pith. sign in

arxiv: 2604.11049 · v1 · submitted 2026-04-13 · 🧮 math.RT · math.NT

Algorithms on the Pyasetskii involution on local Langlands parameters of classical groups

Alexander Hazeltine , Chi-Heng Lo This is my paper

Pith reviewed 2026-05-10 15:51 UTC · model grok-4.3

classification 🧮 math.RT math.NT MSC 22E50
keywords Pyasetskii involutionlocal Langlands parametersclassical groupsSp(2n)SO(2n+1)O(2n)Aubert-Zelevinsky involutionalgorithm
0
0 comments X

The pith

The paper provides an algorithm to compute the Pyasetskii involution for Sp_{2n}, SO_{2n+1} and O_{2n} by combining existing methods.

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

This paper aims to establish a practical algorithm for determining the Pyasetskii involution on local Langlands parameters associated to the classical groups Sp_{2n}, SO_{2n+1}, and O_{2n}. It does so by integrating Moeglin-Waldspurger's approach for the general linear group with Lanard-Mínguez's handling of bad parity cases in the Aubert-Zelevinsky involution. Additionally, it offers a geometric interpretation for the bad parity aspect of the latter algorithm. A reader might care because such involutions play a role in the local Langlands correspondence, potentially aiding in the classification and understanding of representations of these groups over local fields.

Core claim

We give an algorithm to compute the Pyasetskii involution for Sp_{2n}, SO_{2n+1} and O_{2n}. The algorithm is a combination of Moeglin-Waldspurger's algorithm for the Pyasetskii involution for GL_n and Lanard-Mínguez's algorithm for the Aubert-Zelevinsky involution of bad parity representations for classical groups. In particular, we give a geometric interpretation of the bad parity case of Lanard-Mínguez's algorithm.

What carries the argument

A combined algorithmic procedure that merges the Pyasetskii involution computation from GL_n with the Aubert-Zelevinsky involution for bad parity representations, including a geometric reinterpretation of the latter.

If this is right

  • The algorithm allows explicit computation of the involution for parameters of these classical groups.
  • It provides a way to handle cases of bad parity through geometric means.
  • Users can now apply this to verify involution properties or compute dual parameters in the Langlands correspondence.
  • Extends the reach of computable involutions from GL_n to the specified classical groups.

Where Pith is reading between the lines

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

  • This algorithm could be implemented in software to generate examples for testing broader conjectures in representation theory.
  • The geometric interpretation might suggest similar geometric models for other involutions on Langlands parameters.
  • One could test the algorithm on small n where known values exist to confirm correctness.

Load-bearing premise

The combined algorithms and geometric interpretation correctly define and compute the Pyasetskii involution without discrepancies or loss of key properties.

What would settle it

A specific local Langlands parameter for Sp_4 or SO_5 where applying the algorithm yields an output that does not match known values or fails to satisfy involution properties.

read the original abstract

We give an algorithm to compute the Pyasetskii involution for $\mathrm{Sp}_{2n}$, $\mathrm{SO}_{2n+1}$ and $\mathrm{O}_{2n}$. The algorithm is a combination of Moeglin-Waldspurger's algorithm for the Pyasetskii involution for $\mathrm{GL}_n$ ([MW86]) and Lanard-M${\'i}$nguez's algorithm for the Aubert-Zelevinsky involution of bad parity representations for classical groups ([LM25]). In particular, we give a geometric interpretation of the bad parity case of Lanard-M${\'i}$nguez's algorithm.

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

0 major / 3 minor

Summary. The paper constructs an explicit algorithm to compute the Pyasetskii involution on local Langlands parameters for the groups Sp_{2n}, SO_{2n+1} and O_{2n}. The algorithm is obtained by composing Moeglin-Waldspurger's algorithm for the Pyasetskii involution on GL_n with Lanard-Mínguez's algorithm for the Aubert-Zelevinsky involution on bad-parity representations of classical groups, together with a geometric reinterpretation of the bad-parity case of the latter.

Significance. If the composite procedure is shown to coincide with the Pyasetskii involution, the result supplies a concrete computational tool for an important operation in the local Langlands correspondence for classical groups. The geometric reinterpretation of the bad-parity case may also clarify the relationship between the two involutions and could be useful for further structural questions in the theory.

minor comments (3)
  1. §2.3: the definition of the geometric map on the bad-parity parameters is stated only for the split orthogonal case; an explicit extension to the non-split case (or a reference to where it is handled) would improve readability.
  2. The paper cites [MW86] and [LM25] but does not include a short self-contained summary of the input/output formats of those algorithms; adding a one-paragraph recap in §1.2 would make the combination easier to verify.
  3. Figure 1 (p. 7) uses a non-standard shading convention for the nilpotent orbits; a legend or caption clarification would prevent misreading.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the manuscript, including the recommendation for minor revision. We appreciate the recognition of the algorithm's construction via composition of existing methods and the potential utility of the geometric reinterpretation.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper constructs an explicit algorithm for the Pyasetskii involution on Sp_{2n}, SO_{2n+1} and O_{2n} by combining two externally published procedures: Moeglin-Waldspurger's algorithm for GL_n ([MW86]) and Lanard-Mínguez's algorithm for the Aubert-Zelevinsky involution on bad-parity representations ([LM25]), together with a geometric reinterpretation of the latter's bad-parity case. These source algorithms are independent prior work with no author overlap. No step reduces by definition to its own output, no fitted parameter is relabeled as a prediction, and no load-bearing uniqueness theorem or ansatz is imported via self-citation. The derivation is therefore a procedural synthesis of independent external results and remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the correctness of the two cited source algorithms and on standard properties of the local Langlands correspondence and involutions; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The Moeglin-Waldspurger algorithm correctly computes the Pyasetskii involution for GL_n.
    Invoked as the base case for the combination.
  • domain assumption The Lanard-Mínguez algorithm correctly computes the Aubert-Zelevinsky involution for bad-parity representations of classical groups.
    Invoked as the second ingredient and reinterpreted geometrically.

pith-pipeline@v0.9.0 · 5410 in / 1375 out tokens · 34217 ms · 2026-05-10T15:51:50.971831+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

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

  1. [1]

    Adams, N

    J. Adams, N. Arancibia Robert, and P. Mezo, Equivalent definitions of Arthur packets for real classical groups, Mem. Amer. Math. Soc. 300 (2024), no. 1503, v+110

    work page 2024

  2. [2]

    Adams, D

    J. Adams, D. Barbasch and D. Vogan, The Langlands classification and irreducible characters for real reductive groups. (Birkhäuser Boston, Inc., Boston, MA,1992)

    work page 1992

  3. [3]

    Arthur, Unipotent automorphic representations: conjectures

    J. Arthur, Unipotent automorphic representations: conjectures. Ast \'e risque . pp. 13-71 (1989), Orbites unipotentes et repr \' e sentations, II

    work page 1989

  4. [4]

    Atobe and A

    H. Atobe and A. M \'i nguez, The explicit Aubert-Zelevinsky duality. Preprint (2020), arXiv:2008.05689

    work page arXiv 2020

  5. [5]

    Atobe, On an algorithm to compute derivatives

    H. Atobe, On an algorithm to compute derivatives. arXiv:2006.02638

    work page arXiv 2006

  6. [6]

    Atobe, Construction of local A-packets

    H. Atobe, Construction of local A-packets. Journal f\"ur die reine und angewandte Mathematik (Crelles Journal) , to appear,

    work page

  7. [7]

    Atobe, W

    H. Atobe, W. T. Gan, A. Ichino, T. Kaletha, A. M \'i nguez, and S. W. Shin, Local Intertwining Relations and Co-tempered A-packets of Classical Groups (2024), arXiv:2410.13504

    work page arXiv 2024

  8. [8]

    Aubert, Dualit \' e dans le groupe de Grothendieck de la cat \' e gorie des repr \' e sentations lisses de longueur finie d'un groupe r \' e ductif p-adique

    A. Aubert, Dualit \' e dans le groupe de Grothendieck de la cat \' e gorie des repr \' e sentations lisses de longueur finie d'un groupe r \' e ductif p-adique. Trans. Amer. Math. Soc. 347 (1995) 2179-2189

    work page 1995

  9. [9]

    Aubert, Y

    A. Aubert and Y. Xu, The Explicit Local Langlands Correspondence for G_2 . arXiv:2208.12391

    work page arXiv

  10. [10]

    Cunningham, A

    C. Cunningham, A. Fiori, and N. Kitt, Appearance of the Kashiwara-Saito singularity in the representation theory of p -adic _ 16 (2021), Pacific Journal of Mathematics , to appear. arXiv:2103.04538

    work page arXiv 2021

  11. [11]

    Cunningham, A

    C. Cunningham, A. Fiori, A. Moussaoui, J. Mracek, and B. Xu, Arthur packets for p-adic groups by way of microlocal vanishing cycles of perverse sheaves, with examples. Mem. Amer. Math. Soc. 276 (2022), ix+216

    work page 2022

  12. [12]

    Cunningham, A

    C. Cunningham, A. Fiori, A. Moussaoui, and Q. Zhang, Generic representations and ABV-packets for p -adic groups, Preprint, in preparation

    work page

  13. [13]

    Cunningham, A

    C. Cunningham, A. Fiori, and Q. Zhang, Toward the endoscopic classification of unipotent representations of p-adic G_2 , preprint, arXiv: 2101.04578

    work page arXiv

  14. [14]

    Cunningham, A

    C. Cunningham, A. Fiori, and Q. Zhang, Arthur packets for G_2 and perverse sheaves on cubics, Adv. Math. 395 (2022), 108074, 74pp

    work page 2022

  15. [15]

    Cunningham, A

    C. Cunningham, A. Hazeltine, B. Liu, C.-H. Lo, M. Ray, and B. Xu, On Vogan's conjecture on Arthur packets of classical groups over non-Archimedean local fields. In preparation

    work page

  16. [16]

    Cunningham and M

    C. Cunningham and M. Ray, Proof of Vogan's conjecture on Arthur packets: irreducible parameters of p-adic general linear groups. Manuscripta Math. 173, 1073-1097 (2024)

    work page 2024

  17. [17]

    Cunningham and M

    C. Cunningham and M. Ray, Proof of Vogan's conjecture on Arthur packets for GL_n over p -adic fields , manuscripta math. 177 (2026), no. 2

    work page 2026

  18. [18]

    W. T. Gan, B. H. Gross, and D. Prasad, Symplectic local root numbers, central critical L values, and restriction problems in the representation theory of classical groups. Ast \'e risque . pp. 1-109 (2012), Sur les conjectures de Gross et Prasad. I

    work page 2012

  19. [19]

    W. T. Gan and G. Savin, The local Langlands conjecture for G_2 , Forum Math. Pi 11:e28 (2023), 1-42

    work page 2023

  20. [20]

    Harris and R

    M. Harris and R. Taylor, The geometry and cohomology of some simple Shimura varieties. (Princeton University Press, 2001)

    work page 2001

  21. [21]

    Hazeltine, D

    A. Hazeltine, D. Jiang, B. Liu, C.-H. Lo, and Q. Zhang, Arthur representations and unitary dual for classical groups , arXiv:2410.11806v3

    work page arXiv

  22. [22]

    Hazeltine, B

    A. Hazeltine, B. Liu, and C.-H. Lo, On the local theta correspondence and intersection of local Arthur packets for even orthogonal groups. Preprint

    work page

  23. [23]

    Hazeltine, B

    A. Hazeltine, B. Liu, C.-H. Lo, and Q. Zhang, The closure ordering conjecture on local Arthur packets of classical groups. J. Reine Angew. Math. . 823 pp. 1-60 (2025)

    work page 2025

  24. [24]

    Henniart, Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique

    G. Henniart, Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique. Invent. Math. 139 (2000), 439-455

    work page 2000

  25. [25]

    Ishimoto, The endoscopic classification of representations of non-quasi-split odd special orthogonal groups

    H. Ishimoto, The endoscopic classification of representations of non-quasi-split odd special orthogonal groups. Int. Math. Res. Not. IMRN ., 10939-11012 (2024)

    work page 2024

  26. [26]

    Knight and A

    H. Knight and A. Zelevinsky, Representations of quivers of type A and the multisegment duality. Adv. Math. 117, 273-293 (1996)

    work page 1996

  27. [27]

    An Algorithm for Aubert–Zelevinsky Duality\a la M{\oe}glin–Waldspurger.arXiv preprint arXiv:2509.13231, 2025

    Thomas Lanard and Alberto M \'i nguez, An algorithm for Aubert-Zelevinsky duality \`a la M glin-Waldspurger. (2025), arXiv:2509.13231

    work page internal anchor Pith review arXiv 2025

  28. [28]

    C.-H. Lo. Vogan's conjecture on local Arthur packets of p-adic GL_n and a combinatorial lemma. Pacific J. Math. 333, 331-356 (2024)

    work page 2024

  29. [29]

    Matringe, Local converse theorems and Langlands parameters

    N. Matringe, Local converse theorems and Langlands parameters. (2024), arXiv:2409.20240

    work page arXiv 2024

  30. [30]

    M glin, Image des op \'e rateurs d'entrelacements normalis \'e s et p \^ o les des s \'e ries d'Eisenstein

    C. M glin, Image des op \'e rateurs d'entrelacements normalis \'e s et p \^ o les des s \'e ries d'Eisenstein. Adv. Math. 228 (2011), 1068-1134

    work page 2011

  31. [31]

    M glin, D

    C. M glin, D. Renard, Sur les paquets d'Arthur des groupes classiques et unitaires non quasi-d \'e ploy \'e s. Relative Aspects In Representation Theory, Langlands Functoriality And Automorphic Forms . 2221 pp. 341-361 (2018)

    work page 2018

  32. [32]

    M glin and J

    C. M glin and J. Waldspurger, Sur l'involution de Zelevinski. J. Reine Angew. Math. 372 pp. 136-177 (1986)

    work page 1986

  33. [33]

    Mok, Endoscopic classification of representations of quasi-split unitary groups

    C. Mok, Endoscopic classification of representations of quasi-split unitary groups. Mem. Amer. Math. Soc. . Mem. Amer. Math. Soc. 235 (2015), no.1108, vi+248 pp

    work page 2015

  34. [34]

    14, 20 FUNCTORIALITY AND THE THETA CORRESPONDENCE 45

    C. Riddlesden, Combinatorial Approach to ABV-packets for GLn. (2023). arXiv:2304.09598

    work page arXiv 2023

  35. [35]

    Scholze, The local Langlands correspondence for _n over p-adic fields

    P. Scholze, The local Langlands correspondence for _n over p-adic fields. Invent. Math. 192, (2013), 663-715

    work page 2013

  36. [36]

    Steinberg, Regular elements of semisimple algebraic groups

    R. Steinberg, Regular elements of semisimple algebraic groups. Inst. Hautes \'E tudes Sci. Publ. Math. , 49-80 (1965)

    work page 1965

  37. [37]

    On reducibility and unitarizability for classical p-adic groups, some general results

    Tadić, M. On reducibility and unitarizability for classical p-adic groups, some general results. Canad. J. Math. 61, 427-450 (2009)

    work page 2009

  38. [38]

    Vogan, The local Langlands conjecture

    D. Vogan, The local Langlands conjecture. Representation Theory Of Groups And Algebras . 145 (1993), 305-379

    work page 1993

  39. [39]

    Xu, On the cuspidal support of discrete series for p-adic quasisplit Sp(N) and SO(N)

    B. Xu, On the cuspidal support of discrete series for p-adic quasisplit Sp(N) and SO(N). Manuscripta Math. 154, 441-502 (2017)

    work page 2017

  40. [40]

    Xu, On M glin's parametrization of Arthur packets for p-adic quasisplit Sp(N) and SO(N)

    B. Xu, On M glin's parametrization of Arthur packets for p-adic quasisplit Sp(N) and SO(N). Canad. J. Math. 69, (2017), 890-960

    work page 2017

  41. [41]

    Zelevinsky

    A. Zelevinsky. Induced representations of reductive p-adic groups. II. On irreducible representations of GL(n). Ann. Sci. \'E cole Norm. Sup. (4) . 13, 165-210 (1980)

    work page 1980

  42. [42]

    Zelevinsky

    A. Zelevinsky. The p-adic analogue of the Kazhdan-Lusztig conjecture. Funktsional. Anal. I Prilozhen. 15, 9-21, 96 (1981)

    work page 1981