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On $\mathbb J$-strata with Parahoric Stabilizers in Affine Deligne-Lusztig Varieties

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Pith's one-line read J-strata with parahoric stabilizers in basic affine Deligne-Lusztig varieties are parametrized by small cocharacters, with cardinality matching a Weyl group orbit subset of μ.

desk verdict The paper gives a bijection from J-strata with parahoric stabilizers to small cocharacters in basic affine Deligne-Lusztig varieties and shows the count matches a subset of the Weyl orbit of μ. read the letter →

arxiv 2606.03062 v1 pith:E5JXBWWV submitted 2026-06-02 math.AG math.NT

Ryosuke Shimada This is my paper · ORCID
classification math.AGmath.NT
keywords affineDeligne-LusztigvarietiesJ-stratificationparahoricstabilizerssmallcocharactersWeylgrouporbitminusculecocharacter
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 examines the J-stratification on basic affine Deligne-Lusztig varieties for a minuscule cocharacter μ. It builds a natural bijection that identifies the J-strata whose stabilizers in the Frobenius-twisted centralizer are parahoric with combinatorial objects called small cocharacters. It further establishes that the number of these strata equals the size of a designated subset inside the Weyl group orbit of μ. This supplies a combinatorial handle on the strata inside the setup of p-adic reductive groups.

What carries the argument

Natural bijection from J-strata with parahoric stabilizers to small cocharacters.

What would settle it

An explicit J-stratum whose parahoric stabilizer does not map to any small cocharacter, or a direct count showing the two sets have different sizes, would refute the parametrization and cardinality claim.

Watch

Extended reading notes

Core claim

We construct a natural bijection between the J-strata with parahoric stabilizers and small cocharacters, and prove that the cardinality of these sets equals that of a certain subset of the Weyl group orbit of μ.

Load-bearing premise

The J-stratification of Chen-Viehmann applies to basic affine Deligne-Lusztig varieties for minuscule μ, with the Frobenius-twisted centralizer group and parahoric stabilizers following standard behavior.

Editorial extensions

If this is right

  • The parahoric J-strata admit a complete combinatorial classification via small cocharacters.
  • Their total number is governed by the Weyl group action on the cocharacter μ.
  • A link exists to the weakly fully Hodge-Newton decomposability condition studied by Chen-Tong.

Reading between the lines

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

  • The bijection may supply a counting tool for points in the basic locus of associated Shimura varieties.
  • The same combinatorial reduction could be tested on non-minuscule cocharacters or on other stratifications of the affine Grassmannian.
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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 / 2 minor

Summary. The paper studies the J-stratification of basic affine Deligne-Lusztig varieties for a minuscule cocharacter μ, introduced by Chen-Viehmann. It parametrizes the J-strata whose stabilizers in the Frobenius-twisted centralizer group are parahoric by constructing a natural bijection to combinatorial invariants called small cocharacters. It further proves that the cardinality of these sets equals that of a certain subset of the Weyl group orbit of μ, and discusses a relationship with the weakly fully Hodge-Newton decomposability of Chen-Tong.

Significance. If the bijection and cardinality equality hold, the results supply a combinatorial parametrization of selected J-strata that may serve as a tool for studying basic loci in Shimura varieties, extending the Chen-Viehmann framework. The explicit link to a subset of the Weyl orbit of μ and the connection to Hodge-Newton decomposability are potentially useful for explicit computations in the theory of affine Deligne-Lusztig varieties.

minor comments (2)
  1. The abstract introduces 'small cocharacters' as combinatorial invariants without indicating whether this notion is defined in the paper or drawn from prior literature; a clear definition or reference in §1 or §2 would improve readability.
  2. The relationship with Chen-Tong's weakly fully Hodge-Newton decomposability is mentioned but not detailed in the abstract; specifying the precise statement or theorem number where this is discussed would clarify the contribution.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for summarizing our results on the J-stratification of basic affine Deligne-Lusztig varieties and for noting their potential utility in studying basic loci in Shimura varieties. The significance assessment is appreciated. No major comments appear in the report, so we have no individual points requiring response or revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity detected

full rationale

The paper's central results consist of constructing a bijection parametrizing certain J-strata by small cocharacters and proving a cardinality equality with a subset of the Weyl orbit of μ. These steps are presented as new constructions building on the externally introduced J-stratification of Chen-Viehmann (and a relation to Chen-Tong), with no equations or claims reducing by definition to fitted inputs, self-citations, or ansatzes from the authors' own prior work. The derivation chain remains self-contained against the cited external benchmarks.

Assumptions & free parameters 0 free parameters · 2 assumptions · 1 invented entities

Work rests on the prior definition of the J-stratification and standard facts about minuscule cocharacters, Weyl groups, and parahoric subgroups in p-adic groups; introduces small cocharacters as a new labeling device.

assumptions (2)
  • domain assumption J-stratification of basic affine Deligne-Lusztig varieties for minuscule μ as defined by Chen-Viehmann
    The paper studies this specific stratification.
  • standard math Standard properties of Frobenius-twisted centralizer groups and parahoric stabilizers in reductive groups over local fields
    Invoked when defining the strata and their stabilizers.
invented entities (1)
  • small cocharacters
    purpose: Combinatorial invariants providing the bijection for parametrizing the parahoric J-strata
    Newly employed here to label the strata; no independent evidence supplied in abstract.

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Pith. "Pith review of On $\mathbb J$-strata with Parahoric Stabilizers in Affine Deligne-Lusztig Varieties." pith.science (2026). https://pith.science/paper/E5JXBWWV

@misc{pith2026260603062,
  author       = {Pith},
  title        = {Pith review of: On $\mathbb J$-strata with Parahoric Stabilizers in Affine Deligne-Lusztig Varieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E5JXBWWV}},
  note         = {Machine review of arXiv:2606.03062}
}
abstract

In this paper, we study the $\mathbb J$-stratification of basic affine Deligne-Lusztig varieties for a minuscule cocharacter $\mu$. This stratification was introduced by Chen-Viehmann and has been expected to serve as an interesting tool for studying basic loci in Shimura varieties. We parametrize the $\mathbb J$-strata whose stabilizers in the Frobenius-twisted centralizer group are parahoric by constructing a natural bijection to combinatorial invariants called small cocharacters. We further prove that the cardinality of these sets is equal to that of a certain subset of the Weyl group orbit of $\mu$. A relationship with the weakly fully Hodge-Newton decomposability of Chen-Tong is also discussed.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

55 extracted references · 10 canonical work pages

  1. [1]

    Armstrong, N

    D. Armstrong, N. A. Loehr, and G. S. Warrington,Sweep maps: a continuous family of sorting algorithms, Adv. Math.284(2015), 159–185

    2015

  2. [2]

    Bhatt and P

    B. Bhatt and P. Scholze,Projectivity of the Witt vector affine Grassmannian, Invent. Math.209(2017), no. 2, 329–423

    2017

  3. [3]

    M. T. L. Bizley,Derivation of a new formula for the number of minimal lattice paths from(0,0)to(km, kn)having justtcontacts with the linemy=nx and having no points above this line; and a proof of Grossman’s formula for the number of paths which may touch but do not rise above this line, J. Inst. Actuar.80(1954), 55–62

    1954

  4. [4]

    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 Sci- entifiques et Industrielles [Current Scientific and Industrial Topics], vol. No. 1337, Hermann, Paris, 1968. 50

    1968

  5. [5]

    M. Chen, M. Kisin, and E. Viehmann,Connected components of affine Deligne- Lusztig varieties in mixed characteristic, Compos. Math.151(2015), no. 9, 1697–1762

    2015

  6. [6]

    Chen and J

    M. Chen and J. Tong,Weakly admissible locus and Newton stratification in p-adic Hodge theory, Amer. J. Math.148(2026), no. 1, 1–67

    2026

  7. [7]

    Chen and E

    M. Chen and E. Viehmann,Affine Deligne-Lusztig varieties and the action of J, J. Algebraic Geom.27(2018), no. 2, 273–304

    2018

  8. [8]

    A. J. de Jong and F. Oort,Purity of the stratification by Newton polygons, J. Amer. Math. Soc.13(2000), no. 1, 209–241

    2000

Show all 55 references
  1. [9]

    M. Fox, B. Howard, and N. Imai,Rapoport–Zink spaces of type GU(2, n - 2), Algebraic Geometry13(2026), no. 2, 193–240

    2026

  2. [10]

    Fox and N

    M. Fox and N. Imai,The supersingular locus of the Shimura variety ofGU(2, n− 2), arXiv:2108.03584, to appear in Peking Math. J

  3. [11]

    Q. R. Gashi,On a conjecture of Kottwitz and Rapoport, Ann. Sci. ´Ec. Norm. Sup´ er. (4)43(2010), no. 6, 1017–1038

    2010

  4. [12]

    G¨ ortz,Stratifications of affine Deligne-Lusztig varieties, Trans

    U. G¨ ortz,Stratifications of affine Deligne-Lusztig varieties, Trans. Amer. Math. Soc.372(2019), no. 7, 4675–4699

    2019

  5. [13]

    G¨ ortz and X

    U. G¨ ortz and X. He,Basic loci of Coxeter type in Shimura varieties, Camb. J. Math.3(2015), no. 3, 323–353

    2015

  6. [14]

    G¨ ortz, X

    U. G¨ ortz, X. He, and S. Nie,Fully Hodge-Newton decomposable Shimura vari- eties, Peking Math. J.2(2019), no. 2, 99–154

    2019

  7. [15]

    G¨ ortz and C.-F

    U. G¨ ortz and C.-F. Yu,Supersingular Kottwitz-Rapoport strata and Deligne- Lusztig varieties, J. Inst. Math. Jussieu9(2010), no. 2, 357–390

    2010

  8. [16]

    Ann.353(2012), no

    ,The supersingular locus in Siegel modular varieties with Iwahori level structure, Math. Ann.353(2012), no. 2, 465–498

    2012

  9. [17]

    M. J. Greenberg,Schemata over local rings. II, Ann. of Math. (2)78(1963), 256–266

    1963

  10. [18]

    Hamacher,The dimension of affine Deligne-Lusztig varieties in the affine Grassmannian, Int

    P. Hamacher,The dimension of affine Deligne-Lusztig varieties in the affine Grassmannian, Int. Math. Res. Not. IMRN (2015), no. 23, 12804–12839

    2015

  11. [19]

    Hamacher and E

    P. Hamacher and E. Viehmann,Irreducible components of minuscule affine Deligne-Lusztig varieties, Algebra Number Theory12(2018), no. 7, 1611–1634

    2018

  12. [20]

    Hartl and E

    U. Hartl and E. Viehmann,The Newton stratification on deformations of local G-shtukas, J. Reine Angew. Math.656(2011), 87–129. 51

    2011

  13. [21]

    Math.229 (2012), no

    ,Foliations in deformation spaces of localG-shtukas, Adv. Math.229 (2012), no. 1, 54–78

    2012

  14. [22]

    He,A subalgebra of 0-Hecke algebra, J

    X. He,A subalgebra of 0-Hecke algebra, J. Algebra322(2009), no. 11, 4030– 4039

    2009

  15. [23]

    ,Geometric and homological properties of affine Deligne-Lusztig vari- eties, Ann. of Math. (2)179(2014), no. 1, 367–404

    2014

  16. [24]

    Howard and G

    B. Howard and G. Pappas,On the supersingular locus of theGU(2,2)Shimura variety, Algebra Number Theory8(2014), no. 7, 1659–1699

    2014

  17. [25]

    Karemaker and C.-F

    V. Karemaker and C.-F. Yu,Supersingular Ekedahl-Oort strata and Oort’s con- jecture, arXiv:2406.19748 (2024)

    2024

  18. [26]

    Katsura and F

    T. Katsura and F. Oort,Families of supersingular abelian surfaces, Compositio Math.62(1987), no. 2, 107–167

    1987

  19. [27]

    Kottwitz and M

    R. Kottwitz and M. Rapoport,Minuscule alcoves forGL n andGSp 2n, Manuscripta Math.102(2000), no. 4, 403–428

    2000

  20. [28]

    R. E. Kottwitz,Isocrystals with additional structure, Compositio Math.56 (1985), no. 2, 201–220

    1985

  21. [29]

    Math.108(1992), no

    ,On theλ-adic representations associated to some simple Shimura va- rieties, Invent. Math.108(1992), no. 3, 653–665

    1992

  22. [30]

    Kret,The basic stratum of some simple Shimura varieties, Math

    A. Kret,The basic stratum of some simple Shimura varieties, Math. Ann.356 (2013), no. 2, 487–518

    2013

  23. [31]

    Math.20(2015), 237–267

    ,Combinatorics of the basic stratum of some simple Shimura varieties, Doc. Math.20(2015), 237–267

    2015

  24. [32]

    Kudla and M

    S. Kudla and M. Rapoport,Special cycles on unitary Shimura varieties I. Un- ramified local theory, Invent. Math.184(2011), no. 3, 629–682

    2011

  25. [33]

    I. G. Macdonald,Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, vol. 157, Cambridge University Press, Cambridge, 2003

    2003

  26. [34]

    Nie,Irreducible components of affine Deligne-Lusztig varieties, arXiv:1809.03683v7 (2021)

    S. Nie,Irreducible components of affine Deligne-Lusztig varieties, arXiv:1809.03683v7 (2021)

    2021

  27. [35]

    2, 433–510

    ,Irreducible components of affine Deligne-Lusztig varieties, Cambridge Journal of Mathematics10(2022), no. 2, 433–510

    2022

  28. [36]

    Pappas and M

    G. Pappas and M. Rapoport,Twisted loop groups and their affine flag varieties, Adv. Math.219(2008), no. 1, 118–198, With an appendix by T. Haines and Rapoport. 52

    2008

  29. [37]

    Rapoport and T

    M. Rapoport and T. Zink,Period spaces forp-divisible groups, Annals of Math- ematics Studies, vol. 141, Princeton University Press, Princeton, NJ, 1996

    1996

  30. [38]

    Rapoport,A guide to the reduction modulopof Shimura varieties, Ast´ erisque (2005), no

    M. Rapoport,A guide to the reduction modulopof Shimura varieties, Ast´ erisque (2005), no. 298, 271–318

    2005

  31. [39]

    Schremmer,Generic Newton points and cordial elements, arXiv:2205.02039 (2022)

    F. Schremmer,Generic Newton points and cordial elements, arXiv:2205.02039 (2022)

    2022

  32. [40]

    Schremmer and E

    F. Schremmer and E. Viehmann,Affine Deligne-Lusztig varieties beyond the minute case, arXiv:2511.08879 (2025)

    2025

  33. [41]

    Serre,Alg` ebre locale

    J.-P. Serre,Alg` ebre locale. Multiplicit´ es, Lecture Notes in Mathematics, vol. 11, Springer-Verlag, Berlin-New York, 1965, Cours au Coll` ege de France, 1957– 1958, r´ edig´ e par Pierre Gabriel, Seconde ´ edition, 1965

    1965

  34. [42]

    Shimada,The Ekedahl-Oort stratification and the semi-module stratification, arXiv:2309.03371, to appear in Canad

    R. Shimada,The Ekedahl-Oort stratification and the semi-module stratification, arXiv:2309.03371, to appear in Canad. J. Math

  35. [43]

    ,Basic loci of positive Coxeter type forGL n, arXiv:2402.13216 (2024)

    2024

  36. [44]

    Shimada and T

    R. Shimada and T. Takamatsu,On the supersingular locus of the Siegel modular variety of genus3or4, arXiv:2403.19505, to appear in Ann. Inst. Fourier (Grenoble)

  37. [45]

    Steinberg,Endomorphisms of linear algebraic groups, Memoirs of the Amer- ican Mathematical Society, vol

    R. Steinberg,Endomorphisms of linear algebraic groups, Memoirs of the Amer- ican Mathematical Society, vol. No. 80, American Mathematical Society, Prov- idence, RI, 1968

    1968

  38. [46]

    Takaya,Equidimensionality of affine Deligne-Lusztig varieties in mixed char- acteristic, Adv

    Y. Takaya,Equidimensionality of affine Deligne-Lusztig varieties in mixed char- acteristic, Adv. Math.465(2025), Paper No. 110153, 22

    2025

  39. [47]

    Thomas and N

    H. Thomas and N. Williams,Sweeping up zeta, Selecta Math. (N.S.)24(2018), no. 3, 2003–2034

    2018

  40. [48]

    Viehmann,The dimension of some affine Deligne-Lusztig varieties, Ann

    E. Viehmann,The dimension of some affine Deligne-Lusztig varieties, Ann. Sci. ´Ecole Norm. Sup. (4)39(2006), no. 3, 513–526

    2006

  41. [49]

    Algebraic Geom.17(2008), no

    ,Moduli spaces ofp-divisible groups, J. Algebraic Geom.17(2008), no. 2, 341–374

    2008

  42. [50]

    ,Oort’s conjecture on automorphisms of generic supersingular abelian varieties, arXiv:2603.06033 (2026)

    2026 arXiv

  43. [51]

    Vollaard and T

    I. Vollaard and T. Wedhorn,The supersingular locus of the Shimura variety of GU(1, n−1)II, Invent. Math.184(2011), no. 3, 591–627. 53

    2011

  44. [52]

    Xiao and X

    L. Xiao and X. Zhu,Cycles on Shimura varieties via geometric Satake, arXiv:1707.05700 (2017)

    2017 arXiv

  45. [53]

    Zhang,On arithmetic fundamental lemmas, Invent

    W. Zhang,On arithmetic fundamental lemmas, Invent. Math.188(2012), no. 1, 197–252

    2012

  46. [54]

    Zhou and Y

    R. Zhou and Y. Zhu,Twisted orbital integrals and irreducible components of affine Deligne-Lusztig varieties, Camb. J. Math.8(2020), no. 1, 149–241

    2020

  47. [55]

    Zhu,Affine Grassmannians and the geometric Satake in mixed characteristic, Ann

    X. Zhu,Affine Grassmannians and the geometric Satake in mixed characteristic, Ann. of Math. (2)185(2017), no. 2, 403–492. 54

    2017

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