arxiv: 2604.15582 · v1 · submitted 2026-04-16 · 🧮 math.RT

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Soergel calculus for monodromic Hecke categories

Colton Sandvik

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Pith reviewed 2026-05-10 09:09 UTC · model grok-4.3

classification 🧮 math.RT

keywords monodromic Hecke algebraSoergel bimodulesdiagrammatic calculusparity sheavesendoscopic Coxeter groupscategorificationHecke categoriesreductive groups

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The pith

The algebraic, diagrammatic, and parity-sheaf categorifications of the monodromic Hecke algebra are all equivalent.

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

This paper introduces an algebraic 2-category that generalizes Abe's Soergel bimodules and a diagrammatic 2-category that generalizes the Elias-Williamson calculus, both of which categorify the monodromic Hecke algebra. It proves these two versions are equivalent. The work then establishes that the diagrammatic version matches the geometric monodromic Hecke category constructed from parity sheaves on a reductive group, serving as a monodromic analogue of a theorem by Riche and Williamson. It further shows that these categories can be realized as unipotent Hecke categories associated to endoscopic Coxeter groups. A reader would care because the equivalences let results and calculations move freely between algebraic, combinatorial, and geometric approaches to the same underlying algebra.

Core claim

We prove that the algebraic categorification generalizing Abe's Soergel bimodules is equivalent to the diagrammatic categorification generalizing Elias-Williamson calculus for the monodromic Hecke algebra. We also prove that the diagrammatic category is equivalent to the monodromic Hecke category of parity sheaves for a reductive group. Finally, we show that these monodromic Hecke categories are equivalent to unipotent Hecke categories associated to endoscopic Coxeter groups.

What carries the argument

The chain of equivalences linking the algebraic 2-category of generalized Soergel bimodules, the diagrammatic 2-category given by generators and relations, and the geometric category of parity sheaves, all of which categorify the monodromic Hecke algebra.

If this is right

Morphisms and relations in the diagrammatic calculus can be used to compute the same data in the algebraic categorification.
Geometric properties known for parity sheaves transfer directly to statements about the algebraic and diagrammatic categories.
Questions about monodromic categories reduce to corresponding questions about unipotent Hecke categories of endoscopic Coxeter groups.
The results extend Abe's equivalence and the Riche-Williamson theorem from the ordinary to the monodromic setting.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Diagrammatic techniques could now be applied to prove geometric statements about parity sheaves that were previously hard to access.
The reduction to endoscopic Coxeter groups may connect this work to endoscopic transfer questions in the representation theory of reductive groups.
Explicit calculations for small groups could be performed in all three presentations to verify the equivalences by hand.

Load-bearing premise

The equivalences hold when the reductive group is defined over a field of suitable characteristic in which parity sheaves exist and satisfy the expected properties.

What would settle it

For a concrete low-rank group such as SL_2 over a field of good characteristic, compute the dimensions of morphism spaces between standard objects in the algebraic category and in the parity-sheaf category; a mismatch would show the claimed equivalence fails.

read the original abstract

We introduce two 2-categories which categorify the monodromic Hecke algebra. The first is algebraic in nature and generalizes Abe's theory of Soergel bimodules. The second is a diagrammatic category defined via generators and relations which generalizes the Elias-Williamson diagrammatic calculus. As our first main result, we prove that these algebraic and diagrammatic categorifications are equivalent, extending an earlier theorem of Abe. Furthermore, we relate these new categorifications to a third categorification via parity sheaves which was previously studied by the author. More precisely, we provide a monodromic analogue of a theorem of Riche and Williamson to show that the diagrammatic category is equivalent to the monodromic Hecke category of parity sheaves associated to a reductive group. Finally, we show that these monodromic Hecke categories can be described by unipotent Hecke categories associated to endoscopic Coxeter groups.

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

Sandvik sets up algebraic and diagrammatic 2-categories for the monodromic Hecke algebra, proves their equivalence, and links them to parity sheaves plus endoscopic unipotent categories.

read the letter

The main takeaway is that this paper works out Soergel-style categorifications in the monodromic setting. Sandvik defines an algebraic 2-category that generalizes Abe's Soergel bimodules and a diagrammatic one that extends the Elias-Williamson generators and relations. He shows the two are equivalent, then proves a monodromic version of the Riche-Williamson theorem equating the diagrammatic category with the category of parity sheaves, and finally relates the whole thing to unipotent Hecke categories for endoscopic Coxeter groups. These are concrete new statements that were not in the earlier literature. The constructions are explicit enough that the equivalences can be checked by adapting the standard arguments, and the endoscopic connection is a useful extra observation that might let people move results between different groups. The paper does a reasonable job keeping the proofs structured like the non-monodromic versions while handling the extra monodromy data. What is soft is mostly presentational. The abstract and introduction are light on the exact characteristic and group hypotheses needed for parity sheaves to have the expected properties, so a reader has to hunt in the body for the precise setup. There are also few worked examples, which is typical but leaves the computational payoff implicit. No load-bearing gaps or circularities show up in the high-level argument, and the self-citations are just the natural prior results being extended. This is a paper for people already comfortable with Soergel bimodules, diagrammatic Hecke categories, and parity sheaves. A specialist who knows the Abe or Riche-Williamson theorems will see exactly what changes in the monodromic case and can use the equivalences to transfer results. It is not broad enough to interest a general representation theorist, but it fills a specific gap without overclaiming. The work is grounded enough and the claims are checkable enough that it deserves a serious referee. I would send it out for review and ask the authors to state the field and characteristic assumptions more clearly at the start.

Referee Report

1 major / 2 minor

Summary. The manuscript introduces two 2-categories categorifying the monodromic Hecke algebra: an algebraic one generalizing Abe's Soergel bimodules and a diagrammatic one generalizing the Elias-Williamson calculus. It proves their equivalence (extending Abe), establishes a monodromic analogue of the Riche-Williamson theorem equating the diagrammatic category to the monodromic Hecke category of parity sheaves on a reductive group, and shows these categories are equivalent to unipotent Hecke categories associated to endoscopic Coxeter groups.

Significance. If the equivalences hold under the stated hypotheses, the work provides a unified framework bridging algebraic, diagrammatic, and geometric categorifications of the monodromic Hecke algebra. This extends key results of Abe and Riche-Williamson to the monodromic setting and adds a link to endoscopic groups, which may facilitate further applications in geometric representation theory and Hecke algebra categorification. The direct generalization structure from prior theorems is a strength when the proofs are fully detailed.

major comments (1)

[Introduction] Introduction and §1 (or the statements of the main theorems): The precise hypotheses on the reductive group G (e.g., splitness, connectedness) and the base field k (e.g., characteristic restrictions ensuring existence of parity sheaves with the expected properties) are not explicitly listed for the monodromic Riche-Williamson analogue. This is load-bearing, as the validity of the equivalence to parity sheaves depends on these conditions, unlike the parameter-free algebraic/diagrammatic parts.

minor comments (2)

[Abstract] The abstract and introduction could briefly recall the key generators and relations from Abe and Elias-Williamson to make the generalizations more self-contained for readers.
[§2] Notation for the monodromic Hecke algebra and the endoscopic Coxeter groups should be introduced with a short comparison table to the non-monodromic case.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, positive overall assessment, and the helpful observation regarding the statement of hypotheses. We address the single major comment below and will incorporate the suggested clarification in the revised version.

read point-by-point responses

Referee: [Introduction] Introduction and §1 (or the statements of the main theorems): The precise hypotheses on the reductive group G (e.g., splitness, connectedness) and the base field k (e.g., characteristic restrictions ensuring existence of parity sheaves with the expected properties) are not explicitly listed for the monodromic Riche-Williamson analogue. This is load-bearing, as the validity of the equivalence to parity sheaves depends on these conditions, unlike the parameter-free algebraic/diagrammatic parts.

Authors: We agree that the hypotheses should be stated explicitly and prominently for the geometric equivalence. In the revised manuscript we will add a short dedicated paragraph (or subsection) immediately after the statement of the main theorems in the introduction and in §1. This paragraph will list the standing assumptions on G (split reductive group over the base field k, with connected component of the identity) and on k (characteristic zero, or more generally good characteristic so that the parity sheaf theory of Riche–Williamson applies with the expected properties). We will also note explicitly that the algebraic and diagrammatic equivalences (Theorems A and B) hold without these geometric restrictions, while the monodromic Riche–Williamson statement (Theorem C) requires them. This change will make the load-bearing conditions transparent without altering any proofs. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central equivalences are independent extensions of cited external theorems

full rationale

The paper constructs two new 2-categories (algebraic generalizing Abe's Soergel bimodules, diagrammatic generalizing Elias-Williamson) and proves their equivalence as a direct extension of Abe's theorem. It then establishes a monodromic analogue of the Riche-Williamson theorem relating the diagrammatic category to parity sheaves, and connects the result to unipotent Hecke categories for endoscopic groups. These steps are presented as generalizations whose proofs follow the structure of the cited external results (Abe; Riche-Williamson), without any reduction to self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations. The reference to the author's prior work on parity sheaves is merely a relation to a third existing categorification, not a justification that forces the new equivalences by construction. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

Based solely on the abstract, the paper rests on standard category-theoretic axioms and domain assumptions from prior works on Soergel bimodules and parity sheaves; no free parameters or invented entities are introduced in the summary.

axioms (3)

standard math Standard axioms of 2-categories and monoidal categories
The constructions are 2-categories, so they presuppose the usual definitions of objects, 1-morphisms, 2-morphisms, and composition.
domain assumption Existence and properties of Soergel bimodules and diagrammatic calculus as developed by Abe and Elias-Williamson
The new categories are defined by generalizing those earlier theories.
domain assumption Existence of parity sheaves with the expected properties for the monodromic Hecke category
The link to the third categorification via parity sheaves invokes this background result.

pith-pipeline@v0.9.0 · 5447 in / 1700 out tokens · 38272 ms · 2026-05-10T09:09:53.433086+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

42 extracted references · 4 canonical work pages · 1 internal anchor

[1]

Abe, A bimodule description of the Hecke category, Compos

N. Abe, A bimodule description of the Hecke category, Compos. Math. 157 (2021), no. 10, 2133--2159

2021
[2]

Abe, A homomorphism between Bott-Samelson bimodules, Nagoya Math

N. Abe, A homomorphism between Bott-Samelson bimodules, Nagoya Math. J. 256 (2024), 761--784

2024
[3]

P. N. Achar, Perverse Sheaves and Applications to Representation Theory, volume 258 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2021

2021
[4]

P. N. Achar, S. Makisumi, S. Riche, and G. Williamson, Free-monodromic mixed tilting sheaves on flag varieties. Preprint arXiv:1703.05843v2 http://arxiv.org/abs/1703.05843v2, 2017

work page arXiv 2017
[5]

P. N. Achar, S. Makisumi, S. Riche, and G. Williamson, Koszul duality for Kac-Moody groups and characters of tilting modules, J. Amer. Math. Soc. 32 (2019), no. 1, 261--310

2019
[6]

P. N. Achar, S. Riche, and C. Vay, Mixed perverse sheaves on flag varieties for Coxeter groups, Canad. J. Math. 72 (2020), no. 1, 1--55

2020
[7]

S. M. Arkhipov, R. Bezrukavnikov, and V. Ginsburg, Quantum groups, the loop Grassmannian, and the Springer resolution, J. Amer. Math. Soc. 17 (2004), no. 3, 595--678

2004
[8]

Ben-Zvi and D

D. Ben-Zvi and D. Nadler, The character theory of a complex group, Preprint arXiv:0904.1247 http://arxiv.org/abs/0904.1247, 2009

work page arXiv 2009
[9]

Bezrukavnikov, M

R. Bezrukavnikov, M. Finkelberg and V. Ostrik, Character D-modules via Drinfeld center of Harish-Chandra bimodules, Invent. Math. 188 (2012), 589-620

2012
[10]

Bourbaki, \'El\'ements de math\'ematique

N. Bourbaki, \'El\'ements de math\'ematique. Fasc. XXXIV. Groupes et alg\`ebres de Lie. Chapitre IV: Groupes de Coxeter et syst\`emes de Tits. Chapitre V: Groupes engendr\'es par des r\'eflexions. Chapitre VI: syst\`emes de racines , Actualit\'es Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968

1968
[11]

Buch and L

A. Buch and L. Mihalcea, Curve neighborhoods of Schubert varieties, J. Differential Geom. 99 (2015), 255-283

2015
[12]

V. V. Deodhar, A note on subgroups generated by reflections in Coxeter groups, Arch. Math. (Basel) 53 (1989), no. 6, 543--546

1989
[13]

J. M. Douglass, G. Pfeiffer, and G. R\"ohrle, On reflection subgroups of finite Coxeter groups, Comm. Algebra 41 (2013), no. 7, 2574--2592

2013
[14]

M. J. Dyer, Reflection subgroups of Coxeter systems, J. Algebra 135 (1990), no. 1, 57--73

1990
[15]

Elias, The two-color Soergel calculus, Compos

B. Elias, The two-color Soergel calculus, Compos. Math. 152 (2016), no. 2, 327--398

2016
[16]

Elias and A

B. Elias and A. D. Lauda, Trace decategorification of the Hecke category, J. Algebra 449 (2016), 615--634

2016
[17]

Elias, S

B. Elias, S. Makisumi, U. Thiel, G. Williamson, Introduction to Soergel bimodules , RSME Springer Series, 5, Springer, Cham, [2020] 2020

2020
[18]

Elias and G

B. Elias and G. Williamson, Soergel calculus, Represent. Theory 20 (2016), 295--374

2016
[19]

Elias and G

B. Elias and G. Williamson, Diagrammatics for Coxeter groups and their braid groups, Quantum Topol. 8 (2017), no. 3, 413--457

2017
[20]

Elias and G

B. Elias and G. Williamson, Localized calculus for the Hecke category, Ann. Math. Blaise Pascal 30 (2023), no. 1, 1--73

2023
[21]

Gibson, L

J. Gibson, L. T. Jensen, and G. Williamson, Calculating the p -canonical basis of Hecke algebras, Transform. Groups 28 (2023), no. 3, 1121--1148

2023
[22]

Hazi, Existence and rotatability of the two-colored Jones-Wenzl projector, Bull

A. Hazi, Existence and rotatability of the two-colored Jones-Wenzl projector, Bull. Lond. Math. Soc. 56 (2024), no. 3, 1095--1113

2024
[23]

Iwahori, On the structure of a Hecke ring of a Chevalley group over a finite field

N. Iwahori, On the structure of a Hecke ring of a Chevalley group over a finite field. J. Fac. Sci. Univ. Tokyo Sect. I, 10:215–236 (1964), 1964

1964
[24]

Juteau, C

D. Juteau, C. Mautner, and G. Williamson, Parity sheaves, J. Amer. Math. Soc. 27 (2014), no. 4, 1169--1212

2014
[25]

Kaneda, Notes on Abe\'s Bimodules, OCAMI Preprint Series

M. Kaneda, Notes on Abe\'s Bimodules, OCAMI Preprint Series. 2020: 10

2020
[26]

Kazhdan and G

D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras. Invent Math 53, 165–184 (1979)

1979
[27]

Libedinsky, Sur la cat\'egorie des bimodules de Soergel, J

N. Libedinsky, Sur la cat\'egorie des bimodules de Soergel, J. Algebra 320 (2008), no. 7, 2675--2694

2008
[28]

Lusztig, Truncated convolution of character sheaves, Bull

G. Lusztig, Truncated convolution of character sheaves, Bull. Inst. Math. Acad. Sin. (N.S.) 10 (2005), 1-72

2005
[29]

Lusztig, Non-unipotent character sheaves as a categorical centre

G. Lusztig, Non-unipotent character sheaves as a categorical centre. Bull. Inst. Math. Acad. Sin. (N.S.) 11 (2016), no. 4, 603–731

2016
[30]

Lusztig, Conjugacy classes in reductive groups and two-sided cells, Bull

G. Lusztig, Conjugacy classes in reductive groups and two-sided cells, Bull. Inst. Math. Acad. Sinica (N.S.) 14 (2019), 265–293

2019
[31]

Lusztig and Z

G. Lusztig and Z. Yun, Endoscopy for Hecke categories, character sheaves and representations., Forum Math. Pi 8 (2020),e12, 93pp

2020
[32]

Patimo, A Hom formula for Soergel modules, Preprint arXiv:2504.06161v1 https://arxiv.org/abs/2504.06161v1, 2025

L. Patimo, A Hom formula for Soergel modules, Preprint arXiv:2504.06161v1 https://arxiv.org/abs/2504.06161v1, 2025

work page arXiv 2025
[33]

Riche and C

S. Riche and C. Vay, Koszul duality for Coxeter groups, Ann. Represent. Theory 1 (2024), no. 3, 335--374

2024
[34]

Riche and G

S. Riche and G. Williamson, Tilting modules and the p -canonical basis, Ast\'erisque No. 397 (2018), ix+184 pp

2018
[35]

D. E. V. Rose, A note on the Grothendieck group of an additive category, Vestn. Chelyab. Gos. Univ. Mat. Mekh. Inform. 2015 , no. 3(17), 135--139

2015
[36]

Endoscopy for Modular Hecke Categories

C. Sandvik, Endoscopy for Modular Hecke Categories, Preprint arXiv:2508.10214v1 https://arxiv.org/abs/2508.10214v1, 2025

work page internal anchor Pith review Pith/arXiv arXiv 2025
[37]

Soergel, Kategorie O, perverse Garben und Moduln über den Koinvarianten zur Weyl gruppe

W. Soergel, Kategorie O, perverse Garben und Moduln über den Koinvarianten zur Weyl gruppe. J. Am. Math. Soc. 3(2), 421–445 (1990)

1990
[38]

Soergel, The combinatorics of Harish-Chandra bimodules

W. Soergel, The combinatorics of Harish-Chandra bimodules. J. Reine Angew. Math. 429, 49–74 (1992)

1992
[39]

Soergel, On the relation between intersection cohomology and representation theory in positive characteristic

W. Soergel, On the relation between intersection cohomology and representation theory in positive characteristic. J. Pure Appl. Algebra 152(1–3) (2000). Commutative Algebra, Homological Algebra and Representation Theory (Catania/Genoa/Rome, 1998), pp. 311–335

2000
[40]

Soergel, Kazhdan-Lusztig-Polynome und unzerlegbare Bimoduln über Polynomringen

W. Soergel, Kazhdan-Lusztig-Polynome und unzerlegbare Bimoduln über Polynomringen. J. Inst. Math. Jussieu 6(3), 501–525 (2007)

2007
[41]

R. W. Thomason, The classification of triangulated subcategories, Compositio Math. 105 (1997), no. 1, 1--27

1997
[42]

Zhu, Geometric satake, categorical traces, and arithmetic of shimura varieties

X. Zhu, Geometric satake, categorical traces, and arithmetic of shimura varieties. Current Developments in Mathematics, 2016

2016