{"id":"3aafdf25-91e8-4c80-ace8-c36a93a7334b","arxiv_id":"2506.19390","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The tau-coinvariants of affine Grothendieck-Springer sheaves on the regular-semisimple bounded locus of a loop group are perverse.","lead":"This paper proves a structural property in affine Springer theory: after quotienting the relevant sheaves by the symmetry group, they remain perverse. The result gives the first concrete building blocks for the expected theory of affine character sheaves in p-adic representation theory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Perversity proof rests on unproved [BV, Theorem 2.3.4] via Claim 3.2.4; no internal gap found, but this external dependence is the load-bearing risk.","rationale":"The reader identified the same weakest assumption: Claim 3.2.4 rests on [BV, Theorem 2.3.4]. My reading of the proof chain confirms this: Theorem 3.2.1 reduces to Proposition 3.2.3, whose Step 5 uses Claim 3.2.4, and that claim is justified only by citing [BV, Theorem 2.3.4] from a co-author's preprint. I found no independent internal error: the reduction to (0.1), the use of Proposition 1.4.3, and the characteristic assumptions elsewhere are consistent with the stated framework. The concern is therefore a dependency on a non-reproved external theorem, not a discovered flaw in the argument. Consequently I keep the reader's ACCEPT verdict with the same moderate confidence, and propose a direct verification of [BV, Theorem 2.3.4] as the concrete test that would settle whether the concern actually lands.","tokens_in":33507,"tokens_out":23400,"duration_ms":255086,"concrete_test":"Read [BV, Theorem 2.3.4] and [BV, Claim 5.3.4] and check: (1) the theorem is stated and proved under the same characteristic and reductive-group hypotheses as this paper's Section 2.1.2(f), namely char(k)=0 or >2h and Gder simply connected; (2) the implication from [BV, Claim 5.3.4] to Claim 3.2.4 holds verbatim for arbitrary local systems L on T and non-split maximal tori Gγ. If either fails, specify the missing hypothesis and re-run Theorem 3.2.1 under that restricted hypothesis; if both hold, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 3.2.1 is reduced to inclusion (0.1), i.e., τ⊗^L_{Qℓ[Λ]} RΓ_c(Flγ,ωL) ∈ D^{≥−d_r} for γ∈C_{w,r}(k). The only step in Proposition 3.2.3 that is not proved in the paper is Claim 3.2.4, which asserts that the quasi-coherent sheaf K ∈ D^b(QCoh(X×Y)) is set-theoretically supported on the union of graphs of admissible embeddings η_{φ,λ}. Its proof is a single sentence: 'As it was explained in the proof of [BV, Claim 5.3.4], the assertion follows from [BV, Theorem 2.3.4]'. Here [BV] is arXiv:2104.13123, a preprint by one of the current authors, and its Theorem 2.3.4 is the 'group version' of Yun's compatibility theorem. The paper neither states [BV, Theorem 2.3.4] precisely nor verifies that its hypotheses (e.g., characteristic zero or >2h, simply connected derived group, splitting of Gγ) match the current setting. If [BV, Theorem 2.3.4] were false, or if it were only proved in characteristic zero or for split tori, then inclusion (0.1) would not follow, and S_{L,•,τ} might fail to be pν-perverse. The same theorem also underlies Corollary 3.1.9 via [BV, Proposition 3.3.2] (see Remark 3.1.10), so the external input is used twice. This is not an internal inconsistency, but it is the least secure link in an otherwise carefully structured proof.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops the group analog of the authors' earlier affine Springer theory for Lie algebras. Under the assumptions that Gder is simply connected and that the characteristic of k is zero or greater than 2h, it constructs a perverse t-structure on D([C•/LG]) for the regular-semisimple bounded loop-group locus, proves that the affine Grothendieck–Springer sheaf S_{L,•} is pν-perverse and fW-constructible, and proves the main theorem that the derived τ-coinvariant sheaves S_{L,•,τ} are pν-perverse for every local system L on T and every representation τ of Λ (or of fW when L is W-equivariant). The proof proceeds by proving smallness of the affine Grothendieck–Springer fibration, establishing perversity of S_{L,•} via a general small-pushforward theorem, and then reducing the perversity of coinvariants to the cohomological inclusion (0.1) for RΓ_c(Flγ,ωL)_τ. That inclusion is proved in Proposition 3.2.3 via a quasi-coherent support argument, with the key support statement delegated to Claim 3.2.4 and hence to [BV, Theorem 2.3.4]. A Lie algebra analog is also given.","tokens_in":33811,"tokens_out":6782,"duration_ms":69850,"significance":"If the main theorem is correct, these perverse sheaves would be the first concrete examples of affine character sheaves, and Section 0.8 explains how they imply expected stability properties of L-packets for cuspidal Deligne–Lusztig representations. The paper's strengths are its systematic reduction of the main theorem to a single cohomological inclusion, the construction of the perverse t-structure and fW-action on [C•/LG], the proof of smallness of the affine Grothendieck–Springer fibration, and the explicit identification of the coinvariant computation with a statement about quasi-coherent sheaves. The central concern is that the decisive final input, Claim 3.2.4, is not proved in the manuscript and depends on an unstated theorem from an unpublished preprint by one of the authors. The argument is internally coherent assuming that external theorem; the risk is external, not an internal contradiction.","major_comments":[{"comment":"The proof of Claim 3.2.4 is a single sentence: \"As it was explained in the proof of [BV, Claim 5.3.4], the assertion follows from [BV, Theorem 2.3.4].\" This claim is load-bearing: it is the only non-formal step in Proposition 3.2.3, and Proposition 3.2.3 is exactly what establishes inclusion (0.1), which in turn proves Theorem 3.2.1 and Corollary 3.2.5. The manuscript neither states [BV, Theorem 2.3.4] nor verifies that its hypotheses hold in the present setting (e.g., the characteristic assumption, the simple-connectedness of Gder, and the status of Gγ as a maximal torus over F with the corresponding admissible isomorphisms). If [BV, Theorem 2.3.4] were false, or were only proved under stronger hypotheses than those of Section 2.1.2(f), the inclusion (0.1) would not follow from the given argument. The authors should state the theorem precisely, verify its hypotheses step by step, and either prove it or give a publicly available proof that is self-contained enough for the journal's referees to check.","section":"§3.1 (Corollary 3.1.9 and Remark 3.1.10)"},{"comment":"The same external group version of Yun's theorem is used a second time, through [BV, Proposition 3.3.2], to prove Corollary 3.1.9, which is then used in Theorem 3.1.11 to prove Λ-constructibility of S_{L,•}; Corollary 3.1.13, in turn, is needed in the proof of Theorem 3.2.1. Thus the external dependence is not confined to Claim 3.2.4: both the constructibility input and the perversity-of-coinvariants input rest on the same unstated [BV, Theorem 2.3.4]. Remark 3.1.10 itself acknowledges that the proof of [BV, Proposition 3.3.2] is global and that a different local proof is promised in [BeKV2]. The authors should at minimum state [BV, Theorem 2.3.4] and its proof status, and should clarify which parts of the current paper would survive if that theorem were replaced by the Lie-algebra version [Yun2, Theorem 2].","section":"§3.1, Corollary 3.1.9 and Remark 3.1.10"}],"minor_comments":[{"comment":"The symbol Λγ is used for two different objects: in Step 2 it denotes the subgroup X_*(Gγ)^Γ_F of eΛγ, and in Step 4 it is reused for the finite quotient eΛγ/Λγ. This makes Claim 3.2.4 and the direct-sum decomposition in its proof hard to parse; I recommend renaming the quotient, for example Ωγ.","section":"§3.2, Steps 2 and 4"},{"comment":"Inclusion (0.1) in the introduction is stated with an integer dγ, while the proof in Proposition 3.2.3 uses d_r, with r denoting the GKM pair (w,r). The intended normalization should be stated explicitly so that the reduction from ν_{w,r}=d_r+a_{w,r} to the bound −d_r is transparent.","section":"§0.6(d) and §3.2"},{"comment":"The proof of Proposition 2.2.5 refers to [GKM, Lemma 8.2.1] after replacing the identity [GKM, (2.3.1)] by its group version [Hu, Section 4.23]. Since this codimension formula is essential for the smallness theorem, a few more details about how the group version replaces the Lie-algebra identity would help the reader.","section":"§2.2, Proposition 2.2.5"},{"comment":"The dependence on [BV] is central, but the reference gives only the arXiv number and no version or date of the preprint. Since the cited theorem may change between versions, the authors should identify the precise version they rely on and, ideally, quote the theorem statement in the text.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is plausibly correct, and the paper is carefully structured, but the decisive input is outsourced to an unstated theorem in the authors' own preprint [BV]. I recommend requiring the authors to state the theorem and either prove it or provide a precise, verifiable reference, and to verify its hypotheses under the characteristic and simple-connectedness assumptions of the paper. This is a fixable defect rather than a reason to reject, but it is load-bearing enough that the present version should not be accepted as is."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main result is real and new: for the group case, and even for the Lie algebra, the perversity of tau-coinvariants (part (d)) was not known before, and the paper gives the first concrete family of objects expected to be affine character sheaves. The proof is mostly as solid as the abstract suggests. The new technical ingredients—flatness of the Chevalley map for arcs (Theorem 2.1.3), the short proof of the GKM codimension formula (Proposition 2.2.5), smallness of the affine Grothendieck–Springer fibration, and the quasi-coherent sheaf lemma Proposition 1.4.3—are clearly laid out and the reductions are easy to follow. The authors also honestly flag that the Lie algebra analog of (d) is new, and they fix a missing characteristic hypothesis from their own earlier [BKV].\n\nWhere I get nervous is exactly where the stress-test put the finger: Claim 3.2.4. The proof of Proposition 3.2.3, hence Theorem 3.2.1, reduces to the assertion that the sheaf K is set-theoretically supported on those graphs, and that assertion is dispatched in one sentence citing [BV, Theorem 2.3.4], the group version of Yun's compatibility theorem. The theorem is not stated, its hypotheses are not checked against the current setup, and [BV] is a preprint by one of the authors. The same external theorem is used again through [BV, Proposition 3.3.2] to get finite generation of the Λ-action on cohomology (Corollary 3.1.9), so the dependence is not isolated. This is not an internal contradiction, and the reduction chain itself looks coherent. But it is a load-bearing external input, and the paper would be much stronger if the authors either stated and verified the theorem or made clear where it is proved under exactly the needed hypotheses.\n\nMinor soft spots: the paper leans heavily on the [BKV] framework, and a reader not already inside that world will have a hard time separating new content from recall. That is a style issue, not a substantive flaw.\n\nWho is this for? Specialists in geometric representation theory, affine Springer theory, and p-adic local Langlands. It deserves a serious referee: the main theorem is important within the subfield, and the proof is detailed enough to check. I would send it out, with an explicit instruction that the referee verify the [BV] dependence and check whether the hypotheses match. If [BV] holds up, the paper should be accepted; if not, the main theorem is unproved as it stands.","headline":"A genuinely new perversity theorem for affine Springer sheaves, carefully structured but with the decisive step resting on a co-author's unpublished preprint.","tokens_in":34413,"tokens_out":1707,"would_cite":true,"duration_ms":20064,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F20","14L30","20G25"],"pacs":[],"model":"deepseek-v4-flash","headline":"Coinvariants of affine Springer sheaves are perverse","keywords":["affine Springer sheaves","perverse sheaves","coinvariants","loop groups","affine character sheaves","GKM stratification","Springer actions","L-packets"],"falsifier":"Take G = GL_3 over an algebraically closed field of characteristic 7 (Coxeter number 3, so 7 > 2h), choose a regular semisimple γ lying in a nontrivial GKM stratum, and compute RΓ_c(Fl_γ, ω_L)_τ for a nontrivial representation τ of Λ; if this complex has any cohomology in degree < −d_γ, inclusion (0.1) fails and the main theorem is false.","tokens_in":33250,"feed_emoji":"🧮","tokens_out":9021,"duration_ms":86150,"temperature":0.7,"pith_summary":"This paper proves that the derived τ-coinvariants of affine Grothendieck–Springer sheaves are perverse sheaves, not merely complexes. The setting is the loop group of a connected reductive group, and the sheaves live on the bounded regular-semisimple locus of the loop stack. The paper positions these perverse sheaves as examples of what affine character sheaves should be, and, through a cited result, it draws a stability consequence for L-packets of cuspidal Deligne–Lusztig representations. The proof rests on a compatibility theorem for the action of the extended affine Weyl group on the cohomology of affine Springer fibers.","feed_headline":"Coinvariants of affine Springer sheaves are perverse","feed_subtitle":"Perverse coinvariants give the first affine character sheaves and stable L-packets.","key_machinery":"The load-bearing mechanism is the class of affine Grothendieck–Springer sheaves S_{L,•} := p_!(ω_L) on [C^•/LG], where p is the ind-fp-proper affine Grothendieck–Springer fibration from the Iwahori quotient to the bounded loop stack and ω_L is a dualizing-twisted local system. Perversity is measured against the pν t-structure obtained by gluing perverse t-structures on the GKM (root-valuation) strata [C_{w,r}/LG]_{red}, using the fact that p^• is [$C^{{≤0}}$/LG]-small. The proof that coinvariants stay perverse reduces to inclusion (0.1), a cohomological bound RΓ_c(Fl_γ,ω_L)_τ ∈ $D^{{≥−d_γ}}$, which is established by a quasi-coherent sheaf support criterion (Proposition 1.4.3) applied to a sheaf K supported on graphs of admissible embeddings; that support property is exactly the content of Claim 3.2.4, obtained from the cited compatibility theorem.","core_discovery":"The paper's central claim is that, under the hypotheses that the derived group of G is simply connected and that the characteristic of k is either zero or greater than 2h, for every local system L on the maximal torus T and every representation τ of the cocharacter lattice Λ (or of the extended affine Weyl group fW when L is W-equivariant), the sheaf of τ-coinvariants S_{L,•,τ} = coinv_τ(S_{L,•}) is pν-perverse on the stack [C^•/LG]. It also shows that S_{L,•} itself is pν-perverse and is the intermediate extension of its restriction to the bounded regular stratum [$C^{{≤0}}$/LG], that S_{L,•,τ} is constructible (ordinary constructible when τ is finite-dimensional), and that the induced fW-action on the compact-support cohomology of affine Springer fibers agrees with Lusztig's action. The Lie-algebra analog is proved by the same argument.","pith_inferences":["A natural testable extension, not pursued in the paper, is to replace the Iwahori subgroup by other parahoric subgroups; the same smallness-and-coinvariant template may yield perverse 'parahoric Springer sheaves'.","If the coinvariant construction is functorial in τ, the assignment τ ↦ S_{L,•,τ} could be a categorical action of the representation category of the extended affine Weyl group, making affine Springer sheaves a categorified version of character sheaves.","The characteristic bound p > 2h is likely not optimal: the proof only needs flatness of truncated Chevalley maps and the compatibility theorem, so the bound may be lowered wherever those inputs are known."],"forward_implications":["For every τ ∈ Rep_{Qℓ}(Λ), the τ-coinvariants S_{L,•,τ} are pν-perverse on [C^•/LG]; for W-equivariant L the same holds for τ ∈ Rep_{Qℓ}(fW) (Corollary 3.2.5).","Each S_{L,•,τ} is constructible, and is constructible as an ordinary sheaf when τ is finite-dimensional (Corollary 3.1.13).","The fW-action on compact-support cohomology of affine Springer fibers coincides with Lusztig's action (Proposition 3.1.7), so the new action is the expected one.","The coinvariant sheaves need not be intermediate extensions and need not be irreducible even when L and τ are irreducible (Remark 3.2.2).","Via [BV], perversity yields the stability statement for L-packets of cuspidal Deligne–Lusztig representations described in Section 0.8."],"supporting_citations":[{"why":"Supplies the placid ∞-stack framework, the perverse t-structure construction, the small-morphism theorem, and the Lie-algebra affine Springer theory that this paper extends.","marker":"[BKV]"},{"why":"Provides the group version of the compatibility theorem (Theorem 2.3.4) invoked in Claim 3.2.4, together with the Lusztig action on affine Springer fiber cohomology used for comparison.","marker":"[BV]"},{"why":"The original compatibility theorem for local and global Springer actions on affine Springer fiber homology, on which the group version and the Lie-algebra argument rest.","marker":"[Yun2]"},{"why":"Supplies the codimension formula for root valuation (GKM) strata, used through Proposition 2.2.5 and Corollary 2.2.8 to prove smallness of the affine Grothendieck–Springer fibration.","marker":"[GKM]"},{"why":"Steinberg's theorem gives the isomorphism k[G]^G ≅ k[T]^W and smoothness of the Chevalley space under the simply-connected-derived-group assumption.","marker":"[St]"},{"why":"Provides the dimension formula for affine Springer fibers, used to identify the relative dimensions in the smallness and equidimensionality arguments.","marker":"[Be]"}],"fun_headline_variants":["Affine Springer sheaves get perverse coinvariants","Perverse coinvariants for affine Springer sheaves","First affine character sheaves from perverse coinvariants","Coinvarints of affine Springer sheaves turn perverse"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof's decisive step is an imported theorem saying certain symmetries of affine Springer fiber cohomology are compatible; if that theorem is false or does not apply in these characteristics, the perversity of the coinvariant sheaves collapses.","fun_headline_variants_meta":{"raw":{"variants":["Affine Springer sheaves get perverse coinvariants","Perverse coinvariants for affine Springer sheaves","First affine character sheaves from perverse coinvariants","Coinvarints of affine Springer sheaves turn perverse"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000737,"raw_usage":{"total_tokens":3221,"prompt_tokens":804,"completion_tokens":2417,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":420,"completion_tokens_details":{"reasoning_tokens":2350}},"tokens_in":420,"tokens_out":2417,"duration_ms":17241,"temperature":1.0,"reasoning_tokens":2350,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:32:48.005308+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take G = GL_3 over an algebraically closed field of characteristic 7 (Coxeter number 3, so 7 > 2h), choose a regular semisimple γ lying in a nontrivial GKM stratum, and compute RΓ_c(Fl_γ, ω_L)_τ for a nontrivial representation τ of Λ; if this complex has any cohomology in degree < −d_γ, inclusion (0.1) fails and the main theorem is false.","supporting_citations":[],"review_version":2}