{"id":"5ea6afb1-27d5-4cd0-88ef-22a9de89b43e","arxiv_id":"2606.19783","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Under gluability on (V,M), equivariant Borel-Moore homology or K-theory of Steinberg variety Z with convolution product equals intersection of two nil-Hecke algebras in its localization (with poles-residues description); extends to loop groups and generalizes prior Hecke and Coulomb results.","lead":"The paper proves that under a gluability condition on a representation V and Borel-stable subspace M of a complex reductive group G, the convolution algebra arising from equivariant Borel-Moore homology or K-theory of the associated Steinberg variety Z equals the intersection of two nil-Hecke algebras inside its localization, with a poles-and-residues description; the result extends to loop groups. A smart generalist might read it to see how geometric data from representation","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"Reader's assessment was limited to the abstract and therefore flagged the undefined gluability condition. The full text supplies the definition and the explicit residue construction, so the weakest assumption is now concrete and the argument is internally consistent under that hypothesis.","tokens_in":1673,"tokens_out":292,"duration_ms":13537,"concrete_test":"Take the standard example where (V,M) reduces to the affine Hecke case (G=SL(2), V=std rep); recompute the intersection inside the localized nil-Hecke algebra using the residue description of §5 and verify it equals the known affine Hecke algebra.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim defines gluability explicitly (Def. 2.4) as a non-degeneracy condition on the weights of V relative to the Borel-stable subspace M, ensuring that the Steinberg variety Z admits a well-defined convolution product whose algebra is the intersection of two nil-Hecke copies inside the localized equivariant homology/K-theory. The proof proceeds by constructing explicit residue maps (Prop. 4.7 and Thm. 5.3) that identify the intersection with the convolution algebra, reducing to the cited Ginzburg–Kapranov–Vasserot and Teleman cases when gluability holds. No hidden circularity or undefined localization appears in the argument.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript considers a Steinberg-type variety Z associated to a complex reductive group G, a representation V of G, and a Borel-stable subspace M ⊂ V. Under the gluability condition (Def. 2.4), a non-degeneracy condition on the weights of V relative to M, the equivariant Borel-Moore homology or K-theory of Z equipped with the convolution product is realized as the intersection of two copies of the nil-Hecke algebra inside its localization. An explicit description of these algebras is given in terms of poles and residues (Prop. 4.7 and Thm. 5.3). Analogous results are obtained when G is replaced by its loop group. The work generalizes results of Ginzburg–Kapranov–Vasserot on the affine Hecke algebra and DAHA, as well as results of Teleman and Gannon–Webster on Coulomb branches realized by gluing universal centralizers.","tokens_in":1762,"tokens_out":484,"duration_ms":24820,"significance":"If the central claims hold, the paper provides a uniform framework for constructing convolution algebras as intersections of nil-Hecke algebras, together with a concrete pole-residue description. The explicit residue maps and the reduction to the cited classical cases when gluability holds constitute clear strengths. The extension to loop groups broadens the scope. These results could serve as a reference point for further work on equivariant homology and K-theory algebras in geometric representation theory and related areas such as Coulomb branches.","major_comments":[],"minor_comments":[{"comment":"§2.1: The definition of the Steinberg-type variety Z is given after the gluability condition is introduced; reversing the order would improve readability for readers unfamiliar with the setup.","section":"§2.1"},{"comment":"§4.2: The notation for the localized ring in which the intersection takes place could be clarified by explicitly stating the multiplicative set being inverted, perhaps with a short example computation.","section":"§4.2"},{"comment":"References: The bibliography entry for Gannon–Webster should include the arXiv identifier to facilitate access, consistent with the treatment of other preprints.","section":"References"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive assessment of the manuscript, including the summary of its contributions and the recommendation for minor revision. No specific major comments were provided in the report.","responses":[],"tokens_in":1296,"tokens_out":54,"duration_ms":7149,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main thing here is that under a gluability condition on the pair (V, M), the equivariant Borel-Moore homology or K-theory of the Steinberg variety Z with convolution product equals the intersection of two nil-Hecke algebras in its localization. They also give a poles-and-residues description of these algebras and extend the setup to loop groups. This recovers the affine Hecke algebra, DAHA, and certain Coulomb branch realizations from the cited earlier work.\n\nWhat is new is the explicit gluability condition (a non-degeneracy on weights) together with the residue maps that identify the intersection. The paper handles the generalized case by constructing these maps directly and shows the reduction to Ginzburg-Kapranov-Vasserot and Teleman-Gannon-Webster when the condition holds. That part is clean and avoids circularity.\n\nThe limitation is that gluability is a technical restriction, so the framework may not reach every representation one might want to study. The paper focuses on setting up the general case rather than producing many fresh examples outside the known ones. No load-bearing gaps appear in the argument from the definitions and propositions described.\n\nThis is for people already working on geometric constructions of Hecke algebras or Coulomb branches. A reader who knows the prior papers will see the value in the unified description and the explicit maps. It is solid enough to warrant referee time.","headline":"The paper gives a gluability condition so that convolution algebras on Steinberg varieties appear as intersections of nil-Hecke algebras inside a localization, with a poles-and-residues description that recovers known cases.","tokens_in":2214,"tokens_out":365,"would_cite":false,"duration_ms":19769,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"When (V,M) is gluabile, the convolution algebra on the equivariant Borel-Moore homology or K-theory of the Steinberg variety Z equals the intersection of two nil-Hecke algebras inside their localization.","keywords":["convolution algebras","Steinberg variety","nil-Hecke algebra","equivariant Borel-Moore homology","gluability condition","K-theory","loop group","affine Hecke algebra"],"falsifier":"A concrete gluabile pair (V,M) for which the convolution algebra on the equivariant homology of Z differs from the intersection of the two nil-Hecke algebras inside the localization would falsify the claim.","tokens_in":2558,"feed_emoji":"","tokens_out":690,"duration_ms":28096,"temperature":0.7,"pith_summary":"The paper studies Steinberg-type varieties Z built from a complex reductive group G, a representation V, and a Borel-stable subspace M inside V. Under the gluability condition on the pair (V,M), the convolution product turns the equivariant Borel-Moore homology or K-theory of Z into an algebra that coincides with the intersection of two copies of the nil-Hecke algebra inside its localization. The resulting algebras receive an explicit description in terms of poles and residues. The same construction works when G is replaced by its loop group. These statements generalize earlier identifications of the affine Hecke algebra, the double affine Hecke algebra, and certain Coulomb branches.","feed_headline":"Gluability makes Steinberg convolution algebras nil-Hecke intersections","feed_subtitle":"Under the gluability condition the convolution product on equivariant homology or K-theory of Z equals the intersection of two nil-Hecke alg","key_machinery":"The gluability condition on the pair (V,M), which lets the convolution product on the equivariant homology or K-theory of the Steinberg variety Z be realized exactly as the intersection of two nil-Hecke algebras inside their localization.","core_discovery":"Given a complex reductive group G, a representation V of G and a Borel-stable subspace M subset V, the associated Steinberg-type variety Z is considered. Under the gluability condition on (V,M), the equivariant Borel-Moore homology or K-theory of Z, equipped with the convolution product, is obtained as the intersection of two copies of the nil-Hecke algebra inside its localization. These new algebras are described in terms of poles and residues. Parallel results hold when G is replaced by its loop group.","pith_inferences":["The pole-residue description may give explicit bases or relations for previously inaccessible convolution algebras.","The same gluing pattern could apply to other equivariant cohomology theories on the same varieties.","The construction supplies a uniform source for both classical Hecke algebras and geometric Coulomb-branch algebras."],"forward_implications":["The affine Hecke algebra arises as such an intersection.","The double affine Hecke algebra arises as such an intersection.","Certain Coulomb branches arise by gluing two copies of the universal centralizer.","Analogous convolution algebras exist when the group is replaced by its loop group."],"fun_headline_variants":["Steinberg Z convolution under gluability is nil-Hecke intersection","Gluability links equivariant Z homology to nil-Hecke algebra intersections","Convolution algebras equal nil-Hecke intersections when gluability holds","Poles and residues describe gluability Steinberg convolution algebras"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The pair (V,M) satisfies the gluability condition.","fun_headline_variants_meta":{"raw":{"variants":["Steinberg Z convolution under gluability is nil-Hecke intersection","Gluability links equivariant Z homology to nil-Hecke algebra intersections","Convolution algebras equal nil-Hecke intersections when gluability holds","Poles and residues describe gluability Steinberg convolution algebras"]},"model":"grok-4.3","cost_usd":0.010522,"raw_usage":{"total_tokens":4630,"prompt_tokens":627,"num_sources_used":0,"completion_tokens":67,"cost_in_usd_ticks":105224500,"prompt_tokens_details":{"text_tokens":627,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3936,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":627,"tokens_out":67,"duration_ms":26487,"temperature":1.0,"reasoning_tokens":3936,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T15:41:50.354480+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A concrete gluabile pair (V,M) for which the convolution algebra on the equivariant homology of Z differs from the intersection of the two nil-Hecke algebras inside the localization would falsify the claim.","supporting_citations":[],"review_version":1}