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arxiv: 2605.15296 · v1 · pith:G5P4ABSHnew · submitted 2026-05-14 · 🧮 math.RT

Complex Weyl correspondence and harmonic representation of SU(p,q)

Benjamin Cahen This is my paper

Pith reviewed 2026-05-19 15:48 UTC · model grok-4.3

classification 🧮 math.RT
keywords harmonic representationcomplex Weyl correspondenceFock spaceSU(p,q)Heisenberg grouprepresentation theory
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The pith

The paper gives explicit formulas for the complex Weyl symbols of the harmonic representation operators of SU(p,q) on the Fock space.

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

The author studies the harmonic representation of SU(p,q) by relating it to the complex Weyl correspondence on the Fock space. Explicit formulas are derived for the symbols of the operators in this representation. The same is done for the extended version involving the Heisenberg group. This matters because it turns the abstract action of the group into explicit functions that can be used in symbol-based calculations.

Core claim

We study the harmonic representation of SU(p,q) in connection to the complex Weyl correspondence on the Fock space. In particular, we give explicit formulas for the complex Weyl symbols of the harmonic representation operators. Similar results are also obtained for the extended harmonic representation of the semi-direct product of the (2p+2q+1)-dimensional Heisenberg group by SU(p,q).

What carries the argument

The complex Weyl correspondence, which provides a way to associate symbols to operators on the Fock space.

If this is right

  • Formulas allow explicit computation of symbols for all harmonic representation operators.
  • The method applies similarly to the extended representation with the Heisenberg group.
  • This establishes a direct link between the representation and the symbol calculus.

Where Pith is reading between the lines

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

  • The explicit symbols may simplify the study of intertwining operators or matrix coefficients in this representation.
  • It opens the possibility of applying Weyl symbol techniques to problems in non-compact group representations.

Load-bearing premise

The harmonic representation of SU(p,q) is realized on the Fock space in a manner compatible with the complex Weyl correspondence.

What would settle it

Deriving the symbol from the formula for a particular operator and then using the correspondence to recover the operator, and finding it does not match the original representation operator.

read the original abstract

We study the harmonic representation of $SU(p,q)$ in connection to the complex Weyl correspondence on the Fock space. In particular, we give explicit formulas for the complex Weyl symbols of the harmonic representation operators. Similar results are also obtained for the extended harmonic representation of the semi-direct product of the $(2p+2q+1)$-dimensional Heisenberg group by $SU(p,q)$.

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 / 2 minor

Summary. The manuscript studies the harmonic representation of SU(p,q) in connection with the complex Weyl correspondence on Fock space. It derives explicit formulas for the complex Weyl symbols of the harmonic representation operators and obtains analogous results for the extended harmonic representation of the semidirect product of the (2p+2q+1)-dimensional Heisenberg group with SU(p,q).

Significance. If the derivations hold, the work supplies concrete, usable formulas that connect the harmonic representation (via its standard realization on Fock space and the holomorphic discrete series) to the complex Weyl symbol calculus. This is a genuine contribution to explicit computations in the representation theory of indefinite unitary groups and their oscillator-type extensions; the absence of free parameters or ad-hoc fitting in the construction is a strength.

minor comments (2)
  1. The abstract states the main results but does not indicate the key identifications (Fock space with holomorphic discrete series, compatibility with the complex Weyl correspondence) that the derivations rely upon; a single sentence clarifying these standard inputs would improve accessibility.
  2. Notation for the Fock space inner product, the complex Weyl symbol map, and the action of SU(p,q) should be collected in a single preliminary section or table to avoid repeated re-definition across sections.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript and for recommending minor revision. No specific major comments were listed in the report, so we will prepare a revised version incorporating any minor editorial or technical clarifications as needed.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained on standard definitions

full rationale

The paper derives explicit formulas for complex Weyl symbols of harmonic representation operators of SU(p,q) on Fock space, together with the semidirect product case. These rest on standard identifications of Fock space with the holomorphic discrete series and on the known action of the complex Weyl correspondence, without any reduction of the claimed formulas to fitted parameters, self-definitions, or load-bearing self-citations. The abstract and construction supply independent content from prior independent results, with no steps that equate outputs to inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the prior existence and standard properties of the harmonic representation of SU(p,q) and the complex Weyl correspondence on Fock space; no free parameters, new entities, or ad-hoc axioms are mentioned in the abstract.

axioms (1)
  • domain assumption The harmonic representation of SU(p,q) exists and acts on the Fock space
    The paper studies this representation in connection with the Weyl correspondence.

pith-pipeline@v0.9.0 · 5571 in / 1083 out tokens · 43116 ms · 2026-05-19T15:48:58.457178+00:00 · methodology

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Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

  1. [1]

    Arazy and H

    J. Arazy and H. Upmeier, Invariant symbolic calculi and eigenvalues of invariant operators on sym- metric domains , Function spaces, interpolation theory and related topics (Lund, 2000) 151-211, de Gruyter, Berlin, 2002

    work page 2000

  2. [2]

    Arazy and H

    J. Arazy and H. Upmeier, Weyl Calculus for Complex and Real Symmetric Domains , Harmonic analysis on complex homogeneous domains and Lie groups (Rome, 2001). Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 13, no 3-4 (2002), 165–181

    work page 2001

  3. [3]

    F. A. Berezin, Quantization, Math. USSR Izv. 8, 5 (1974), 1109-1165

    work page 1974

  4. [4]

    F. A. Berezin, Quantization in complex symmetric domains, Math. USSR Izv. 9, 2 (1975), 341-379

    work page 1975

  5. [5]

    R. J. Blattner and J. H. Rawnsley, Quantization of the action of U(k, l ) on R2(k+l), J. Funct. Anal. 50 (1983), 188–214

    work page 1983

  6. [6]

    Cahen, Berezin Quantization and Holomorphic Representations , Rend

    B. Cahen, Berezin Quantization and Holomorphic Representations , Rend. Sem. Mat. Univ. Padova 129 (2013), 277-297

    work page 2013

  7. [7]

    Cahen, Stratonovich-Weyl correspondence for the diamond group, Riv

    B. Cahen, Stratonovich-Weyl correspondence for the diamond group, Riv. Mat. Univ. Parma 4 (2013), 197–213

    work page 2013

  8. [8]

    Cahen, Weyl calculus on the Fock space and Stratonovich-Weyl correspondence for Heisenberg motion groups, Rend

    B. Cahen, Weyl calculus on the Fock space and Stratonovich-Weyl correspondence for Heisenberg motion groups, Rend. Semin. Mat. Univ. Politec. Torino 76 (2018), 63-79

    work page 2018

  9. [9]

    Cahen, The complex Weyl calculus as a Stratonovich-Weyl correspondence for the diamond group , Tsukuba J

    B. Cahen, The complex Weyl calculus as a Stratonovich-Weyl correspondence for the diamond group , Tsukuba J. Math. 44 (2020), 121-137

    work page 2020

  10. [10]

    Cahen, A note on the harmonic representation of SU (p, q), Nihonkai Math

    B. Cahen, A note on the harmonic representation of SU (p, q), Nihonkai Math. J. 33 (2022), 61-79

    work page 2022

  11. [11]

    Cahen, Complex Weyl symbols of metaplectic operators: an elementary approach , Rend

    B. Cahen, Complex Weyl symbols of metaplectic operators: an elementary approach , Rend. Istit. Mat. Univ. Trieste 55 (2023), Art. No. 5, 27 pp

    work page 2023

  12. [12]

    Cahen, Complex Weyl symbols of the extended metaplectic representation operators , in prepara- tion

    B. Cahen, Complex Weyl symbols of the extended metaplectic representation operators , in prepara- tion

    work page

  13. [13]

    and Robert, D., Coherent states and applications in mathematical physics, Theo- retical and Mathematical Physics, Springer, Dordrecht, 2012

    Combescure, M. and Robert, D., Coherent states and applications in mathematical physics, Theo- retical and Mathematical Physics, Springer, Dordrecht, 2012

    work page 2012

  14. [14]

    and Robert, D., Quadratic quantum Hamiltonians revisited , Cubo 8 (2006), 61-86

    Combescure, M. and Robert, D., Quadratic quantum Hamiltonians revisited , Cubo 8 (2006), 61-86

    work page 2006

  15. [15]

    M. G. Davidson, The harmonic representation of U(p, q) and its connection with the generalized unit disk, Pacific J. Math. 129 (1987), 33-55

    work page 1987

  16. [16]

    Folland, Harmonic Analysis in Phase Space, Princeton Univ

    B. Folland, Harmonic Analysis in Phase Space, Princeton Univ. Press, 1989

    work page 1989

  17. [17]

    Gracia-Bond` ıa, J. M.,Generalized Moyal quantization on homogeneous symplectic spaces , Deforma- tion theory and quantum groups with applications to mathematical physics (Amherst, MA, 1990), 93–114, Contemp. Math., 134, Amer. Math. Soc., Providence, RI, 1992

    work page 1990

  18. [18]

    H¨ ormander, The analysis of linear partial differential operators, Vol

    L. H¨ ormander, The analysis of linear partial differential operators, Vol. 3, Section 18.5, Springer- Verlag, Berlin, Heidelberg, New-York, 1985

    work page 1985

  19. [19]

    H¨ ormander,Symplectic classification of quadratic forms, and general Mehler formulas , Math

    L. H¨ ormander,Symplectic classification of quadratic forms, and general Mehler formulas , Math. Z. 219 (1995), 413-449

    work page 1995

  20. [20]

    Howe, The oscillator semigroup

    R. Howe, The oscillator semigroup . The mathematical heritage of Hermann Weyl (Durham, NC, 1987), 61–132, Proc. Sympos. Pure Math., 48, Amer. Math. Soc., Providence, RI, 1988

    work page 1987

  21. [21]

    Kashiwara and M

    M. Kashiwara and M. Vergne, On the Segal-Shale-Weil Representations and Harmonic Polynomials , Inventiones Math. 44 (1978), 1-47

    work page 1978

  22. [22]

    Luo, S., Polar decomposition and isometric integral transforms, Int. Transf. Spec. Funct. 9, 4 (2000), 313–324

    work page 2000

  23. [23]

    J. D. Lorch, An integral transform and ladder representations of U(p, q), Pacific J. Math. 186 (1998), 89-109

    work page 1998

  24. [24]

    Neeb, Holomorphy and Convexity in Lie Theory, de Gruyter Expositions in Mathematics, Vol

    K-H. Neeb, Holomorphy and Convexity in Lie Theory, de Gruyter Expositions in Mathematics, Vol. 28, Walter de Gruyter, Berlin, New-York 2000

    work page 2000

  25. [25]

    Ørsted and G

    B. Ørsted and G. Zhang, Weyl Quantization and Tensor Products of Fock and Bergman Spaces , Indiana Univ. Math. J. 43, 2 (1994), 551-583

    work page 1994

  26. [26]

    Peetre, The Fock bundle

    J. Peetre, The Fock bundle . Analysis and partial differential equations, 301–326, Lecture Notes in Pure and Appl. Math., 122, Dekker, New York, 1990

    work page 1990

  27. [27]

    Peetre, Some calculations related to Fock space and the Shale-Weil representation , Integral Equa- tions Operator Theory 12 (1989), 67–81

    J. Peetre, Some calculations related to Fock space and the Shale-Weil representation , Integral Equa- tions Operator Theory 12 (1989), 67–81. 16 BENJAMIN CAHEN

    work page 1989

  28. [28]

    Sternberg and J

    S. Sternberg and J. A. Wolf, Hermitian Lie algebras and metaplectic representations. I. Trans. Amer. Math. Soc. 238 (1978), 1–43

    work page 1978

  29. [29]

    M. E. Taylor, Noncommutative Harmonic Analysis, Mathematical Surveys and Monographs 22, American Mathematical Society, Providence, Rhode Island 1986

    work page 1986

  30. [30]

    and Upmeier, H., Berezin transform and invariant differential operators , Commun

    Unterberger, A. and Upmeier, H., Berezin transform and invariant differential operators , Commun. Math. Phys. 164, 3 (1994), 563-597. Universit´e de Lorraine, Site de Metz, UFR-MIM, D ´epartement de math ´ematiques, Bˆatiment A, 3 rue Augustin Fresnel, BP 45112, 57073 METZ Cedex 03, France. Email address : benjamin.cahen@univ-lorraine.fr

    work page 1994