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arxiv: 2506.22986 · v2 · submitted 2025-06-28 · 🧮 math.RT

Howe duality over finite fields III: Full computation and the Gurevich-Howe conjectures

Sophie Kriz This is my paper

Pith reviewed 2026-05-19 08:18 UTC · model grok-4.3

classification 🧮 math.RT
keywords Howe dualityoscillator representationfinite symplectic groupsfinite orthogonal groupsGurevich-Howe conjecturesunipotent representationsrepresentation theory of finite groups
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The pith

The oscillator representation over finite fields restricts completely to all occurring dual pairs of symplectic and orthogonal groups, yielding recursive constructions of their irreducible representations.

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

This paper provides a full description of the restriction of the oscillator representation to products of dual pairs involving symplectic and orthogonal groups over finite fields. It includes a dictionary matching the occurring tensor products to those identified in Pan's work. Using this, it constructs a recursive way to build all irreducible complex representations of these finite groups and gives a formula for characters of unipotent cuspidal representations. It also proves the rank and exhaustion conjectures of Gurevich and Howe specifically for type C groups. A reader would care because this organizes the representation theory of these groups in a systematic, computable manner rather than through isolated cases.

Core claim

We give a complete description of the restriction of the oscillator representation over a finite field to products of dual pairs of symplectic and orthogonal groups in all cases that occur. We also provide a dictionary with the notation of S.-Y. Pan, who identified which tensor products of irreducible representations occur with non-zero multiplicity. As an application, we give a recursive construction of all irreducible complex representations of finite symplectic and orthogonal groups and a recursive formula for the characters of unipotent cuspidal representations. We also give a proof of the Gurevich-Howe rank and exhaustion conjectures for type C groups.

What carries the argument

The oscillator representation restricted to dual pairs, with a multiplicity dictionary to Pan's tensor product identifications.

If this is right

  • A complete multiplicity dictionary for the decompositions in all dual pair cases.
  • Recursive construction of all irreducible complex representations of finite symplectic and orthogonal groups.
  • Recursive formula for the characters of unipotent cuspidal representations.
  • Proof of the Gurevich-Howe rank and exhaustion conjectures for type C groups.

Where Pith is reading between the lines

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

  • The completion of this series allows for a unified treatment of Howe duality over finite fields across types.
  • Implementing the recursive construction for small finite fields could verify against existing character tables of symplectic groups.
  • Extensions might include similar computations for other dual pair types or connections to modular representation theory.

Load-bearing premise

The oscillator representation and dual pair identifications from the previous papers in the series are correctly defined and the correspondence with Pan's multiplicity results holds without omissions.

What would settle it

An explicit computation of the restriction for a small dual pair, such as the symplectic group Sp(2) and orthogonal group over F_5, showing whether the predicted irreducible constituents and multiplicities match the recursive construction output.

Figures

Figures reproduced from arXiv: 2506.22986 by Sophie Kriz.

Figure 1
Figure 1. Figure 1: These are the top row Young diagrams corresponding to the symbols (69), in the case of α = (1 ≤ 3 ≤ 4), N′ ρ − a = 6. The boxes highlighted gray show the skew semistandard tableau which, after removal from each Young diagram recovers the original α. At each i, just enough boxes are added to a row of the (i + 1)th Young diagram to not be obtainable from the (i + 2)th. appearing in the ith term of the altern… view at source ↗
read the original abstract

In this third paper in a series on type I Howe duality for finite fields, we give a complete description of the restriction of the oscillator representation over a finite field to products of dual pairs of symplectic and orthogonal groups in all cases that occur. We also provide a dictionary with the notation of S.-Y. Pan, who identified which tensor products of irreducible representations occur with non-zero multiplicity. As an application, we give a recursive construction of all irreducible complex representations of finite symplectic and orthogonal groups and a recursive formula for the characters of unipotent cuspidal representations. We also give a proof of the Gurevich-Howe rank and exhaustion conjectures for type C groups.

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. This third paper in a series on type I Howe duality over finite fields claims to give a complete description of the restriction of the oscillator representation to all occurring products of dual pairs of symplectic and orthogonal groups. It supplies a dictionary relating the multiplicities to the notation of S.-Y. Pan, derives a recursive construction of all irreducible complex representations of finite symplectic and orthogonal groups together with a recursive formula for the characters of unipotent cuspidal representations, and proves the Gurevich-Howe rank and exhaustion conjectures for type C groups.

Significance. If the derivations and case-by-case verifications hold, the work would complete the explicit Howe correspondence for finite fields of odd and even characteristic, furnish concrete recursive tools for constructing representations and characters, and resolve longstanding conjectures. These results would be of substantial value to the representation theory of finite classical groups and to applications in the Langlands program over finite fields.

major comments (2)
  1. [§2, §5] §2 and the recursive construction in §5: the completeness claim for the restriction description and the recursive formulas for all irreducibles rest on the oscillator representation and dual-pair identifications established in the preceding papers of the series; without an explicit cross-reference or independent verification that every occurring dual pair (including small-rank orthogonal cases and even q) is covered by the multiplicity dictionary, the recursion may miss families of representations.
  2. [§6] The proof of the Gurevich-Howe conjectures in §6: the rank and exhaustion statements are deduced from the multiplicity dictionary and the recursive character formula; if any multiplicity in the dictionary with Pan’s work is incomplete for a single family, the exhaustion argument fails to cover all unipotent cuspidal representations.
minor comments (2)
  1. [§1] Notation for the dual pairs and the oscillator representation should be restated briefly in §1 or an appendix so that the paper can be read independently of the earlier installments.
  2. The tables or explicit multiplicity lists that accompany the dictionary with Pan’s work would benefit from a uniform indexing scheme that makes the correspondence with the recursive construction immediately visible.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive major comments. We address each point below and have revised the text to improve explicitness and cross-references while preserving the logical structure of the series.

read point-by-point responses

  1. Referee: [§2, §5] §2 and the recursive construction in §5: the completeness claim for the restriction description and the recursive formulas for all irreducibles rest on the oscillator representation and dual-pair identifications established in the preceding papers of the series; without an explicit cross-reference or independent verification that every occurring dual pair (including small-rank orthogonal cases and even q) is covered by the multiplicity dictionary, the recursion may miss families of representations.

    Authors: We agree that explicit cross-references strengthen the presentation. In the revised manuscript we have inserted a new paragraph at the end of §2 that enumerates every dual pair that arises, with precise citations to the oscillator-restriction theorems of Papers I and II. This enumeration explicitly includes the small-rank orthogonal cases (GO(2), GO(4), etc.) and the even-characteristic setting. The multiplicity dictionary of §5 is derived directly from these identifications; we have added a short verification remark confirming that no occurring pair is omitted. Consequently the recursive construction of all irreducibles in §5 rests on a fully referenced foundation. revision: yes

  2. Referee: [§6] The proof of the Gurevich-Howe conjectures in §6: the rank and exhaustion statements are deduced from the multiplicity dictionary and the recursive character formula; if any multiplicity in the dictionary with Pan’s work is incomplete for a single family, the exhaustion argument fails to cover all unipotent cuspidal representations.

    Authors: The exhaustion argument in §6 proceeds by induction on rank, using the multiplicity dictionary to generate all unipotent cuspidals from lower-rank ones. We have added a clarifying lemma that records the exact matching between our dictionary and Pan’s classification for every family, thereby confirming that the inductive step covers the complete set. The rank conjecture is invoked only after this completeness is established. While the referee is correct that an incomplete multiplicity would break exhaustion, the dictionary is complete by construction from the Howe correspondence proved in the series; the added lemma makes this dependence fully transparent. revision: partial

Circularity Check

0 steps flagged

Series dependence on prior definitions does not reduce new computations or conjecture proofs to self-citation by construction.

full rationale

The paper is the third in a series and explicitly relies on the oscillator representation and dual-pair identifications established in prior installments. However, the central contributions—a complete restriction description, multiplicity dictionary with Pan's work, recursive constructions of irreducibles, character formulas, and proofs of the Gurevich-Howe rank and exhaustion conjectures—are presented as arising from independent case-by-case analysis and explicit computations in the current work. No equation or claim is shown to be equivalent to its inputs by definition, no fitted parameter is relabeled as a prediction, and no load-bearing uniqueness theorem is imported solely via overlapping-author citation without external verification. The derivation chain therefore remains self-contained against the new results even while citing foundational objects from the series.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper builds on the oscillator representation and dual-pair framework established in the first two papers of the series; no new free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption The oscillator representation and type I dual pairs for finite fields are defined and behave as in the preceding papers of the series.
    The complete description and recursive constructions presuppose the correctness of the setup from earlier installments.

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Works this paper leans on

46 extracted references · 46 canonical work pages

  1. [1]

    Adams, A

    J. Adams, A. Moy. Unipotent representations and reductive dual pairs over finite fields. Trans. Amer. Math. Soc., 340 (1993), 309–321

    work page 1993

  2. [2]

    Aubert, W

    A.-M. Aubert, W. Kra´ skiewicz, T. Przebinda. Howe correspondence and Springer correspondence for dual pairs over a finite field, In: Lie Algebras, Lie Superalgebras, Vertex Algebras and Related Topics, Proc. Sympos. Pure Math., 92, Amer. Math. Soc., Providence, RI, 2016, pp. 17-44. 60 SOPHIE KRIZ

    work page 2016

  3. [3]

    Aubert, J

    A.-M. Aubert, J. Michel, R. Rouquier. Correspondance de Howe pour les groupes r´ eductifs sur les corps finis.Duke Math. J. , 83 (1996), 353-397

    work page 1996

  4. [4]

    R. Carter. Centralizers of semisimple elements in the finite classical groups Proc. London Math. Soc., 42 (1981), 1-41

    work page 1981

  5. [5]

    C. W. Curtis, I. Reiner. Methods of representation theory. Vol. II. With ap- plications to finite groups and orders. Pure Appl. Math. (N.Y.) Wiley-Intersci. Publ. John Wiley & Sons, Inc., New York, 1987. xviii+951 pp

    work page 1987

  6. [6]

    P. Deligne. La cat´ egorie des repr´ esentations du groupe sym´ etriqueSt, lorsque t n’est pas un entier naturel. Algebraic groups and homogeneous spaces, 209-273, Tata Inst. Fund. Res. Stud. Math., 19, Tata Inst. Fund. Res., Mumbai, 2007

    work page 2007

  7. [7]

    P. Deligne. Cat´ egories Tensorielles.Mosc. Math. J . 2 2 (2002) pp. 227-248

    work page 2002

  8. [8]

    Deligne, J

    P. Deligne, J. Milne: Cat´ egories Tannakiennes, Grothendieck Festschrift, vol. II, Birkh¨ auser Progress in Math. 87, 1990, pp. 111-195

    work page 1990

  9. [9]

    Deligne, private communication

    P. Deligne, private communication

    work page

  10. [10]

    Epequin Chavez

    J. Epequin Chavez. Extremal unipotent representations for the finite Howe correspondence, J. Algebra 535 (2019), 480-502

    work page 2019

  11. [11]

    Etingof, S

    P. Etingof, S. Gelaki, D. Nikshych, V. Ostrik: Tensor categories. Math. Surveys Monogr., 205 American Mathematical Society, Providence, RI, 2015, xvi+343 pp

    work page 2015

  12. [12]

    W.T. Gan, S. Takeda. A proof of the Howe duality conjecture. J. Amer. Math. Soc., 29 (2016), 473-493

    work page 2016

  13. [13]

    S.S. Gelbart. Examples of dual reductive pairs, In: Automorphic Forms, Rep- resentations and L-functions , Oregon State Univ., Corvallis, OR, 1977, Proc. Sympos. Pure Math., 33, Amer. Math. Soc., Providence, RI, 1979, pp. 287-296

    work page 1977

  14. [14]

    G´ erardin

    P. G´ erardin. Weil representations associated to finite fields. J. Algebra , 46 (1977), 54-101

    work page 1977

  15. [15]

    Gurevich, R

    S. Gurevich, R. Howe. Rank and duality in representation theory,Jpn. J. Math. 15 (2020), 223-309

    work page 2020

  16. [16]

    Gurevich, R

    S. Gurevich, R. Howe. Small representations of finite classical groups, Progr. Math., 323 Birkh¨ auser/Springer, Cham, 2017, 209-234

    work page 2017

  17. [17]

    Harman, A

    N. Harman, A. Snowden. Oligomorphic groups and tensor categories. arXiv:2204.04526, 2022. 61

    work page arXiv 2022

  18. [18]

    R. Howe. Invariant Theory and Duality for Classical Groups over Finite Fields, with Applications to their Singular Representation Theory, preprint, Yale Uni- versity

    work page

  19. [19]

    R. Howe. On the character of Weil’s representation, Trans. Amer. Math. S. 177 (1973), 287-298

    work page 1973

  20. [20]

    R. Howe. The oscillator semigroup over finite fields, to appear in Symme- try in Geometry and Analysis, Volume 1: Festschrift in Honor of Toshiyuki Kobayashi, 2025

    work page 2025

  21. [21]

    Howe, θ-series and invariant theory, In: Automorphic Forms, Representa- tions and L-Functions, Oregon State Univ., Corvallis, OR, 1977,Proc

    R. Howe, θ-series and invariant theory, In: Automorphic Forms, Representa- tions and L-Functions, Oregon State Univ., Corvallis, OR, 1977,Proc. Sympos. Pure Math., 33, Amer. Math. Soc., Providence, RI, 1979, pp. 275-285

    work page 1977

  22. [22]

    N. M. Katz. Larsen’s alternative, moments, and the monodromy of Lefschetz pencils. Contributions to automorphic forms, geometry, and number theory , 521-560. Johns Hopkins University Press, Baltimore, MD, 2004

    work page 2004

  23. [23]

    N. M. Katz, P. H. Tiep. Moments, exponential sums, and monodromy groups. Available at https://web.math.princeton.edu/∼nmk/kt24-70.pdf

    work page

  24. [24]

    N. M. Katz, P. H. Tiep. On a Conjecture of Miyamoto, preprint

    work page

  25. [25]

    F. Knop. A construction of semisimple tensor categories. C. R. Math. Acad. Sci. Paris C . 343, 2006

    work page 2006

  26. [26]

    F. Knop. Tensor Envelopes of Regular Categories. Adv. Math. 214, 2007

    work page 2007

  27. [27]

    S. Kriz. Quantum Delannoy categories, 2023. Available at https://krizsophie.github.io/QuantumDelannoyCategory23111.pdf

    work page 2023

  28. [28]

    S. Kriz. Oscillator Representations and Semisimple Pre-Tannakian Categories,

    work page

  29. [29]

    Available at https://krizsophie.github.io/WeilShale240315.pdf

    work page

  30. [30]

    S. Kriz. Interpolated equivariant schemes, 2025. Available at https://krizsophie.github.io/InterpolatedSchemes25019.pdf

    work page 2025

  31. [31]

    S. Kriz. Howe duality over finite fields I: The two stable ranges, preprint, 2025

    work page 2025

  32. [32]

    S. Kriz. Howe duality over finite fields II: Explicit stable computations, preprint, 2025

    work page 2025

  33. [33]

    S. S. Kudla. On the local theta-correspondence. Invent. Math., 83 (1986), 229- 255. 62 SOPHIE KRIZ

    work page 1986

  34. [34]

    S. S. Kudla, J.J. Millson. The theta correspondence and harmonic forms. I, Math. Ann., 274 (1986), 353-378

    work page 1986

  35. [35]

    D. Liu, Z. Wang. Remarks on the theta correspondence over finite fields.Pacific J. Math. 306 (2020), 587-609

    work page 2020

  36. [36]

    G. Lusztig. Irreducible representations of finite classical groups. Invent. Math. 43 (1977), 125-175

    work page 1977

  37. [37]

    Lusztig: Characters of Reductive Groups over a Finite Field

    G. Lusztig: Characters of Reductive Groups over a Finite Field. Ann. of Math. Stud., 107 Princeton University Press, Princeton, NJ, 1984, xxi+384 pp

    work page 1984

  38. [38]

    Montealegre-Mora, D

    F. Montealegre-Mora, D. Gross. Rank-deficient representations in the theta correspondence over finite fields arise from quantum codes. Rep. Theory, 25 (2021), 193-223

    work page 2021

  39. [39]

    Moeglin, M.-F

    C. Moeglin, M.-F. Vign´ eras, J.-L. Waldspurger:Correspondances de Howe sur un corps p-adique . Lecture Notes in Math., 1291, Springer-Verlag, 1987

    work page 1987

  40. [40]

    S.-Y. Pan. Howe correspondence of unipotent characters for a finite symplectic/even-orthogonal dual pair. Am. J. Math. , John Hopkins Univ. Press, 146 (2024), 813-869

    work page 2024

  41. [41]

    S.-Y. Pan. Lusztig correspondence and Howe correspondence for finite reduc- tive dual pairs. Math. Ann. 390 (2024), 4657-4699

    work page 2024

  42. [42]

    Prasad, Weil representation, Howe duality, and the theta correspondence, In: Theta Functions: From the Classical to the Modern, CRM Proc

    D. Prasad, Weil representation, Howe duality, and the theta correspondence, In: Theta Functions: From the Classical to the Modern, CRM Proc. Lecture Notes, 1, Amer. Math. Soc., Providence, RI, 1993, pp. 105-127

    work page 1993

  43. [43]

    Srinivasan

    B. Srinivasan. Weil representations of classical groups. Invent. Math. , 51 (1979), 143-153

    work page 1979

  44. [44]

    Waldspurger: D´ emonstration d’une conjecture de dualit´ e de Howe dans le cas p-adique, p ̸= 2

    J.-L. Waldspurger: D´ emonstration d’une conjecture de dualit´ e de Howe dans le cas p-adique, p ̸= 2. Festschrift in Honor of I. I. Piatetski-Shapiro on the Occasion of His Sixtieth Birthday, Part I, Israel Math. Conf. Proc. , 2 (1990), 267-324

    work page 1990

  45. [45]

    A. Weil. Sur certains groupes d’operateurs unitaires, Acta Math. 111 (1964), 143-211

    work page 1964

  46. [46]

    Z. Yun. Theta correspondence and relative Langlands. Harvard Arithmetic Quantum Field Theory Conference, March 29, 2024, https://www.youtube.com/watch?v=Bb6aTlzNDV4

    work page 2024