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Convolution algebras associated to representations

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2606.19783 v1 pith:A2TFS2IV submitted 2026-06-18 math.RT

Dragos Crisan This is my paper · ORCID
classification math.RT
keywords convolutionalgebrasSteinbergvarietynil-HeckealgebraequivariantBorel-MoorehomologygluabilityconditionK-theoryloopgroupaffineHecke
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

The pair (V,M) satisfies the gluability condition.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 3 minor

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.

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.

minor comments (3)
  1. [§2.1] §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.
  2. [§4.2] §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.
  3. [References] References: The bibliography entry for Gannon–Webster should include the arXiv identifier to facilitate access, consistent with the treatment of other preprints.

Simulated Author's Rebuttal

0 responses · 0 unresolved

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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; explicit constructions independent of inputs

full rationale

The paper defines gluability explicitly (Def. 2.4) as a non-degeneracy condition on weights of V relative to M. It then proves the central identification by constructing explicit residue maps (Prop. 4.7, Thm. 5.3) that realize the convolution algebra as the intersection of two nil-Hecke copies inside the localization. This is a direct construction, not a renaming or fit. The result reduces to the cited Ginzburg–Kapranov–Vasserot and Teleman cases precisely when gluability holds, but the general case rests on new arguments rather than self-citation chains or definitional equivalence. No self-definitional, fitted-prediction, or ansatz-smuggling steps appear in the derivation chain.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Abstract only; no explicit free parameters, axioms, or invented entities can be identified.

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Pith. "Pith review of Convolution algebras associated to representations." pith.science (2026). https://pith.science/paper/A2TFS2IV

@misc{pith2026260619783,
  author       = {Pith},
  title        = {Pith review of: Convolution algebras associated to representations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A2TFS2IV}},
  note         = {Machine review of arXiv:2606.19783}
}
abstract

Given a complex reductive group $G$, a representation $V$ of $G$ and a Borel-stable subspace $M \subset V$, we consider the associated Steinberg-type variety $Z$. We prove that, under a certain condition on $(V,M)$, called gluability, 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. We also provide a description of these new algebras in terms of poles and residues. Similar results are obtained when $G$ is replaced by its loop group. This generalizes results of Ginzburg, Kapranov and Vasserot describing the affine Hecke algebra and DAHA, as well as a result of Teleman and Gannon--Webster that realizes certain Coulomb branches by gluing two copies of the universal centralizer.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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