mu$ for parabolic Zastava spaces would be a direct test.","Because the formula $P_\\mu = q^{-1}P(\\mu,q^2)$ depends only on the root system, it yields a coefficient-field-independent table of stalk polynomials; comparing explicit modular computations for small groups, such as $\\mathrm{SL}_3$ in characteristic 2, against this table would catch any error in the chain of imported theorems."],"forward_implications":["For every stratum of the natural stratification, the cohomology sheaves of $\\mathrm{IC}$ on $\\overline{\\mathrm{Bun}}_B$ and on $Z^\\mu$ are locally constant, so these complexes are constructible with respect to those stratifications.","The dimensions of stalks and costalks of $\\mathrm{IC}$ on Drinfeld's compactifications and on Zastava schemes are independent of the coefficient field, as long as its characteristic is good for $G$.","With coefficients in a principal ideal domain in which all bad primes are invertible, the stalk and costalk cohomology is free over the PID and has the same ranks as in the field case.","The Zastava intersection cohomology complexes are even with respect to the natural stratification, while the $\\mathrm{IC}$ complexes on $\\overline{\\mathrm{Bun}}_B$ are even or odd according to the parity of $(g-1)\\dim(B)$.","The modular description gives the same root-system combinatorics as the characteristic-0 answer, so the $q$-analogue of Kostant's partition function governs the singularities in all good characteristics."],"supporting_citations":[{"why":"Companion preprint whose Theorem 5.8 computes the costalks of the Gaitsgory sheaf; this is the main input for Corollary 5.11.","marker":"[2]"},{"why":"Gaitsgory's characteristic-0 construction of the semi-infinite intersection cohomology sheaf and the strategy for the isomorphism with $\\pi_!\\mathrm{IC}_{\\mathrm{Bun}_U}$.","marker":"[17]"},{"why":"Establishes the characteristic-0 description of stalks of $\\mathrm{IC}$ on Drinfeld compactifications and provides factorization and stratification results used throughout.","marker":"[10]"},{"why":"Supplies the factorization property of Zastava schemes used to pass from closed strata to all strata.","marker":"[16]"},{"why":"Provides the $\\mathbb{P}^1$ constructions used in the proof of Proposition 5.15, reducing the key vanishing statement to $\\mathbb{A}^1$.","marker":"[15]"},{"why":"Defines the $q$-analogue of Kostant's partition function that enters the final formula.","marker":"[28]"},{"why":"States the Mirković–Vilonen conjecture on stalks of standard spherical perverse sheaves, on which the companion costalk computation depends.","marker":"[30]"},{"why":"One of the two proofs that the Mirković–Vilonen conjecture holds in good characteristic, justifying the restriction to good primes.","marker":"[4]"},{"why":"The other proof of the Mirković–Vilonen conjecture in good characteristic.","marker":"[29]"}],"fun_headline_variants":["Stalk sizes on Drinfeld spaces ignore coefficient field","Drinfeld stalk dimensions: same for any good characteristic","Good characteristic: Drinfeld stalks match characteristic 0","Field-independent stalk formulas for Drinfeld compactifications","Kostant's partition function unifies Drinfeld stalk sizes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central formula inherits its only serious input from the companion theorem on the costalks of the Gaitsgory sheaf, and that theorem rests on the Mirković–Vilonen conjecture, which is known only in good characteristic; if that companion theorem is wrong or its hypotheses are missed, the stalk formula for Zastava schemes collapses.","fun_headline_variants_meta":{"raw":{"variants":["Stalk sizes on Drinfeld spaces ignore coefficient field","Drinfeld stalk dimensions: same for any good characteristic","Good characteristic: Drinfeld stalks match characteristic 0","Field-independent stalk formulas for Drinfeld compactifications","Kostant's partition function unifies Drinfeld stalk sizes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000646,"raw_usage":{"total_tokens":2931,"prompt_tokens":874,"completion_tokens":2057,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":490,"completion_tokens_details":{"reasoning_tokens":1975}},"tokens_in":490,"tokens_out":2057,"duration_ms":11656,"temperature":1.0,"reasoning_tokens":1975,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:37:15.021022+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For $G = \\mathrm{SL}_3$ and $\\mu$ the sum of the two simple coroots, compute the cohomology of the stalks of $\\mathrm{IC}_{Z^\\mu}$ along the closed stratum over a field of characteristic 2, which is good for $\\mathrm{SL}_3$, and compare the ranks with the coefficients of $q^{-1}P(\\mu,q^2)$. The formula fixes the ranks completely, so any deviation would disprove the central claim.","supporting_citations":[{"cited_title":"Semiinfinite sheaves on affine flag varieties","cited_arxiv_id":"2503.18412","evidence_quote":"Companion preprint whose Theorem 5.8 computes the costalks of the Gaitsgory sheaf; this is the main input for Corollary 5.11."},{"cited_title":"Gaitsgory,The semi-infinite intersection cohomology sheaf, Adv","cited_arxiv_id":null,"evidence_quote":"Gaitsgory's characteristic-0 construction of the semi-infinite intersection cohomology sheaf and the strategy for the isomorphism with $\\pi_!\\mathrm{IC}_{\\mathrm{Bun}_U}$."},{"cited_title":"Braverman, M","cited_arxiv_id":null,"evidence_quote":"Establishes the characteristic-0 description of stalks of $\\mathrm{IC}$ on Drinfeld compactifications and provides factorization and stratification results used throughout."},{"cited_title":"Finkelberg and I","cited_arxiv_id":null,"evidence_quote":"Supplies the factorization property of Zastava schemes used to pass from closed strata to all strata."},{"cited_title":"Feigin, M","cited_arxiv_id":null,"evidence_quote":"Provides the $\\mathbb{P}^1$ constructions used in the proof of Proposition 5.15, reducing the key vanishing statement to $\\mathbb{A}^1$."},{"cited_title":"Lusztig,Singularities, character formulas, and aq-analogue of weight multiplicities, Anal- ysis and topology on singular spaces, II, III (Luminy, 1981), Ast´ erisque, no","cited_arxiv_id":null,"evidence_quote":"Defines the $q$-analogue of Kostant's partition function that enters the final formula."},{"cited_title":"Mirkovi´ c and K","cited_arxiv_id":null,"evidence_quote":"States the Mirković–Vilonen conjecture on stalks of standard spherical perverse sheaves, on which the companion costalk computation depends."},{"cited_title":"Achar and L","cited_arxiv_id":null,"evidence_quote":"One of the two proofs that the Mirković–Vilonen conjecture holds in good characteristic, justifying the restriction to good primes."},{"cited_title":"Mautner and S","cited_arxiv_id":null,"evidence_quote":"The other proof of the Mirković–Vilonen conjecture in good characteristic."}],"review_version":1}