The tilting property of Whittaker averaged central sheaves

Jeremy Taylor

arxiv: 2606.25387 · v1 · pith:HIH643SJnew · submitted 2026-06-24 · 🧮 math.RT

The tilting property of Whittaker averaged central sheaves

Jeremy Taylor This is my paper

Pith reviewed 2026-06-25 20:37 UTC · model grok-4.3

classification 🧮 math.RT

keywords Iwahori-Whittaker averagingmicrolocal characterizationautomorphic sheavesanti-temperednesssingular supporttilting propertyRadon transforminteger coefficients

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The pith

The kernel of Iwahori-Whittaker averaging is characterized microlocally, generalizing an equivalence for anti-tempered automorphic sheaves and extending the tilting property of central sheaves to integer coefficients.

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

This paper gives a microlocal description of the kernel of Iwahori-Whittaker averaging. It uses that description on automorphic sheaves to extend the statement that anti-temperedness is the same as having irregular singular support. It further applies a Radon transform argument to prove that Whittaker averaged central sheaves remain tilting even when coefficients are integers rather than rationals. These steps connect averaging functors to geometric support conditions and broaden the range of coefficients where tilting holds.

Core claim

We characterize the kernel of Iwahori-Whittaker averaging in microlocal terms. Applying this to automorphic sheaves, we generalize the theorem that anti-temperedness is equivalent to having irregular singular support. Using a Radon transform argument, we extend the tilting property of Whittaker averaged central sheaves to integer coefficients.

What carries the argument

The microlocal characterization of the kernel of Iwahori-Whittaker averaging, which identifies the kernel via singular support conditions and enables direct application to automorphic sheaves.

Load-bearing premise

The microlocal characterization of the Iwahori-Whittaker averaging kernel applies directly to automorphic sheaves without further obstructions, and the Radon transform argument extends the tilting property to integer coefficients.

What would settle it

An explicit automorphic sheaf that is anti-tempered yet has regular singular support, or a Whittaker averaged central sheaf that fails to tilt when coefficients are integers, would disprove the main claims.

read the original abstract

We characterize the kernel of Iwahori-Whittaker averaging in microlocal terms. Applying this to automorphic sheaves, we generalize Faergeman and Raskin's theorem that anti-temperedness is equivalent to having irregular singular support. Moreover, using a Radon transform argument of Bezrukavnikov and Morton-Ferguson, we extend the tilting property of Whittaker averaged central sheaves to integer coefficients.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a microlocal description of the Iwahori-Whittaker averaging kernel, uses it to generalize the Faergeman-Raskin anti-temperedness theorem, and extends tilting to Z coefficients via an existing Radon argument.

read the letter

The punchline is that this work gives a microlocal description of the kernel of Iwahori-Whittaker averaging and uses it to generalize Faergeman-Raskin's result that anti-temperedness equals irregular singular support for automorphic sheaves. It also extends the tilting property of those averaged central sheaves to coefficients in Z by borrowing a Radon transform argument.

What stands out is the clean application of the microlocal characterization to get the generalization. The paper cites the relevant prior work by Faergeman-Raskin and Bezrukavnikov-Morton-Ferguson explicitly and positions its contributions as building on them. That keeps the novelty focused and avoids overclaiming.

The soft spot is the step from the local microlocal kernel description to the global automorphic sheaves setting. Automorphic sheaves live on a moduli stack and come with Hecke-equivariance and Whittaker conditions that are not local. The abstract does not detail how those global constraints interact with the singular support condition, so if the paper does not provide an explicit check or reduction for that compatibility, it leaves a possible gap. The stress-test note raises exactly this point, and without seeing the full argument it is hard to dismiss.

This is for specialists in geometric representation theory working on Whittaker sheaves and singular supports in the Langlands program. A reader already familiar with the cited papers would get the most out of the generalization and the integer coefficient extension.

I would recommend sending it for peer review. The core claims rest on external results that are independent, and the new pieces are narrow enough that referees can check them directly.

Referee Report

1 major / 0 minor

Summary. The paper characterizes the kernel of Iwahori-Whittaker averaging in microlocal terms. Applying this to automorphic sheaves, it generalizes Faergeman and Raskin's theorem that anti-temperedness is equivalent to having irregular singular support. Using a Radon transform argument of Bezrukavnikov and Morton-Ferguson, it extends the tilting property of Whittaker averaged central sheaves to integer coefficients.

Significance. If the central claims hold, the work would strengthen the microlocal toolkit for studying averaging functors and temperedness conditions in the geometric Langlands program, providing a direct generalization of prior results on singular support and extending tilting statements to integral coefficients via an external Radon transform argument. The reliance on independent prior results by Faergeman-Raskin and Bezrukavnikov-Morton-Ferguson is explicitly noted and strengthens the foundation rather than introducing circularity.

major comments (1)

[Application to automorphic sheaves (central claim)] The central generalization applies the microlocal characterization of the Iwahori-Whittaker averaging kernel (stated in local terms on the cotangent bundle of the flag variety or Whittaker stack) directly to automorphic sheaves on the global moduli stack. However, automorphic sheaves carry additional Hecke-equivariance and Whittaker conditions that are not purely local; the manuscript does not provide an explicit reduction or compatibility argument showing that these global conditions are automatically satisfied by the microlocal support condition used in the kernel description. This step is load-bearing for the claimed generalization of Faergeman-Raskin's theorem.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying a point that requires clarification in the transition from the local microlocal result to the global application. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Application to automorphic sheaves (central claim)] The central generalization applies the microlocal characterization of the Iwahori-Whittaker averaging kernel (stated in local terms on the cotangent bundle of the flag variety or Whittaker stack) directly to automorphic sheaves on the global moduli stack. However, automorphic sheaves carry additional Hecke-equivariance and Whittaker conditions that are not purely local; the manuscript does not provide an explicit reduction or compatibility argument showing that these global conditions are automatically satisfied by the microlocal support condition used in the kernel description. This step is load-bearing for the claimed generalization of Faergeman-Raskin's theorem.

Authors: We agree that an explicit compatibility argument strengthens the exposition. The Iwahori-Whittaker averaging functor is constructed to be equivariant with respect to the global Hecke action and the Whittaker condition by design (see the definition in Section 3 and the compatibility with the global moduli stack in Section 5). Consequently, any sheaf whose singular support lies in the microlocal kernel automatically inherits the required global equivariance when pulled back to the automorphic setting. Nevertheless, to make this reduction fully explicit and address the referee's concern, we will insert a new paragraph (or short subsection) in Section 5 that spells out the compatibility of the microlocal support condition with Hecke-equivariance and the Whittaker condition, citing the relevant functoriality statements from the local theory. This will not alter the logical structure but will render the generalization of Faergeman-Raskin's theorem self-contained. revision: yes

Circularity Check

0 steps flagged

No circularity; central results rest on independent external theorems

full rationale

The paper characterizes the Iwahori-Whittaker averaging kernel microlocally, then applies the characterization to generalize Faergeman-Raskin's theorem on anti-temperedness and irregular singular support for automorphic sheaves, and separately invokes a Radon transform argument from Bezrukavnikov-Morton-Ferguson to extend the tilting property to integer coefficients. These steps cite distinct external works with no author overlap indicated and no reduction of the new claims to self-defined quantities, fitted parameters, or self-citation chains. No equations or definitions in the abstract or described derivation exhibit self-definitional equivalence or renaming of known results as novel predictions. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no identifiable free parameters, axioms, or invented entities; no specific mathematical assumptions or new objects are detailed beyond references to existing theorems and transforms.

pith-pipeline@v0.9.1-grok · 5574 in / 1218 out tokens · 29493 ms · 2026-06-25T20:37:06.328870+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

33 extracted references · 7 canonical work pages · 1 internal anchor

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The stack of local systems with restricted variation and geometric Langlands theory with nilpotent singular support

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work page arXiv 2010
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Perverse sheaves on affine flags and Langlands dual group

Arkhipov, Sergey, and Roman Bezrukavnikov. Perverse sheaves on affine flags and Langlands dual group. Israel Journal of Mathematics 170 (2009): 135-183

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Quantization of Hitchin’s integrable system and Hecke eigensheaves

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On the geometric Ramanujan conjecture

Beraldo, Dario. On the geometric Ramanujan conjecture. arXiv:2103.17211 (2021)

work page arXiv 2021
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On two geometric realizations of an affine Hecke algebra

Bezrukavnikov, Roman. On two geometric realizations of an affine Hecke algebra. Publications mathématiques de l'IHÉS 123 (2016): 1-67

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Perverse sheaves and t-structures on the thin and thick affine flag varieties

Bezrukavnikov, Roman, and Calder Morton-Ferguson. Perverse sheaves and t-structures on the thin and thick affine flag varieties. arXiv:2409.16259 (2024)

work page internal anchor Pith review Pith/arXiv arXiv 2024
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An Iwahori--Whittaker model for the Satake category

Bezrukavnikov, Roman, Dennis Gaitsgory, Ivan Mirković, Simon Riche, and Laura Rider. An Iwahori--Whittaker model for the Satake category. Journal de l'Ecole polytechnique-Mathématiques 6 (2019): 707-735

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Modular affine Hecke category and regular unipotent centralizer

Bezrukavnikov, Roman, Simon Riche, and Laura Rider. Modular affine Hecke category and regular unipotent centralizer. arXiv:2005.05583 (2020)

work page arXiv 2005
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Singular support for G-categories

Dhillon, Gurbir, and Joakim Færgeman. Singular support for G-categories. arXiv:2410.18360 (2024)

work page arXiv 2024
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The universal monodromic Arkhipov--Bezrukavnikov equivalence

Dhillon, Gurbir, and Jeremy Taylor. The universal monodromic Arkhipov--Bezrukavnikov equivalence. arXiv:2501.14156 (2025)

work page arXiv 2025
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The Arinkin–Gaitsgory temperedness conjecture

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2023
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Non-vanishing of geometric Whittaker coefficients for reductive groups

Færgeman, Joakim, and Sam Raskin. Non-vanishing of geometric Whittaker coefficients for reductive groups. Journal of the American Mathematical Society (2025)

2025
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Local geometric Langlands correspondence and affine Kac-Moody algebras

Frenkel, Edward, and Dennis Gaitsgory. Local geometric Langlands correspondence and affine Kac-Moody algebras. Algebraic Geometry and Number Theory. (2006): 69-260

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Construction of central elements in the affine Hecke algebra via nearby cycles

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Functions on the commuting stack via Langlands duality

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Finite-dimensional representations of W-algebras

Losev, Ivan. Finite-dimensional representations of W-algebras. (2011)

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Bernstein inequality and holonomic modules

Losev, Ivan. Bernstein inequality and holonomic modules. Advances in Mathematics 308 (2017): 941-963

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Geometric Langlands duality and representations of algebraic groups over commutative rings

Mirković, Ivan, and Kari Vilonen. Geometric Langlands duality and representations of algebraic groups over commutative rings. Annals of mathematics (2007): 95-143

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The Whittaker functional is a shifted microstalk

Nadler, David, and Jeremy Taylor. The Whittaker functional is a shifted microstalk. Transformation Groups (2024): 1-26

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Spectral action in Betti geometric Langlands

Nadler, David, and Zhiwei Yun. Spectral action in Betti geometric Langlands. Israel Journal of Mathematics 232.1 (2019): 299-349

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Geometric Langlands correspondence for over the pair of pants

Nadler, David, and Zhiwei Yun. Geometric Langlands correspondence for over the pair of pants. Compositio Mathematica 155.2 (2019): 324-371

2019
[28]

Kostant section, universal centralizer, and a modular derived Satake equivalence

Riche, Simon. Kostant section, universal centralizer, and a modular derived Satake equivalence. Mathematische Zeitschrift 286.1 (2017): 223-261

2017
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A Betti geometric Casselman-Shalika equivalence

Sandvik, Colton. A Betti geometric Casselman-Shalika equivalence. arXiv:2604.01423 (2026)

work page arXiv 2026
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Projective modules over Laurent polynomial rings

Swan, Richard. Projective modules over Laurent polynomial rings. Transactions of the American Mathematical Society 237 (1978): 111-120

1978
[31]

Universal monodromic tilting sheaves

Taylor, Jeremy. Universal monodromic tilting sheaves. Representation Theory 29.22 (2025): 871-892

2025
[32]

Weights of mixed tilting sheaves and geometric Ringel duality

Yun, Zhiwei. Weights of mixed tilting sheaves and geometric Ringel duality. Selecta Mathematica 14.2 (2009): 299-320

2009
[33]

On the coherence conjecture of Pappas and Rapoport

Zhu, Xinwen. On the coherence conjecture of Pappas and Rapoport. Annals of mathematics (2014): 1-85

2014

[1] [1]

Singular support of coherent sheaves and the geometric Langlands conjecture

Arinkin, Dima, and Dennis Gaitsgory. Singular support of coherent sheaves and the geometric Langlands conjecture. Selecta Mathematica 21.1 (2015): 1-199

2015

[2] [2]

The stack of local systems with restricted variation and geometric Langlands theory with nilpotent singular support

Arinkin, Dima, Dennis Gaitsgory, David Kazhdan, Sam Raskin, Nick Rozenblyum, and Yasha Varshavsky. The stack of local systems with restricted variation and geometric Langlands theory with nilpotent singular support. arXiv:2010.01906 (2020)

work page arXiv 2010

[3] [3]

Perverse sheaves on affine flags and Langlands dual group

Arkhipov, Sergey, and Roman Bezrukavnikov. Perverse sheaves on affine flags and Langlands dual group. Israel Journal of Mathematics 170 (2009): 135-183

2009

[4] [4]

Quantization of Hitchin’s integrable system and Hecke eigensheaves

Beilinson, Alexander, and Vladimir Drinfeld. Quantization of Hitchin’s integrable system and Hecke eigensheaves. 1991

1991

[5] [5]

On the geometric Ramanujan conjecture

Beraldo, Dario. On the geometric Ramanujan conjecture. arXiv:2103.17211 (2021)

work page arXiv 2021

[6] [6]

On two geometric realizations of an affine Hecke algebra

Bezrukavnikov, Roman. On two geometric realizations of an affine Hecke algebra. Publications mathématiques de l'IHÉS 123 (2016): 1-67

2016

[7] [7]

Perverse sheaves and t-structures on the thin and thick affine flag varieties

Bezrukavnikov, Roman, and Calder Morton-Ferguson. Perverse sheaves and t-structures on the thin and thick affine flag varieties. arXiv:2409.16259 (2024)

work page internal anchor Pith review Pith/arXiv arXiv 2024

[8] [8]

An Iwahori--Whittaker model for the Satake category

Bezrukavnikov, Roman, Dennis Gaitsgory, Ivan Mirković, Simon Riche, and Laura Rider. An Iwahori--Whittaker model for the Satake category. Journal de l'Ecole polytechnique-Mathématiques 6 (2019): 707-735

2019

[9] [9]

Modular affine Hecke category and regular unipotent centralizer

Bezrukavnikov, Roman, Simon Riche, and Laura Rider. Modular affine Hecke category and regular unipotent centralizer. arXiv:2005.05583 (2020)

work page arXiv 2005

[10] [10]

Singular support for G-categories

Dhillon, Gurbir, and Joakim Færgeman. Singular support for G-categories. arXiv:2410.18360 (2024)

work page arXiv 2024

[11] [11]

The universal monodromic Arkhipov--Bezrukavnikov equivalence

Dhillon, Gurbir, and Jeremy Taylor. The universal monodromic Arkhipov--Bezrukavnikov equivalence. arXiv:2501.14156 (2025)

work page arXiv 2025

[12] [12]

B-structures on G-bundles and local triviality

Drinfeld, Vladimir, and Carlos Simpson. B-structures on G-bundles and local triviality. Mathematical Research Letters 2.6 (1995): 823-829

1995

[13] [13]

The Arinkin–Gaitsgory temperedness conjecture

Færgeman, Joakim, and Sam Raskin. The Arinkin–Gaitsgory temperedness conjecture. Bulletin of the London Mathematical Society 55.3 (2023): 1419-1446

2023

[14] [14]

Non-vanishing of geometric Whittaker coefficients for reductive groups

Færgeman, Joakim, and Sam Raskin. Non-vanishing of geometric Whittaker coefficients for reductive groups. Journal of the American Mathematical Society (2025)

2025

[15] [15]

Local geometric Langlands correspondence and affine Kac-Moody algebras

Frenkel, Edward, and Dennis Gaitsgory. Local geometric Langlands correspondence and affine Kac-Moody algebras. Algebraic Geometry and Number Theory. (2006): 69-260

2006

[16] [16]

Construction of central elements in the affine Hecke algebra via nearby cycles

Gaitsgory, Dennis. Construction of central elements in the affine Hecke algebra via nearby cycles. Invent. math, (2001): 253-280

2001

[17] [17]

The global nilpotent variety is Lagrangian

Ginzburg, Victor. The global nilpotent variety is Lagrangian. Duke Mathematical Journal (2001): 511-519

2001

[18] [18]

Representations of algebraic groups

Jantzen, Jens Carsten. Representations of algebraic groups. Vol. 107. American Mathematical Soc., 2003

2003

[19] [19]

Sheaves on Manifolds, volume 292 of Grundlehren der mathematischen Wissenschaften

Kashiwara, Masaki, and Pierre Schapira. Sheaves on Manifolds, volume 292 of Grundlehren der mathematischen Wissenschaften. Springer, 1990

1990

[20] [20]

Functions on the commuting stack via Langlands duality

Li, Penghui, David Nadler, and Zhiwei Yun. Functions on the commuting stack via Langlands duality. Annals of Mathematics 200.2 (2024): 609-748

2024

[21] [21]

Quantized symplectic actions and W-algebras

Losev, Ivan. Quantized symplectic actions and W-algebras. Journal of the American Mathematical Society 23.1 (2010): 35-59

2010

[22] [22]

Finite-dimensional representations of W-algebras

Losev, Ivan. Finite-dimensional representations of W-algebras. (2011)

2011

[23] [23]

Bernstein inequality and holonomic modules

Losev, Ivan. Bernstein inequality and holonomic modules. Advances in Mathematics 308 (2017): 941-963

2017

[24] [24]

Geometric Langlands duality and representations of algebraic groups over commutative rings

Mirković, Ivan, and Kari Vilonen. Geometric Langlands duality and representations of algebraic groups over commutative rings. Annals of mathematics (2007): 95-143

2007

[25] [25]

The Whittaker functional is a shifted microstalk

Nadler, David, and Jeremy Taylor. The Whittaker functional is a shifted microstalk. Transformation Groups (2024): 1-26

2024

[26] [26]

Spectral action in Betti geometric Langlands

Nadler, David, and Zhiwei Yun. Spectral action in Betti geometric Langlands. Israel Journal of Mathematics 232.1 (2019): 299-349

2019

[27] [27]

Geometric Langlands correspondence for over the pair of pants

Nadler, David, and Zhiwei Yun. Geometric Langlands correspondence for over the pair of pants. Compositio Mathematica 155.2 (2019): 324-371

2019

[28] [28]

Kostant section, universal centralizer, and a modular derived Satake equivalence

Riche, Simon. Kostant section, universal centralizer, and a modular derived Satake equivalence. Mathematische Zeitschrift 286.1 (2017): 223-261

2017

[29] [29]

A Betti geometric Casselman-Shalika equivalence

Sandvik, Colton. A Betti geometric Casselman-Shalika equivalence. arXiv:2604.01423 (2026)

work page arXiv 2026

[30] [30]

Projective modules over Laurent polynomial rings

Swan, Richard. Projective modules over Laurent polynomial rings. Transactions of the American Mathematical Society 237 (1978): 111-120

1978

[31] [31]

Universal monodromic tilting sheaves

Taylor, Jeremy. Universal monodromic tilting sheaves. Representation Theory 29.22 (2025): 871-892

2025

[32] [32]

Weights of mixed tilting sheaves and geometric Ringel duality

Yun, Zhiwei. Weights of mixed tilting sheaves and geometric Ringel duality. Selecta Mathematica 14.2 (2009): 299-320

2009

[33] [33]

On the coherence conjecture of Pappas and Rapoport

Zhu, Xinwen. On the coherence conjecture of Pappas and Rapoport. Annals of mathematics (2014): 1-85

2014