{"id":"ad468b29-67d3-4694-8e9e-64f121227f95","arxiv_id":"2501.14157","paper_version":1,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors prove the tame local Betti geometric Langlands correspondence, a monoidal equivalence between ind-coherent sheaves on a Steinberg stack and nilpotent-singular-support Betti sheaves on a monodromic Hecke stack.","lead":"This paper proves a long-conjectured equivalence between two mathematical ways to encode symmetries of loop groups, called the tame local Betti geometric Langlands correspondence. The result lets researchers translate problems between spectral geometry and automorphic sheaves, extending a celebrated theorem of Bezrukavnikov.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The argument is a clean reduction to [DT24]; the load-bearing point is an unstated compatibility of the two recalled equivalences with the relevant module structures, without which Prop. 3.1.2 could fail.","rationale":"The paper is a serious, internally coherent proof of a major conjecture, and the high-level strategy is clearly described. The reader's weakest_assumption correctly identifies the dependence on [DT24]. My pass refines this: the single most dangerous point is not merely the existence of the two recalled equivalences, but their mutual compatibility with the module actions. Proposition 3.1.2 uses Theorems 2.5.4 and 2.5.5 to identify an endomorphism category; such an identification requires the χHχ-action on χH to match the QCoh_G(G)-action on QCoh_G(~G) after transport of structure. This is a separate datum from the two theorem statements as written. I found no obvious internal contradiction in the subsequent sections: the localization theorem (Thm. 3.3.2) and the pseudocompactness argument (Prop. 3.4.2) are intricate but plausible, and the cited Morita-theoretic step [BZGO20, Prop. 3.2] is a reasonable tool for the bimodule argument. Remark 1.2.6 explicitly leaves the 2-categorical enhancement underived, but that does not affect the main equivalence of monoidal ∞-categories. I therefore keep the reader's CONDITIONAL verdict: acceptance should be conditioned on verification of the [DT24] compatibility, since the central claim is proven only modulo an unstated but essential property of the cited results.","tokens_in":17261,"tokens_out":23098,"duration_ms":205897,"concrete_test":"Check whether [DT24] establishes the following compatibility: under the equivalence F: χH ≃ QCoh_G(~G) of Thm. 2.5.4(2) and the monoidal equivalence QCoh_G(G) ≃ χHχ of Thm. 2.5.5, the tautological action map χHχ → End(χH) must conjugate to the spectral action QCoh_G(G) → End(QCoh_G(~G)). Concretely, verify the diagram of bimodules is commutative; if this is not proved in [DT24], supply a proof, and if it is false, Theorem 1.2.5 is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.2.5 reduces the main construction to Theorems 2.5.4 and 2.5.5 of [DT24], which are cited without proof. In Proposition 3.1.2, the functor (8) is built by identifying the target End_{χHχ}(χH)^rev with End_{QCoh_G(G)}(QCoh_G(~G))^rev ≃ QCoh_G(~G×_G~G). This identification requires more than the two statements as recalled: Theorem 2.5.4(2) gives an equivalence of categories F: χH ≃ QCoh_G(~G), and Theorem 2.5.5 gives a monoidal equivalence QCoh_G(G) ≃ χHχ, but one must also know that the tautological action of χHχ on χH corresponds, under these equivalences, to the geometric action of QCoh_G(G) on QCoh_G(~G). That compatibility is not stated in the paper and is not a formal consequence of the two equivalences alone. If it fails, the functor (8) need not land in the claimed endomorphism category, and the construction of ι! in Prop. 3.1.2 collapses. The later localization and pseudocompactness arguments (Thm. 3.3.2, Cor. 3.3.7, Prop. 3.4.2) appear coherent assuming this step, so this compatibility---which lives in the unproven companion paper [DT24]---is the least secure link.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a monoidal equivalence between the spectral category IndCoh_G(~G ×_G ~G) and the automorphic category Shv_nilp(Ĩ\\LG/Ĩ), thereby settling the tamely ramified local Betti geometric Langlands correspondence conjectured by Ben-Zvi–Nadler, and specializing to Bezrukavnikov's unipotent theorem. The proof constructs a monoidal functor from the automorphic Hecke category to the spectral side via Iwahori–Whittaker modules, identifies its kernel with the infinitely connective objects, and then passes to compact objects via pseudocompactness arguments. The argument is a detailed reduction to two main theorems from the authors' previous work [DT24].","tokens_in":17599,"tokens_out":11598,"duration_ms":98881,"significance":"If correct, this is a major result in geometric representation theory: it proves a conjecture of Ben-Zvi–Nadler and provides a new route to Bezrukavnikov's affine Hecke category equivalence. The paper is well structured, with modular t-boundedness lemmas and an explicit strategy that avoids older triangulated-level complications. However, the proof is not self-contained and depends heavily on [DT24]; one compatibility statement needed for the main construction is not stated or proved, and one perversity assertion is used without support. These issues are local and fixable, but they are load-bearing.","major_comments":[{"comment":"The passage from the target of (8) to End_{QCoh_G(G)}(QCoh_G(~G))^rev is not justified by Theorems 2.5.4 and 2.5.5 as stated. Theorem 2.5.4(2) gives an equivalence of plain categories χH ≃ QCoh_G(~G), and Theorem 2.5.5 gives a monoidal equivalence QCoh_G(G) ≃ χHχ; however, identifying endomorphism categories requires the additional compatibility that the tautological action of χHχ on χH corresponds, under these equivalences, to the action of QCoh_G(G) on QCoh_G(~G) by pullback along ~G→G. This compatibility is not stated in the recollections of Section 2.5 nor proved in Section 3.1, and it is not a formal consequence of the two recalled equivalences. Since the construction of ι! and all subsequent arguments depend on this step, I ask for an explicit compatibility lemma with a proof or a precise citation to the relevant statement in [DT24].","section":"Section 3.1.1, Proposition 3.1.2, Eq. (8)"},{"comment":"In the proof of left exactness of F, the object χδ ⋆ Z_V ⋆ W_λ is asserted to be perverse for V ∈ Rep(G)^♥ and λ ∈ Λ, and this assertion is used to conclude Hom_{χH}(χδ ⋆ Z_V ⋆ W_λ, ξ) ≥ 0 for ξ ≥ 0. No proof or reference is given for this perversity statement. It is load-bearing for Lemma 3.3.4, which in turn is essential for the 'if' direction of Theorem 3.3.2. The authors should either prove this perversity here or cite the precise result in [DT24] (or a standard reference) that implies it.","section":"Lemma 3.3.4"}],"minor_comments":[{"comment":"The claimed equivalence of (∞,2)-categories is stated without a formal proof, and the text explicitly says the deduction is omitted. Since the abstract and introduction describe the paper as proving the full tame local Betti correspondence, I recommend clarifying that Theorem 1.2.5 is the proved statement and that the 2-categorical enhancement is conditional on the development of the referenced foundations.","section":"Remark 1.2.6"},{"comment":"The sentence 'Moreover, the object π_{w,!}(k[d_w]) is a compact generator of the category ... and left convolution with it 1 yields a t-exact equivalence' contains a stray '1' and is hard to parse; please rephrase.","section":"Section 2.3.3"},{"comment":"In Theorem 2.5.4(1), the map QCoh_G(G)→QCoh_G(~G) through which the composite factors is not explicitly described; for the reader's convenience, please state that it is the pullback along ~G→G.","section":"Section 2.5.3, Theorem 2.5.4"}],"recommendation":"major_revision","confidential_remarks":"The paper's main theorem is essentially a reduction to [DT24], and the missing compatibility mentioned in my first major comment may already be proven there; if so, the authors should state it explicitly and give the reference. The acceptance of this paper should be considered in light of the status of [DT24], since the proof here does not reprove the key inputs."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is the real thing: a monoidal equivalence between IndCoh_G(~G x_G ~G) and Shv_nilp(I\\LG/I), proving the Ben-Zvi-Nadler conjecture for tame Betti geometric Langlands and giving a new proof even in the unipotent case. The paper is clearly written, the strategy is a genuine lift of Kazhdan-Lusztig to the categorical level, and the soft t-boundedness arguments avoid the heavier machinery of Bezrukavnikov's original proof. Novelty is high; as far as I know, no tame family result for non-abelian G existed before. Credit is due for that.\n\nThe soft spots are real but localized. The proof leans on two theorems from the authors' own [DT24]: the universal monodromic Arkhipov-Bezrukavnikov equivalence and the bi-Whittaker equivalence. That reliance is not itself a problem - self-citation is fine when the results are load-bearing and the companion paper is serious. The problem is that Proposition 3.1.2 needs more than those two theorems as stated. To identify End_{chiHchi}(chiH)^rev with End_{QCoh_G(G)}(QCoh_G(~G))^rev, you need to know that the action of chiHchi on chiH corresponds under the equivalences to the geometric action of QCoh_G(G) on QCoh_G(~G). The paper just says \"by Theorems 2.5.4 and 2.5.5\" and moves on. That compatibility is not stated and does not formally follow from the two recalled equivalences alone. If it is proven in [DT24], it should be cited precisely; if not, the construction of iota! could collapse. This is the load-bearing gap, and the referee should ask the authors to spell it out.\n\nThe other soft spot is minor and self-acknowledged: Remark 1.2.6 omits a formal deduction of the (infinity,2)-categorical consequence because a reference for 2-IndCoh with singular supports is missing. That is not needed for the main theorem, and the paper is honest about it.\n\nThe proof structure after Proposition 3.1.2 - colocalization, left completion, anticompletion - appears coherent, and the t-boundedness lemmas are argued in detail. I cannot machine-check the higher algebra, so my confidence is conditional, but the paper deserves a serious referee. This is a major result, and the argument is in good shape except for the unstated module-compatibility. Send it to review, but make sure the referee has [DT24] and asks for a clean statement of that compatibility. I would bring it to reading group and would likely cite it once the companion paper is properly integrated.","headline":"A serious proof of a major conjecture, but the key step inherits an unstated module-compatibility from the authors' companion paper, so referee attention should focus there.","tokens_in":792,"tokens_out":770,"would_cite":true,"duration_ms":25596,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14D24","22E57","14F05","20C08"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves the tame local Betti geometric Langlands correspondence, showing that two sheaf categories attached to any complex reductive group are equivalent as monoidal ∞-categories.","keywords":["tame local Betti geometric Langlands","universal affine Hecke category","Iwahori–Whittaker category","nilpotent singular support","ind-coherent sheaves","monoidal equivalence","Wakimoto sheaves","Steinberg stack"],"falsifier":"Compute the bi-Whittaker category of sheaves on the loop group for $\\mathrm{SL}_2$ and compare it with $\\mathrm{QCoh}_{\\check{G}}(\\check{G})$ under the equivalence claimed in [DT24]; a mismatch would invalidate the main theorem, which depends directly on that input.","tokens_in":17070,"feed_emoji":"🔀","tokens_out":18492,"duration_ms":150086,"temperature":0.7,"pith_summary":"The paper proves that two categorically defined objects attached to a complex reductive group $G$ are monoidally equivalent: the category $\\mathrm{IndCoh}_G(\\widetilde{G}\\times_G \\widetilde{G})$ of ind-coherent sheaves on the Steinberg stack of the Grothendieck–Springer resolution, and the category $\\mathrm{Shv}_{\\mathrm{nilp}}(\\widetilde{I}\\backslash LG/\\widetilde{I})$ of Betti sheaves with nilpotent singular support on the universal monodromic Hecke stack. This is the tame local Betti geometric Langlands correspondence conjectured in [BZN07] and developed further in [BZN18]. Because both categories are monoidal under convolution, the equivalence identifies the two realizations of the universal affine Hecke category, the categorical analogue of the affine Hecke algebra. Restricting to unipotent monodromy recovers the unipotent theorem of [B16], now by a new argument. The proof lifts the classical function-theoretic strategy of [KL87] to the categorical level, matching the tautological spectral module with the Iwahori–Whittaker automorphic module.","feed_headline":"Tame local Betti Langlands correspondence proved","feed_subtitle":"Two sheaf categories attached to any reductive group are shown equivalent, recovering the unipotent theorem.","key_machinery":"The central object is the universal affine Hecke category, realized on the spectral side as $\\mathrm{IndCoh}_G(\\widetilde{G}\\times_G \\widetilde{G})$ and on the automorphic side as $\\mathrm{Shv}_{\\mathrm{nilp}}(\\widetilde{I}\\backslash LG/\\widetilde{I})$. The proof's engine is the matching of two module categories: the tautological module $\\mathrm{QCoh}_G(\\widetilde{G})$ on the spectral side and the Iwahori–Whittaker category on the automorphic side. The load-bearing identifications $\\mathrm{QCoh}_G(\\widetilde{G}) \\simeq \\mathrm{Shv}_{\\mathrm{nilp}}(\\widetilde{I},\\chi\\backslash LG/\\widetilde{I})$ and $\\mathrm{QCoh}_G(G) \\simeq \\mathrm{Shv}_{\\mathrm{nilp}}(\\widetilde{I},\\chi\\backslash LG/\\widetilde{I},\\chi)$ are taken from the authors' earlier work [DT24] and are not reproven here. From these, the commuting actions of the bi-Whittaker category produce a monoidal functor between the two realizations, and a t-structure analysis with the translation sheaves (Wakimoto sheaves) and the finiteness of the finite Weyl group shows the functor is an equivalence up to renormalization.","core_discovery":"The central discovery is that the universal affine Hecke category has a single monoidal incarnation even after the monodromy is allowed to vary tamely around the punctures, not just in the unipotent case where several models agree. The paper establishes the equivalence $\\mathrm{IndCoh}_G(\\widetilde{G}\\times_G \\widetilde{G}) \\simeq \\mathrm{Shv}_{\\mathrm{nilp}}(\\widetilde{I}\\backslash LG/\\widetilde{I})$ as monoidal $\\infty$-categories. The proof constructs a monoidal functor $\\iota^!$ from the automorphic side to the spectral side using the commuting actions of the bi-Whittaker category on the Iwahori–Whittaker category, and then identifies this functor's kernel with the infinitely connective objects via a t-structure analysis. Passing to the Verdier quotient and then to compact objects yields an equivalence that extends by ind-completion to the full statement. In particular, the theorem gives a new proof of the unipotent case of [B16] as a specialization.","pith_inferences":["The same module-matching strategy, if supplied with the corresponding universal monodromic equivalences, could prove the tamely ramified de Rham local correspondence; the authors explicitly expect this approach to be useful there.","The t-boundedness control uses only the finite Weyl group, so the correspondence holds uniformly across all root systems without case-by-case verification; this structural uniformity is not stated as a separate theorem but follows from the proof.","Because the equivalence identifies the Drinfeld center of the automorphic Hecke category with the center of the spectral category, the theorem offers a new way to compute endofunctors and central actions of the affine Hecke category that might inform categorical representation theory beyond geometric Langlands."],"forward_implications":["Specializing the main equivalence to unipotent monodromy recovers the fundamental theorem of [B16] on two geometric realizations of the affine Hecke algebra.","The monoidal equivalence implies an equivalence of (∞,2)-categories $\\mathrm{2}\\text{-}\\mathrm{IndCoh}_{\\mathrm{nilp}}(G/G) \\simeq \\mathrm{Shv}_{\\mathrm{nilp}}(\\widetilde{I}\\backslash LG/\\widetilde{I})\\text{-mod}$, so the tame Hecke action on arbitrary moduli stacks is the same in both realizations.","Base-changing to the formal completion of the invariant-theory quotient at its closed points yields the restricted-variation version of the correspondence and confirms the conjecture stated as [B16, Conjecture 58] over the complex numbers.","The proof offers a new path to the unipotent theorem, using higher algebra in place of the ad hoc tilting-sheaf models of [B16]."],"supporting_citations":[{"why":"Supplies the two main categorical inputs of the proof: the universal monodromic equivalence identifying $\\mathrm{QCoh}_G(\\widetilde{G})$ with the Iwahori–Whittaker category, and the bi-Whittaker equivalence for $\\mathrm{QCoh}_G(G)$.","marker":"[DT24]"},{"why":"Introduced the tame local Betti geometric Langlands correspondence that this paper proves.","marker":"[BZN07]"},{"why":"Developed the Betti Langlands framework and provides the Mellin transform identification of the minimal stratum used in the automorphic preliminaries.","marker":"[BZN18]"},{"why":"Provided the function-theoretic strategy of matching the two algebra actions on the same vector space, which the proof lifts to categories.","marker":"[KL87]"},{"why":"Established the unipotent specialization of the equivalence later generalized in [DT24].","marker":"[AB09]"},{"why":"Gives the identification of endofunctors of quasicoherent sheaves with sheaves on a fiber product, used to pass from endomorphisms to the Steinberg stack.","marker":"[BZFN10]"},{"why":"Supplies the Morita-theoretic criterion for a bimodule to induce a fully faithful adjunction, used in the proof of Proposition 3.2.2.","marker":"[BZGO20]"},{"why":"The unipotent theorem recovered as a specialization; its technical arguments (especially Lemma 38) influenced the present t-structure analysis.","marker":"[B16]"}],"fun_headline_variants":["Tame Betti Langlands equivalence proven","Hecke categories match in tame case","New proof of unipotent Langlands","Tame Langlands: two sheaves identical","Universal Hecke category unified"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The two theorems taken from the authors' earlier work [DT24]—the universal monodromic equivalence between coherent sheaves on $\\widetilde{G}$ and the Iwahori–Whittaker category, and the bi-Whittaker equivalence for $\\mathrm{QCoh}_G(G)$—are load-bearing and are not reproven here; if either is wrong, the main equivalence collapses.","fun_headline_variants_meta":{"raw":{"variants":["Tame Betti Langlands equivalence proven","Hecke categories match in tame case","New proof of unipotent Langlands","Tame Langlands: two sheaves identical","Universal Hecke category unified"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000103,"raw_usage":{"total_tokens":955,"prompt_tokens":801,"completion_tokens":154,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":417,"completion_tokens_details":{"reasoning_tokens":102}},"tokens_in":417,"tokens_out":154,"duration_ms":2142,"temperature":1.0,"reasoning_tokens":102,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T15:19:22.323713+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the bi-Whittaker category of sheaves on the loop group for $\\mathrm{SL}_2$ and compare it with $\\mathrm{QCoh}_{\\check{G}}(\\check{G})$ under the equivalence claimed in [DT24]; a mismatch would invalidate the main theorem, which depends directly on that input.","supporting_citations":[],"review_version":1}