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arxiv: 2605.03036 · v2 · submitted 2026-05-04 · 🧮 math.RT

Pinned Jordan Decomposition of Characters and Depth-Zero Hecke Algebras

Prashant Arote , Manish Mishra This is my paper

Pith reviewed 2026-05-15 07:25 UTC · model grok-4.3

classification 🧮 math.RT MSC 20C3320G40
keywords Jordan decompositionLusztig seriesfinite reductive groupsDeligne-Lusztig charactersClifford theoryunipotent charactersdepth-zero Hecke algebras
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The pith

Pinned canonical bijection matches Lusztig series to unipotent characters of dual centralizers

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

For finite reductive groups G over a finite field with a fixed pinning, the paper constructs a canonical bijection from each Lusztig series E(G,s) to the unipotent characters of the F*-fixed points of the centralizer C_{G*}(s). This refines Lusztig's orbit-valued Jordan decomposition by supplying an explicit matching that continues to work when the centralizer is disconnected because of a non-connected center. The bijection is defined so that it commutes with Deligne-Lusztig induction and with Harish-Chandra series. It is obtained from pinned-normalized preferred extensions of cuspidal unipotent characters together with Clifford theory and Howlett-Lehrer comparisons.

Core claim

For a connected reductive group G over a finite field with a fixed pinning and a semisimple element s in the dual group G*, there is a canonical bijection between the Lusztig series E(G,s) and the unipotent characters of C_{G*}(s)^{F*}. The bijection is characterized by compatibility with Deligne-Lusztig character formulae and Harish-Chandra series. For a class of possibly disconnected reductive groups with abelian component group the construction produces an enriched version that records the connected unipotent Jordan datum, the source Clifford class, and the corresponding projective Clifford label.

What carries the argument

Pinned-normalized preferred extensions of cuspidal unipotent characters, which permit the application of Clifford theory and relative Weyl group comparisons to produce the canonical bijection while preserving Deligne-Lusztig compatibility.

If this is right

  • The construction supplies a pinned canonical form for the finite-field input needed in comparisons of depth-zero Hecke algebra parameters with the unipotent case.
  • For possibly disconnected reductive groups with abelian component groups the enriched decomposition records the connected unipotent Jordan datum, the Clifford class, and the projective Clifford label.
  • When the transported Clifford classes agree with the ordinary ones, the enriched target recovers the usual unipotent characters of the disconnected dual centralizer.

Where Pith is reading between the lines

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

  • The pinned normalization may allow explicit calculation of character degrees inside Lusztig series for low-rank groups whose centralizers are disconnected.
  • The method could be checked for consistency against known character tables of unitary or symplectic groups in small dimensions.
  • The same normalization technique might extend to the study of representations over fields of positive characteristic.

Load-bearing premise

The existence of pinned-normalized preferred extensions of cuspidal unipotent characters together with the dual centralizers admitting the required Clifford theory and Howlett-Lehrer comparisons that preserve compatibility with Deligne-Lusztig formulae.

What would settle it

For a concrete small group such as GU(3,q) and a semisimple element s whose centralizer is disconnected, compute the Lusztig series characters explicitly and check whether the proposed bijection preserves the inner products against Deligne-Lusztig characters or matches the known unipotent character degrees on the centralizer side.

read the original abstract

We construct a pinned canonical Jordan decomposition of characters for finite reductive groups in cases where the relevant dual centralizers may be disconnected. For a connected reductive group \(G\) over a finite field, with a fixed pinning, and for a semisimple element \(s\in G^*\), we construct a canonical bijection between the Lusztig series \(\mathcal E(G,s)\) and the unipotent characters of \(C_{G^*}(s)^{F^*}\). This refines Lusztig's orbit-valued Jordan decomposition for groups with disconnected centre, and is characterized by compatibility with Deligne--Lusztig character formulae and Harish--Chandra series. We also treat a class of possibly disconnected reductive groups with abelian component group whose rational components admit pinning-preserving representatives. In this setting the natural result is an enriched disconnected Jordan decomposition: the target records the connected unipotent Jordan datum, the source Clifford class, and the corresponding projective Clifford label. When the transported Clifford classes agree with the ordinary Clifford classes on the dual-centralizer side, this enriched target recovers the usual unipotent characters of the corresponding disconnected dual centralizer. The construction uses pinned-normalized preferred extensions of cuspidal unipotent characters, Clifford theory, relative Weyl group comparison, Malle's matching, and connected and disconnected Howlett--Lehrer theory. As an application, we give a pinned canonical form of the finite-field input in Ohara's comparison of depth-zero Hecke algebra parameters with the unipotent case.

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

1 major / 2 minor

Summary. The manuscript constructs a pinned canonical Jordan decomposition for characters of finite reductive groups. For a connected reductive group G over a finite field with fixed pinning and semisimple s in G*, it defines a canonical bijection between the Lusztig series E(G,s) and the unipotent characters of C_{G*}(s)^{F*}. The construction proceeds via pinned-normalized preferred extensions of cuspidal unipotent characters, followed by Clifford theory, relative Weyl group comparisons, Malle's matching, and connected/disconnected Howlett-Lehrer theory; it is characterized by compatibility with Deligne-Lusztig formulae and Harish-Chandra series. The work also treats a class of possibly disconnected reductive groups with abelian component group, yielding an enriched disconnected Jordan decomposition that records connected unipotent datum, Clifford class, and projective Clifford label, and applies the result to give a pinned canonical form for the finite-field input in Ohara's depth-zero Hecke algebra comparison.

Significance. If the construction holds, the result supplies a canonical, pinning-based refinement of Lusztig's orbit-valued Jordan decomposition that remains well-defined for disconnected dual centralizers. This strengthens the structural understanding of Lusztig series and Harish-Chandra series in the presence of disconnected centers, and supplies a concrete, choice-free input for comparisons of depth-zero Hecke algebra parameters with the unipotent case. The explicit use of pinned extensions and Clifford comparisons is a methodological strength that could facilitate further applications in the representation theory of finite groups of Lie type.

major comments (1)
  1. [Construction via pinned-normalized extensions (abstract and main construction)] The canonicity of the bijection rests on the uniqueness of pinned-normalized preferred extensions of cuspidal unipotent characters when C_{G*}(s) is disconnected. The abstract asserts that the pinning selects the extension, yet the component group action on cuspidal supports may introduce non-trivial cocycles that shift Clifford labels across rational components. No explicit lemma or verification is indicated showing that the chosen normalization is simultaneously compatible with all components while preserving the Deligne-Lusztig character formulae and the Harish-Chandra series decomposition. This point is load-bearing for the central claim of a canonical map.
minor comments (2)
  1. [Enriched disconnected case] The enriched disconnected Jordan decomposition is described in terms of connected datum, Clifford class, and projective label; an explicit small-group example (e.g., a classical group with disconnected centralizer) would clarify how the enriched target recovers ordinary unipotent characters when the transported classes agree.
  2. [Notation and definitions] Notation for the target of the enriched decomposition and for the projective Clifford labels should be introduced with a short table or diagram to distinguish them from the ordinary unipotent characters of the disconnected centralizer.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the load-bearing nature of the canonicity claim. We address the major comment point by point below and will revise the manuscript accordingly to strengthen the exposition.

read point-by-point responses

  1. Referee: [Construction via pinned-normalized extensions (abstract and main construction)] The canonicity of the bijection rests on the uniqueness of pinned-normalized preferred extensions of cuspidal unipotent characters when C_{G*}(s) is disconnected. The abstract asserts that the pinning selects the extension, yet the component group action on cuspidal supports may introduce non-trivial cocycles that shift Clifford labels across rational components. No explicit lemma or verification is indicated showing that the chosen normalization is simultaneously compatible with all components while preserving the Deligne-Lusztig character formulae and the Harish-Chandra series decomposition. This point is load-bearing for the central claim of a canonical map.

    Authors: We agree that an explicit verification of compatibility under the component group action is necessary to fully substantiate the canonicity. The construction in Sections 3--5 proceeds by using the fixed pinning to select unique preferred extensions of cuspidal unipotent characters, then applying Clifford theory and relative Weyl group comparisons (via Malle's matching and Howlett--Lehrer theory) to ensure the resulting bijection respects Deligne--Lusztig character formulae and Harish-Chandra series. The pinning-preserving representatives for rational components (as treated in the disconnected case) are chosen precisely to control the cocycles. However, we acknowledge that the current text does not isolate this verification in a single lemma. In the revised version we will insert a new Lemma 3.7 that explicitly checks the cocycle compatibility across components, confirms that Clifford labels are not shifted in a way that violates the normalization, and verifies preservation of the Deligne--Lusztig values and Harish-Chandra decomposition. This addition will make the argument self-contained without altering the existing proofs. revision: yes

Circularity Check

0 steps flagged

No significant circularity; construction uses independent external theories

full rationale

The derivation constructs the pinned Jordan decomposition bijection by transporting cuspidal unipotent characters via pinned-normalized preferred extensions, then applying Clifford theory, relative Weyl group comparisons, Malle's matching, and Howlett-Lehrer theory to match Deligne-Lusztig series. These components are drawn from established external results (Lusztig series, Clifford theory, Deligne-Lusztig formulae) rather than being defined in terms of the target bijection. No step reduces the output to a fitted parameter, self-definition, or self-citation chain by construction; the pinning serves only to normalize choices within the given assumptions. The result refines an existing orbit-valued decomposition without tautological equivalence to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard background in the representation theory of reductive groups over finite fields together with domain-specific assumptions about the existence of preferred extensions and pinnings.

axioms (2)
  • domain assumption Finite reductive groups admit fixed pinnings that normalize extensions of cuspidal characters
    The construction is stated to use pinned-normalized preferred extensions throughout.
  • standard math Lusztig series, Deligne-Lusztig induction, and Harish-Chandra series behave compatibly under Clifford theory
    Invoked as the characterizing properties of the bijection.

pith-pipeline@v0.9.0 · 5574 in / 1566 out tokens · 58283 ms · 2026-05-15T07:25:39.511577+00:00 · methodology

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