On Braverman-Kazhdan's asymptotic Hecke algebra for inner forms of $\mathrm{GL}_n$

Stefan Dawydiak

REVIEW 2 major objections 3 minor 10 references

On Braverman-Kazhdan's asymptotic Hecke algebra for inner forms of $\mathrm{GL}_n$

T0 review · 2 major / 3 minor · reviewed 2026-06-26 · grok-4.3

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Pith's one-line read The asymptotic Hecke algebra J(G) for inner forms of p-adic GL_n is defined over the algebraic closure of Q_l

desk verdict This paper extends the asymptotic Hecke algebra to inner forms of GL_n, shows it is defined over bar Q_l, supplies explicit centralizer formulas, and includes a proof of the needed generalized Suzuki theorem. read the letter →

arxiv 2606.21313 v1 pith:TRYMUV75 submitted 2026-06-19 math.RT math.NT

Stefan Dawydiak This is my paper

classification math.RTmath.NT

keywords asymptoticHeckealgebrainnerformsofGL_ncategoricallocalLanglandsBushnell-KutzkotypesSécherre-StevensHochschildhomologyKazhdan-LusztigbijectionSuzukitheorem

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 shows that for inner forms $G$ of p-adic $\mathrm{GL}_n$ the Braverman-Kazhdan asymptotic Hecke algebra $J(G)$ is defined over the algebraic closure of the l-adic numbers. The same holds for the property that a representation of $G$ extends to a module over $J(G)$. This makes both objects available for use in the categorical local Langlands correspondence. The proof uses Bushnell-Kutzko and Sécherre-Stevens types and supplies a generalization of Suzuki's theorem.

What carries the argument

Bushnell-Kutzko and Sécherre-Stevens types for inner forms of $\mathrm{GL}_n$ with a generalization of Suzuki's theorem

What would settle it

Finding an inner form $G$ of $\mathrm{GL}_n$ where $J(G)$ cannot be defined over $\overline{Q}_l$ or where the extension property does not align with the categorical local Langlands correspondence would falsify the main claim.

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Extended reading notes

Core claim

$J(G)$ and the extension property for $G$-representations to $J(G)$-modules are defined over $\overline{Q}_l$ for inner forms $G$ of p-adic $\mathrm{GL}_n$. This is established by working with types and a generalized version of Suzuki's theorem. The algebra also admits explicit formulas for its elements in terms of reductive centralizers of L-parameters and shares Hochschild homology with the algebra of compactly supported smooth functions on $G$.

Load-bearing premise

The constructions and proofs rely on the existence and good properties of Bushnell-Kutzko and Sécherre-Stevens types for the inner forms of $\mathrm{GL}_n$ together with a generalization of Suzuki's theorem.

Editorial extensions

If this is right

Stalks of sheaves on Bun_n corresponding to trivial vector bundles on the L-parameter stack, including the Whittaker sheaf, can be discussed in terms of J(GL_n)-modules.
Explicit formulas for many functions in J(G) are given using the reductive centralizer of L-parameters.
J(G) has the same Hochschild homology as C_c^∞(G) and the Kazhdan-Lusztig bijection appears in the isomorphism.

Reading between the lines

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

If the definition over \overline{Q}_l holds, it may allow direct comparison of J(G)-modules with objects in the categorical correspondence for non-split groups.
The compatibility with Hecke operators could lead to new ways to compute invariants of representations using the asymptotic algebra.
The homology isomorphism suggests that J(G) and C_c^∞(G) are related by a map that preserves more structure than just homology.

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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.

Desk Editor's Note

read the letter

The main point is that Dawydiak carries the Braverman-Kazhdan asymptotic Hecke algebra J(G) over to inner forms of p-adic GL_n and proves it is defined over bar Q_l, so the extendability property for representations makes sense in categorical local Langlands. He also gives explicit formulas for many elements of J(G) via reductive centralizers of L-parameters, shows the Hochschild homology of J(G) equals that of C_c^infty(G), and notes that the Kazhdan-Lusztig bijection appears in the isomorphism. A rudimentary compatibility with Hecke operators is stated, which lets one talk about stalks of certain sheaves on Bun_n in terms of J(GL_n)-modules.

The paper does the useful work of supplying its own proof of the generalized Suzuki theorem rather than leaving that step as an assumption. The route through Bushnell-Kutzko and Secherre-Stevens types is the standard one for these groups, and those types are known to exist for inner forms, so the foundation is not invented here.

The soft spots are modest. The compatibility statement is described as rudimentary, so it is not a full functoriality result. Without the full proofs in hand it is hard to judge the length or technical difficulty of the type-theoretic adaptations for inner forms, but the abstract gives no sign of circularity or unfalsifiable steps. The argument rests on external type theory plus the supplied generalization, which keeps the circularity burden low.

This is for people already working inside p-adic representation theory and the categorical Langlands program for GL_n and its forms. A reader who needs the bar Q_l definition or the explicit centralizer formulas will get concrete value. It is an incremental but self-contained step that fills a documented gap.

I would send it to peer review. The claims are specific, the key generalization is proved in the paper, and the methods are standard enough that referees can check them directly.

Referee Report

2 major / 3 minor

Summary. The paper studies the Braverman-Kazhdan asymptotic Hecke algebra J(G) for inner forms G of p-adic GL_n. It proves that both J(G) and the property that a G-representation extends to a J(G)-module are defined over ar Q_l, enabling their use in the categorical local Langlands correspondence. It establishes a basic compatibility with Hecke operators to interpret stalks of sheaves on Bun_n (including the Whittaker sheaf) via J(GL_n)-modules, gives explicit formulas for many elements of J(G) in terms of reductive centralizers of L-parameters, shows that J(G) has the same Hochschild homology as C_c^∞(G) with the Kazhdan-Lusztig bijection appearing in the isomorphism, and proceeds by means of Bushnell-Kutzko and Sécherre-Stevens types while supplying a proof of a generalization of Suzuki's theorem for GL_n.

Significance. If the central claims hold, the results provide a concrete bridge between the asymptotic Hecke algebra construction and the categorical local Langlands program for inner forms of GL_n. The field-of-definition statement is load-bearing for the categorical applications, the explicit formulas and homology computation supply usable tools, and the supplied proof of the generalized Suzuki theorem makes the argument self-contained on a key technical step. These contributions are proportionate to the scope of the manuscript and strengthen the toolkit for studying representations of inner forms in a geometric context.

major comments (2)

[Introduction / § on generalized Suzuki theorem] The central claim that J(G) is defined over ar Q_l rests on the existence of Bushnell-Kutzko/Sécherre-Stevens types (known for inner forms) together with the paper's generalization of Suzuki's theorem. Because the manuscript supplies an explicit proof of that generalization, the argument is internally self-contained; however, the precise statement of the generalized theorem (including the exact hypotheses on the types and the field of definition) should be isolated as a numbered theorem with a self-contained proof section so that the dependence is transparent.
[Section on compatibility with Hecke operators] The rudimentary compatibility with Hecke operators is used to discuss stalks of sheaves on Bun_n corresponding to trivial vector bundles on the L-parameter stack. The precise functoriality statement (which Hecke operators are involved and how they act on the J(G)-module structure) needs to be stated as a proposition with a clear reference to the relevant diagram or exact sequence, as this step directly supports the interpretation of the Whittaker sheaf.

minor comments (3)

[Title and abstract] Notation for the asymptotic Hecke algebra is introduced as Σ(G) in the title but rendered as J(G) throughout the abstract and body; adopt a single consistent symbol and update the title accordingly.
[Abstract] The phrase 'rudimentary form of compatibility' is informal; replace with a precise description of the limited range of Hecke operators for which compatibility is proved.
[Introduction] References to the original Braverman-Kazhdan construction and to the categorical local Langlands correspondence should include the most recent relevant citations (e.g., works post-2020 on Bun_n and L-parameters) to situate the contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and constructive suggestions. We address each major comment below.

read point-by-point responses

Referee: [Introduction / § on generalized Suzuki theorem] The central claim that J(G) is defined over bar Q_l rests on the existence of Bushnell-Kutzko/Sécherre-Stevens types (known for inner forms) together with the paper's generalization of Suzuki's theorem. Because the manuscript supplies an explicit proof of that generalization, the argument is internally self-contained; however, the precise statement of the generalized theorem (including the exact hypotheses on the types and the field of definition) should be isolated as a numbered theorem with a self-contained proof section so that the dependence is transparent.

Authors: We agree that extracting the generalized Suzuki theorem into a numbered statement with its own self-contained proof section will improve transparency regarding the field-of-definition claim. We will make this structural change in the revised manuscript. revision: yes
Referee: [Section on compatibility with Hecke operators] The rudimentary compatibility with Hecke operators is used to discuss stalks of sheaves on Bun_n corresponding to trivial vector bundles on the L-parameter stack. The precise functoriality statement (which Hecke operators are involved and how they act on the J(G)-module structure) needs to be stated as a proposition with a clear reference to the relevant diagram or exact sequence, as this step directly supports the interpretation of the Whittaker sheaf.

Authors: We will formulate the precise functoriality statement as a numbered proposition, with explicit reference to the relevant diagram or exact sequence, in the section discussing compatibility with Hecke operators. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

Derivation self-contained; no load-bearing reductions to self-definition or self-citation

full rationale

The paper's central claims rest on the known existence of Bushnell-Kutzko and Sécherre-Stevens types (external to this work) together with an explicit generalization of Suzuki's theorem whose proof is supplied inside the manuscript. No equations, constructions, or results are shown to reduce by definition or by self-citation chain to the paper's own inputs; the field-of-definition statement for J(G) is therefore independent of any circular step.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The paper rests on standard facts from type theory for p-adic groups and on the existence of the Braverman-Kazhdan asymptotic Hecke algebra; no new free parameters or invented entities are introduced in the abstract.

assumptions (1)

domain assumption Bushnell-Kutzko and Secherre-Stevens types exist and behave well for inner forms of GL_n
Invoked to generalize Suzuki's theorem and construct the algebra and its modules.

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Cite this review

Pith. "Pith review of On Braverman-Kazhdan's asymptotic Hecke algebra for inner forms of $\mathrm{GL}_n$." pith.science (2026). https://pith.science/paper/TRYMUV75

@misc{pith2026260621313,
  author       = {Pith},
  title        = {Pith review of: On Braverman-Kazhdan's asymptotic Hecke algebra for inner forms of $\mathrmGL_n$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TRYMUV75}},
  note         = {Machine review of arXiv:2606.21313}
}

abstract

We study Braverman-Kazhdan's asymptotic Hecke algebra $\mathcal{J}(G)$ for inner forms $G$ of $p$-adic $\mathrm{GL}_n$. We show that $\mathcal{J}(G)$ and the property for a $G$-representation to extend to a $\mathcal{J}(G)$-module are defined over $\overline{\mathbb{Q}_\ell}$, and hence make sense in the context of the categorical local Langlands correspondence. We show a rudimentary form of compatibility with Hecke operators, allowing us discuss stalks of sheaves on $\mathrm{Bun}_n$ corresponding to the trivial vector bundles on the stack of $L$-parameters, in particular the Whittaker sheaf, in terms of $\mathcal{J}(\mathrm{GL}_n)$-modules. We provide explicit formulas in terms of reductive centralizer of $L$-parameters for many functions in $\mathcal{J}(G)$, and show that $\mathcal{J}(G)$ has the same Hochschild homology as $C_c^\infty(G)$, and that the Kazhdan-Lusztig bijection appears in the isomorphism. We proceed via Bushnell-Kutzko and S\'{e}cherre-Stevens types, generalizing a theorem of Suzuki for $\mathrm{GL}_n$ for which we provide a proof.

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Reference graph

Works this paper leans on

10 extracted references · 7 canonical work pages

[1]

Aubert, P

[ABPS17] A.-M. Aubert, P. F. Baum, R. J. Plymen, and M. Solleveld, Hecke algebras for inner forms of p-adic special linear groups, J. Inst. Math. Jussieu 16 (2017), 351–419. [AFMO24a] J. Adler, J. Fintzen, M. Mishra, and K. Ohara, Reduction to depth zero for tame p-adic groups via Hecke algebra isomorphisms (2024), available at https://arXiv.org/abs/2408....

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[CS19] K. Y. Chan and G. Savin, Bernstein-Zelevinsky derivatives: a Hecke algebra approach , Int. Math. Res. Not. IMRN 2019 (2019), no. 3, 731–760. [Daw21] S. Dawydiak, On Lusztig’s asymptotic Hecke algebra for SL2, Proc. Amer. Math. Soc. 149 (2021), no. 1, 71–88. [Daw23] , The asymptotic Hecke algebra and rigidity, with an appendix by D. Rumynin (2023), ...

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[Shi88] , A two-sided cell in an affine Weyl group, II , J. Lond. Math. Soc. 37 (1988), no. 2, 253–264. [Sol13] M. Solleveld, Hochschild homology of affine Hecke algebras , J. Algebra 384 (2013), 1–35. [Sol19] , On completions of Hecke algebras , Progr. Math. 328 (2019), 207–262. [Sol25] , Standard modules and intertwining operators for reductive p-adic g...

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Pith tools

Reviewed June 26, 2026 · model on record in the stance chip above.

[1] [1]

Aubert, P

[ABPS17] A.-M. Aubert, P. F. Baum, R. J. Plymen, and M. Solleveld, Hecke algebras for inner forms of p-adic special linear groups, J. Inst. Math. Jussieu 16 (2017), 351–419. [AFMO24a] J. Adler, J. Fintzen, M. Mishra, and K. Ohara, Reduction to depth zero for tame p-adic groups via Hecke algebra isomorphisms (2024), available at https://arXiv.org/abs/2408....

work page arXiv 2017

[2] [2]

[CS19] K. Y. Chan and G. Savin, Bernstein-Zelevinsky derivatives: a Hecke algebra approach , Int. Math. Res. Not. IMRN 2019 (2019), no. 3, 731–760. [Daw21] S. Dawydiak, On Lusztig’s asymptotic Hecke algebra for SL2, Proc. Amer. Math. Soc. 149 (2021), no. 1, 71–88. [Daw23] , The asymptotic Hecke algebra and rigidity, with an appendix by D. Rumynin (2023), ...

work page arXiv 2019

[3] [3]

Geometrization of the local Langlands correspondence

[FS21] L. Fargues and P. Scholze, Geometrization of the local Langlands correspondence , Ast´ erisque (2021), to appear, available at https://arxiv.org/abs/2102.13459. [HC84] Harish-Chandra, Harmonic Analysis on Reductive p-adic Groups, Collected Papers IV, 1984, pp. 75–100. [Hen00] G. Henniart, Une preuve simple des conjectures de Langlands pour GL(n) su...

work page Pith review arXiv 2021

[4] [4]

Rationality properties of complex representations of reductive p-adic groups

With an appendix by V.G. Berkovich. [Kar16] V. Karemaker, Hecke algebras for GLn over local fields, Arch. Math. 107 (2016), 341–353. [KL79] D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras , Invent. math. 53 (1979), 165–184. [KL87] , Proof of the Deligne-Langlands conjectures for Hecke algebras , Invent. Math. 87 (1987), 153...

work page Pith review arXiv 2016

[5] [5]

Appendix E by M. O. Ronco. [Lus87a] G. Lusztig, Cells in affine Weyl groups, II , J. Algebra 109 (1987), 536–548. [Lus87b] , Cells in affine Weyl groups, III , J. Fac. Sci. Univ. Tokyo Sect. IA Math 34 (1987), 223–243. [Lus89] , Cells in affine Weyl groups, IV , J. Fac. Sci. Univ. Tokyo Sect. IA, Math. 36 (1989), 297–328. [Lus98] , Bases in equivariant k-...

1987

[6] [6]

Rickard, Derived equivalences as derived functors , J

[Ric91] J. Rickard, Derived equivalences as derived functors , J. Lond. Math. Soc. (2) 43 (1991), 37–48. 34 [S´ ec04]V. S´ echerre,Repr´ esentations lisses deGL(m, D) I: caract` eres simples, Bull. Soc. Math. France 3 (2004), 324–396. [S´ ec05] , Repr´ esentations lisses deGL(m, D) III: types simples , Ann. Scient. ´Ec. Norm. Sup´ er. (4)38 (2005), 951–97...

1991

[7] [7]

[Shi88] , A two-sided cell in an affine Weyl group, II , J. Lond. Math. Soc. 37 (1988), no. 2, 253–264. [Sol13] M. Solleveld, Hochschild homology of affine Hecke algebras , J. Algebra 384 (2013), 1–35. [Sol19] , On completions of Hecke algebras , Progr. Math. 328 (2019), 207–262. [Sol25] , Standard modules and intertwining operators for reductive p-adic g...

work page arXiv 1988

[8] [8]

The definition of the tempered category

[SS99] P. Schneider and U. Stuhler, Appendix “The definition of the tempered category” to: K-types for the tempered components of a p-adic general linear group , J. reine angew. Math 517 (1999), 161–208. [Ste75] R. Steinberg, On a theorem of Pittie , Topology 14 (1975), 173–177. [Suz24] K. Suzuki, Trace Paley-Wiener theorem for Braverman-Kazhdan’s asympto...

work page arXiv 1999