pith. sign in

arxiv: 2604.15079 · v1 · submitted 2026-04-16 · 🧮 math.RT · math.NT

On local integrability results for p-adic reductive groups

Cheng-Chiang Tsai This is my paper

Pith reviewed 2026-05-10 09:15 UTC · model grok-4.3

classification 🧮 math.RT math.NT
keywords local integrabilityp-adic reductive groupsHarish-Chandracharacter expansionsrepresentationslocal charactersintegrable characters
0
0 comments X

The pith

Local character expansions prove local integrability of characters for p-adic reductive groups.

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

The paper gives a short proof of Harish-Chandra's theorem on the local integrability of complex characters of p-adic reductive groups. It uses local character expansions to provide an algebraic version of this integrability. The proof works for coefficients other than the complex numbers and covers some cases not previously addressed in the literature. Additionally, it shows that the characters are locally L to the alpha for some alpha greater than one.

Core claim

Using local character expansions, the paper establishes that the complex characters of p-adic reductive groups are locally integrable, giving an algebraic incarnation of this property that applies to other coefficients as well.

What carries the argument

Local character expansions, which allow an algebraic control over the behavior of characters near singular points to deduce integrability.

If this is right

  • Harish-Chandra's theorem on local integrability is proved concisely.
  • The integrability holds for some non-complex coefficients.
  • Local integrability is verified in additional cases not covered before.
  • Characters satisfy local L^α integrability for specified α > 1.

Where Pith is reading between the lines

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

  • This algebraic approach might enable similar results in related settings with different base fields.
  • The method could simplify checks for integrability in explicit computations of characters.
  • Extending the α bound might connect to other analytic properties of characters.

Load-bearing premise

Local character expansions exist and possess the algebraic properties needed to imply the integrability.

What would settle it

Discovery of a p-adic reductive group character that fails to be locally integrable, or a local character expansion lacking the required properties for integrability.

read the original abstract

We present a short proof, based on local character expansions, of the celebrated theorem of Harish-Chandra about local integrability of complex characters of $p$-adic reductive groups. The proof gives an algebraic incarnation of the local integrability that works for some coefficients different from $\mathbb{C}$, verifies local integrability in cases that appear not covered in the literature, and shows that a character is locally-$L^{\alpha}$ for some specified $\alpha>1$ as in [GGH23].

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 presents a short proof of Harish-Chandra's theorem on the local integrability of complex characters of p-adic reductive groups, relying on local character expansions. It supplies an algebraic formulation of integrability valid for certain coefficient rings other than ℂ, establishes the result in cases apparently absent from the literature, and proves that characters are locally L^α for a specified α > 1, in the spirit of [GGH23].

Significance. If the argument is valid, the work supplies a streamlined algebraic proof of a foundational result in the representation theory of p-adic groups. The algebraic incarnation, applicability to non-complex coefficients, coverage of additional cases, and the explicit L^α bound are concrete strengths that could simplify arguments involving distributions and orbital integrals.

major comments (1)
  1. [Introduction] The proof invokes local character expansions from the literature to deduce integrability algebraically. To underwrite the extensions beyond ℂ-coefficients and the new cases, the manuscript must explicitly identify (in the introduction or the proof section) which algebraic properties of the expansions—such as support on nilpotent orbits or the precise form of the leading terms—are used, and confirm these properties are available independently of analytic integrability results for the coefficient rings in question. This step is load-bearing for the central claim of a short, non-circular algebraic proof.
minor comments (2)
  1. Expand the citation [GGH23] to full bibliographic details in the references section.
  2. Specify the numerical value or range of the exponent α in the local L^α statement, and indicate whether the bound is uniform or depends on the group or representation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive major comment. We address the point below and have revised the paper to incorporate the requested clarification.

read point-by-point responses

  1. Referee: [Introduction] The proof invokes local character expansions from the literature to deduce integrability algebraically. To underwrite the extensions beyond ℂ-coefficients and the new cases, the manuscript must explicitly identify (in the introduction or the proof section) which algebraic properties of the expansions—such as support on nilpotent orbits or the precise form of the leading terms—are used, and confirm these properties are available independently of analytic integrability results for the coefficient rings in question. This step is load-bearing for the central claim of a short, non-circular algebraic proof.

    Authors: We agree that an explicit identification of the algebraic properties is necessary to justify the extensions to non-complex coefficients and the additional cases. In the revised version we have inserted a new paragraph at the end of the introduction (with a cross-reference in Section 2) that enumerates the precise properties drawn from the local character expansions: (i) the expansion is supported on nilpotent orbits, and (ii) the leading terms are given by the indicated linear combination of orbital integrals with coefficients in the ring R. We further state that these properties are established in the cited references by purely algebraic arguments (via the structure of the Hecke algebra and the definition of the distributions over R) and do not rely on any analytic integrability results for the coefficient rings under consideration. This addition makes the non-circular character of the argument fully transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external prior results for expansions.

full rationale

The paper derives local integrability (and extensions to other coefficients and L^α bounds) from the assumed existence and algebraic properties of local character expansions, which are invoked from prior literature without re-derivation. No step equates the target integrability statement to a fitted parameter, self-defined quantity, or load-bearing self-citation chain; the expansions are treated as independent inputs whose algebraic features (e.g., support properties) yield the integrability conclusion. The central claim therefore remains non-circular given those external assumptions, consistent with the modest reader's score and absence of any quoted reduction to inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, invented entities, or ad-hoc axioms are mentioned in the abstract; the argument relies on standard background results about local character expansions in p-adic representation theory.

pith-pipeline@v0.9.0 · 5365 in / 1107 out tokens · 46979 ms · 2026-05-10T09:15:27.551868+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

15 extracted references · 15 canonical work pages

  1. [1]

    Avraham Aizenbud, Dmitry Gourevitch, David Kazhdan, Eitan Sayag, Itay Glazer, and Yotam Hendel, On H arish- C handra's integrability theorem in positive characteristic , 2026, https://arxiv.org/abs/2602.16389v1

    work page arXiv 2026

  2. [2]

    Adler and Jonathan Korman, The local character expansion near a tame, semisimple element, Amer

    Jeffrey D. Adler and Jonathan Korman, The local character expansion near a tame, semisimple element, Amer. J. Math. 129 (2007), no. 2, 381--403. 2306039 (2008a:22020)

    work page 2007

  3. [3]

    (N.S.) 22 (2016), no

    Roman Bezrukavnikov, David Kazhdan, and Yakov Varshavsky, On the depth r B ernstein projector , Selecta Math. (N.S.) 22 (2016), no. 4, 2271--2311. 3573958

    work page 2016

  4. [4]

    Raf Cluckers, Julia Gordon, and Immanuel Halupczok, Local integrability results in harmonic analysis on reductive groups in large positive characteristic, Ann. Sci. \'Ec. Norm. Sup\'er. (4) 47 (2014), no. 6, 1163--1195. 3297157

    work page 2014

  5. [5]

    Hendel, Integrability and singularities of H arish- C handra characters , 2023, https://arxiv.org/abs/2312.01591v2

    Itay Glazer, Julia Gordon, and Yotam I. Hendel, Integrability and singularities of H arish- C handra characters , 2023, https://arxiv.org/abs/2312.01591v2

    work page arXiv 2023

  6. [6]

    16, American Mathematical Society, Providence, RI, 1999, Preface and notes by Stephen DeBacker and Paul J

    Harish-Chandra, Admissible invariant distributions on reductive p -adic groups , University Lecture Series, vol. 16, American Mathematical Society, Providence, RI, 1999, Preface and notes by Stephen DeBacker and Paul J. Sally, Jr. 1702257 (2001b:22015)

    work page 1999

  7. [7]

    Roger Howe, The F ourier transform and germs of characters (case of Gl _ n over a p -adic field) , Math. Ann. 208 (1974), 305--322. 0342645 (49 \#7391)

    work page 1974

  8. [8]

    Friedrich Knop, Are the semi-simple elements in a non-connected reductive algebraic group dense?, MathOverflow, URL:https://mathoverflow.net/q/280881 (version: 2019-03-18)

    work page 2019

  9. [9]

    Kottwitz, Harmonic analysis on reductive p -adic groups and L ie algebras , Harmonic analysis, the trace formula, and S himura varieties, Clay Math

    Robert E. Kottwitz, Harmonic analysis on reductive p -adic groups and L ie algebras , Harmonic analysis, the trace formula, and S himura varieties, Clay Math. Proc., vol. 4, Amer. Math. Soc., Providence, RI, 2005, pp. 393--522. 2192014

    work page 2005

  10. [10]

    McNinch, Nilpotent orbits over ground fields of good characteristic, Math

    George J. McNinch, Nilpotent orbits over ground fields of good characteristic, Math. Ann. 329 (2004), no. 1, 49--85. 2052869

    work page 2004

  11. [11]

    Mohrdieck, Conjugacy classes of non-connected semisimple algebraic groups, Transform

    S. Mohrdieck, Conjugacy classes of non-connected semisimple algebraic groups, Transform. Groups 8 (2003), no. 4, 377--395. 2015256

    work page 2003

  12. [12]

    260, 2003, Special issue celebrating the 80th birthday of Robert Steinberg, pp

    Alexander Premet, Nilpotent orbits in good characteristic and the K empf- R ousseau theory , vol. 260, 2003, Special issue celebrating the 80th birthday of Robert Steinberg, pp. 338--366. 1976699

    work page 2003

  13. [13]

    J. A. Shalika, A theorem on semi-simple p -adic groups , Ann. of Math. (2) 95 (1972), 226--242. 0323957 (48 \#2310)

    work page 1972

  14. [14]

    Cheng-Chiang Tsai, (lecture note on) characters of p -adic groups, 2026, to appear, with an incomplete version at https://chengchiangtsai.github.io/2026Spring/Characters_of_p_adic_groups_April_17th_2026.pdf

    work page 2026

  15. [15]

    Reine Angew

    Marie-France Vign\'eras and Jean-Loup Waldspurger, Premiers r\'eguliers de l'analyse harmonique mod l d'un groupe r\'eductif p -adique , J. Reine Angew. Math. 535 (2001), 165--205. 1837098

    work page 2001