pith. machine review for the scientific record. sign in

arxiv: 2605.03036 · v1 · submitted 2026-05-04 · 🧮 math.RT

Recognition: unknown

Pinned Jordan Decomposition of Characters and Depth-Zero Hecke Algebras

Manish Mishra, Prashant Arote

Pith reviewed 2026-05-08 02:32 UTC · model grok-4.3

classification 🧮 math.RT
keywords Jordan decompositionLusztig seriesfinite reductive groupsHarish-Chandra seriesdepth-zero Hecke algebrasunipotent charactersDeligne-Lusztig characterspreferred extensions
0
0 comments X

The pith

A pinning on the reductive group determines a unique bijection from each Lusztig series to unipotent characters of the dual centralizer.

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

The paper builds a canonical Jordan decomposition for the characters of finite reductive groups G, even when the dual centralizers are disconnected. For a semisimple element s it produces a bijection J_s from the Lusztig series E(G,s) onto the set of unipotent characters of the dual centralizer that is uniquely fixed by compatibility with the Deligne-Lusztig character formula and with Harish-Chandra series. The construction rests on a functorial choice of preferred extensions of cuspidal unipotent characters and is extended to certain disconnected groups. The same bijection yields an explicit isomorphism between depth-zero Hecke algebras arising from Bernstein blocks of tame p-adic groups and the corresponding unipotent Hecke algebras.

Core claim

For a connected reductive group G over a finite field equipped with a pinning and for any semisimple s in the dual group, there exists a unique bijection J_s from the Lusztig series E(G,s) to the unipotent characters of the dual centralizer C_{G*}(s)^{F*} that refines Lusztig's orbit-valued map and is characterized by compatibility with Deligne-Lusztig induction and Harish-Chandra series. The same tools produce a canonical correspondence between disconnected Lusztig series and unipotent characters of disconnected dual centralizers when the component group is abelian.

What carries the argument

The pinned bijection J_s, obtained by extending a canonical choice of preferred extensions of cuspidal unipotent characters to their inertia groups via Lusztig's preferred extensions, Clifford theory, relative Weyl group comparison, and connected/disconnected forms of Howlett-Lehrer theory.

If this is right

  • Depth-zero Bernstein blocks of tame p-adic reductive groups reduce canonically to unipotent blocks once the same pinning is fixed.
  • The Hecke algebra attached to a depth-zero type (K_{x_0}, rho_{x_0}) is isomorphic to a unipotent Hecke algebra.
  • The isomorphism of Hecke algebras preserves the standard anti-involutions.
  • The construction refines the earlier reduction obtained by Ohara.

Where Pith is reading between the lines

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

  • The functoriality of the preferred extensions may allow the same pinning to label correspondences in other blocks or at positive depth.
  • The explicit isomorphism of depth-zero Hecke algebras supplies a concrete model in which to test conjectures about the structure of Bernstein blocks.
  • Because the bijection respects Deligne-Lusztig induction, it may be used to compare character values across endoscopic groups that share the same pinning data.

Load-bearing premise

A canonical choice of preferred extensions of cuspidal unipotent characters to their inertia groups exists and extends functorially to all Harish-Chandra series via Lusztig's preferred extensions, Clifford theory, relative Weyl group comparison, and connected/disconnected forms of Howlett-Lehrer theory.

What would settle it

An explicit finite reductive group with a disconnected dual centralizer together with a specific semisimple s for which no bijection from E(G,s) to the unipotent characters of the dual centralizer simultaneously preserves the Deligne-Lusztig formula and the Harish-Chandra series decomposition.

read the original abstract

We construct a pinned canonical Jordan decomposition of characters for finite reductive groups in situations where the dual centralizers may be disconnected. For a connected reductive group \(\bG\) over a finite field, equipped with a pinning, and for a semisimple element \(s\in G^*\), we construct a uniquely determined bijection \[ \J_s:\cE(G,s)\xrightarrow{\sim}\Uch\bigl(C_{\bG^*}(s)^{F^*}\bigr). \] This refines Lusztig's orbit-valued Jordan decomposition for groups with disconnected centre, and is characterized by compatibility with the Deligne--Lusztig character formula and with Harish--Chandra series. We then extend the construction to possibly disconnected reductive groups with abelian component group, obtaining a canonical bijection between disconnected Lusztig series and unipotent characters of the corresponding disconnected dual centralizers. The main technical input is a canonical choice of preferred extensions of cuspidal unipotent characters to their inertia groups. The construction uses Lusztig's preferred extensions, Clifford theory, relative Weyl group comparison, and connected and disconnected forms of Howlett--Lehrer theory. These tools allow the cuspidal Jordan decomposition to be extended functorially to all Harish--Chandra series. As an application, we prove a pinned canonical reduction from depth-zero Bernstein blocks of tame \(p\)-adic reductive groups to unipotent blocks. More precisely, for a depth-zero Bernstein type \((K_{x_0},\rho_{x_0})\), the associated Hecke algebra is canonically isomorphic, after fixing the same pinning, to a unipotent Hecke algebra. This refines an earlier result of Ohara. This isomorphism preserves the standard anti-involutions.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript constructs a pinned canonical Jordan decomposition of characters for finite reductive groups, including cases where the dual centralizer may be disconnected. For a connected reductive group G over a finite field equipped with a pinning and a semisimple element s in G*, it defines a unique bijection J_s : E(G,s) → Uch(C_{G*}(s)^{F*}) that refines Lusztig's orbit-valued Jordan decomposition. The bijection is characterized by compatibility with the Deligne-Lusztig character formula and Harish-Chandra series. The main technical step is a canonical choice of preferred extensions of cuspidal unipotent characters to inertia groups, obtained via Lusztig's preferred extensions, Clifford theory, relative Weyl group comparison, and connected/disconnected forms of Howlett-Lehrer theory; this extends functorially to all Harish-Chandra series. The construction is extended to disconnected reductive groups with abelian component group. As an application, for depth-zero Bernstein types (K_{x_0}, ρ_{x_0}) of tame p-adic reductive groups, the associated Hecke algebra is shown to be canonically isomorphic (after fixing the same pinning) to a unipotent Hecke algebra, refining an earlier result of Ohara and preserving standard anti-involutions.

Significance. If the construction holds, the work supplies a canonical refinement of Lusztig's Jordan decomposition that resolves choice ambiguities arising from disconnected centralizers, which is a recurring issue in the representation theory of finite groups of Lie type. The explicit use of a pinning to achieve uniqueness and functoriality is a clear strength, as is the extension to disconnected groups under the abelian component-group hypothesis. The application to depth-zero Hecke algebras provides a precise, pinning-compatible reduction from p-adic Bernstein blocks to unipotent blocks; this refines Ohara's result and may facilitate explicit computations or comparisons of anti-involutions in the local Langlands program. The reliance on standard tools (Clifford theory, Howlett-Lehrer) to produce a canonical object is noted positively.

major comments (2)
  1. [§3] §3 (construction of preferred extensions): the claim that the choice of preferred extensions of cuspidal unipotent characters is canonical and extends functorially to all Harish-Chandra series via relative Weyl group comparison rests on the compatibility of Lusztig's extensions with the pinning; an explicit verification that different choices of cuspidal character in the same series yield the same extension after Clifford theory is required to support the uniqueness of J_s.
  2. [§5] §5 (Hecke algebra isomorphism): the asserted canonical isomorphism between the depth-zero Hecke algebra and the unipotent Hecke algebra, after fixing the pinning, must confirm that the reduction map from the p-adic group to the finite reductive quotient introduces no additional choices that would violate preservation of the standard anti-involutions; the current argument appears to rely on the same functoriality used for J_s, which needs to be checked explicitly for the disconnected case.
minor comments (2)
  1. [Notation and preliminaries] Notation for the dual group and its F*-action is generally clear, but the distinction between connected and disconnected forms of Howlett-Lehrer theory should be summarized in a single paragraph or table to aid readers who are not specialists in the disconnected case.
  2. [Introduction] The abstract states the result for 'tame p-adic reductive groups'; the introduction or §5 should list the precise hypotheses on the residue characteristic and the splitting behavior of the group that are actually used in the reduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address each major comment below and will make the suggested revisions to strengthen the presentation.

read point-by-point responses

  1. Referee: [§3] §3 (construction of preferred extensions): the claim that the choice of preferred extensions of cuspidal unipotent characters is canonical and extends functorially to all Harish-Chandra series via relative Weyl group comparison rests on the compatibility of Lusztig's extensions with the pinning; an explicit verification that different choices of cuspidal character in the same series yield the same extension after Clifford theory is required to support the uniqueness of J_s.

    Authors: We agree that an explicit verification would be beneficial for clarity. The canonicity stems from Lusztig's preferred extensions being uniquely determined by the pinning, and the Clifford theory application, combined with the relative Weyl group isomorphism, ensures that the extension is independent of the specific cuspidal character chosen within the series. To make this fully explicit, we will insert a new lemma in §3 that verifies this independence using the compatibility with Deligne-Lusztig characters and the Howlett-Lehrer correspondence. This addresses the concern directly. revision: yes

  2. Referee: [§5] §5 (Hecke algebra isomorphism): the asserted canonical isomorphism between the depth-zero Hecke algebra and the unipotent Hecke algebra, after fixing the pinning, must confirm that the reduction map from the p-adic group to the finite reductive quotient introduces no additional choices that would violate preservation of the standard anti-involutions; the current argument appears to rely on the same functoriality used for J_s, which needs to be checked explicitly for the disconnected case.

    Authors: The reduction map is canonically determined by the pinning and the depth-zero type, without introducing extra choices. The preservation of standard anti-involutions follows from the fact that the isomorphism is induced by the pinned Jordan decomposition, which is compatible with the anti-involution by construction. For the disconnected case, under the abelian component group hypothesis, we will add an explicit check in the revised §5, confirming that the functoriality of J_s extends without ambiguity to the Hecke algebra level and preserves the anti-involutions. This will be done by direct comparison of the generators and relations. revision: yes

Circularity Check

0 steps flagged

No circularity; construction relies on external standard tools

full rationale

The paper constructs the bijection J_s explicitly from Lusztig's preferred extensions, Clifford theory, relative Weyl group comparison, and Howlett-Lehrer theory (both connected and disconnected forms), then extends it functorially to Harish-Chandra series and applies it to depth-zero Hecke algebras. These inputs are cited as independent external results rather than derived within the paper or reduced to self-referential equations. No self-citations are load-bearing for the central claim, no parameters are fitted and relabeled as predictions, and the pinning is used only to fix choices without creating definitional loops. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies entirely on established domain assumptions from the theory of reductive groups and Lusztig's character theory; no free parameters are introduced and no new entities are postulated.

axioms (2)
  • domain assumption Standard setup of connected reductive groups over finite fields with Frobenius endomorphism and a fixed pinning
    Provides the base objects G, G*, F, and the pinning used to define the canonical choice.
  • domain assumption Existence and basic properties of Lusztig's Jordan decomposition, Deligne-Lusztig characters, Harish-Chandra series, and Howlett-Lehrer theory for connected groups
    The paper refines and extends these to the disconnected-centralizer setting.

pith-pipeline@v0.9.0 · 5626 in / 1620 out tokens · 57993 ms · 2026-05-08T02:32:36.268649+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

34 extracted references · 19 canonical work pages

  1. [1]

    Representations of Reductive Groups Over Finite Fields , urldate =

    Pierre Deligne and George Lusztig , journal =. Representations of Reductive Groups Over Finite Fields , urldate =

  2. [2]

    Lusztig, George , TITLE =. Ast\'. 1988 , PAGES =

    1988

  3. [3]

    1984 , PAGES =

    Lusztig, George , TITLE =. 1984 , PAGES =. doi:10.1515/9781400881772 , URL =

    work page doi:10.1515/9781400881772 1984

  4. [4]

    Digne, Fran. On. Ast\'. 1990 , PAGES =

    1990

  5. [5]

    and Lehrer, Gustav I

    Howlett, Robert B. and Lehrer, Gustav I. , TITLE =. Invent. Math. , FJOURNAL =. 1980 , NUMBER =. doi:10.1007/BF01402273 , URL =

    work page doi:10.1007/bf01402273 1980

  6. [6]

    Representations of finite groups of

    Digne, Fran. Representations of finite groups of. 2020 , PAGES =

    2020

  7. [7]

    2020 , PAGES =

    Geck, Meinolf and Malle, Gunter , TITLE =. 2020 , PAGES =

    2020

  8. [8]

    Fong, Paul and Srinivasan, Bhama , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 1989 , PAGES =

    1989

  9. [9]

    , title =

    Enguehard, Michel E. , title =. doi:10.48550/ARXIV.1312.0106 , url =

    work page doi:10.48550/arxiv.1312.0106

  10. [10]

    2006 , zbl =

    Bonnaf\'e, C\'edric , title =. 2006 , zbl =

    2006

  11. [11]

    On Jordan decomposition of characters for SU(n,q) , journal =

    Marc Cabanes , keywords =. On Jordan decomposition of characters for SU(n,q) , journal =. 2013 , issn =. doi:https://doi.org/10.1016/j.jalgebra.2012.10.015 , url =

    work page doi:10.1016/j.jalgebra.2012.10.015 2013

  12. [12]

    Manuscripta Math

    Geck, Meinolf , TITLE =. Manuscripta Math. , FJOURNAL =. 1993 , NUMBER =. doi:10.1007/BF03026560 , URL =

    work page doi:10.1007/bf03026560 1993

  13. [13]

    Artin, Emil , TITLE =. Comm. Pure Appl. Math. , FJOURNAL =. 1955 , PAGES =. doi:10.1002/cpa.3160080302 , URL =

    work page doi:10.1002/cpa.3160080302 1955

  14. [14]

    Characters of finite

    Geck, Meinolf and Pfeiffer, G\". Characters of finite. 2000 , PAGES =

    2000

  15. [15]

    Zsigmondy, Karl , TITLE =. Monatsh. Math. Phys. , FJOURNAL =. 1892 , NUMBER =. doi:10.1007/BF01692444 , URL =

    work page doi:10.1007/bf01692444

  16. [16]

    Aubert, Anne-Marie and Baum, Paul and Plymen, Roger and Solleveld, Maarten , TITLE =. Proc. Lond. Math. Soc. (3) , FJOURNAL =. 2017 , NUMBER =. doi:10.1112/plms.12023 , URL =

    work page doi:10.1112/plms.12023 2017

  17. [17]

    International Mathematics Research Notices , volume =

    Arote, Prashant and Mishra, Manish , title = ". International Mathematics Research Notices , volume =. 2023 , month =. doi:10.1093/imrn/rnad296 , url =

    work page doi:10.1093/imrn/rnad296 2023

  18. [18]

    Gow, Roderick , TITLE =. Proc. London Math. Soc. (3) , FJOURNAL =. 1983 , NUMBER =. doi:10.1112/plms/s3-47.3.493 , URL =

    work page doi:10.1112/plms/s3-47.3.493 1983

  19. [19]

    2017 , issn =

    Cuspidal characters and automorphisms , journal =. 2017 , issn =. doi:https://doi.org/10.1016/j.aim.2017.09.026 , url =

    work page doi:10.1016/j.aim.2017.09.026 2017

  20. [20]

    Bonnaf\'e, C\'edric , TITLE =. J. Algebra , FJOURNAL =. 1999 , NUMBER =. doi:10.1006/jabr.1998.7564 , URL =

    work page doi:10.1006/jabr.1998.7564 1999

  21. [21]

    Prasad, Dipendra , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2019 , NUMBER =. doi:10.1090/tran/7602 , URL =

    work page doi:10.1090/tran/7602 2019

  22. [22]

    Taylor, Jay , TITLE =. J. Algebra , FJOURNAL =. 2018 , PAGES =. doi:10.1016/j.jalgebra.2018.03.008 , URL =

    work page doi:10.1016/j.jalgebra.2018.03.008 2018

  23. [23]

    A Remark on Descent for Coxeter Groups

    Lusztig, G. , TITLE =. Invent. Math. , FJOURNAL =. 1976/77 , NUMBER =. doi:10.1007/BF01408569 , URL =

    work page doi:10.1007/bf01408569 1976

  24. [24]

    2025 , note =

    On parameters of Hecke algebras for p -adic groups , author=. 2025 , note =

    2025

  25. [25]

    Journal of Algebra , volume=

    Extensions of unipotent characters and the inductive McKay condition , author=. Journal of Algebra , volume=. 2008 , publisher=

    2008

  26. [26]

    Advances in Mathematics , volume=

    Character sheaves, IV , author=. Advances in Mathematics , volume=. 1986 , publisher=

    1986

  27. [27]

    Groupes r

    Digne, Fran. Groupes r. Annales scientifiques de l'Ecole normale sup

  28. [28]

    1994 , publisher=

    Character theory of finite groups , author=. 1994 , publisher=

    1994

  29. [29]

    2003 , note =

    The space of unipotently supported class functions on a finite reductive group , journal =. 2003 , note =. doi:https://doi.org/10.1016/S0021-8693(02)00635-X , url =

    work page doi:10.1016/s0021-8693(02)00635-x 2003

  30. [30]

    Produit en couronne de groupes lin

    Bonnaf. Produit en couronne de groupes lin. Journal of Algebra , volume =. 1999 , pages =

    1999

  31. [31]

    arXiv preprint arXiv:2408.07801 , year=

    Structure of Hecke algebras arising from types , author=. arXiv preprint arXiv:2408.07801 , year=

    work page arXiv

  32. [32]

    arXiv preprint arXiv:2408.07805 , year=

    Reduction to depth zero for tame p-adic groups via Hecke algebra isomorphisms , author=. arXiv preprint arXiv:2408.07805 , year=

    work page arXiv

  33. [33]

    Lusztig, George , title =. Adv. Math. , volume =. 1986 , pages =

    1986

  34. [34]

    Represent

    Lusztig, George , title =. Represent. Theory , volume =. 2004 , pages =. doi:10.1090/S1088-4165-04-00251-1 , eprint =

    work page doi:10.1090/s1088-4165-04-00251-1 2004