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Complex Weyl correspondence for a generalized diamond group

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Pith's one-line read For every generic representation of the generalized diamond group $G=\mathbb{R}^m\ltimes H_n$ on the Fock space, the complex Weyl correspondence $W_0$ is a Stratonovich-Weyl correspondence: a $G$-covariant, tracial isomorphism from…

desk verdict Extends the complex Weyl correspondence to the generalized diamond group with explicit symbol formulas and a Stratonovich–Weyl correspondence; the main results hold up, but a conjugate-sign typo in the key extension proof and a small existence gap in one corollary need fixing before the paper is fully rigorous. read the letter →

arxiv 2509.08082 v1 pith:YPSOJJJL submitted 2025-09-09 math.RT

Benjamin Cahen This is my paper · ORCID
classification math.RT MSC 22E4522E7081R0581S1081R30
keywords complexWeylcorrespondenceStratonovich-WeylgeneralizeddiamondgroupFockspaceBargmann-FockrepresentationBerezinquantizationcoadjointorbitsMoyalproduct
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper extends the Stratonovich–Weyl quantization program from the Heisenberg group and the classic diamond group to the whole family of generalized diamond groups $G=\mathbb{R}^m\ltimes H_n$. Its main claim is that, for each generic representation $\pi$ of $G$ on the Fock space $F_\lambda$, the complex Weyl correspondence $W_0$ satisfies the four defining axioms: it maps the identity to 1, respects conjugation, is covariant with respect to $\pi$, and is isometric/tracial. The paper also gives explicit formulas for the Berezin and complex Weyl symbols of all representation operators and their Lie-algebra counterparts, and uses them to recover known Moyal-product formulas and to compute star exponentials of quadratic polynomials. If the main claim is right, the generalized diamond groups become one of the few families of solvable groups for which quantization on coadjoint orbits is realized by closed, computable maps.

What carries the argument

The machinery has three components. The first is the Fock-space model: $F_\lambda$ consists of holomorphic functions on $\mathbb{C}^n$ square-integrable against $e^{-\lambda|z|^2/2}d\mu_\lambda(z)$, with coherent states $e_z(w)=\exp(\lambda\bar z w/2)$. The second is the integral formula $W_0(A)(z)=2^n\int k_A(z+w,z-w)\exp\big(\tfrac\lambda2(-z\bar z-w\bar w+z\bar w-\bar z w)\big)d\mu_\lambda(w)$ for the complex Weyl symbol, which extends $W_0$ beyond trace-class operators and is the unitary part of the polar decomposition of the Berezin map. The third is the equivariant moment map $\psi$ above, whose covariance under the coadjoint action lifts the $G$-action on $\mathbb{C}^n$ to the orbit $O(\xi_0)$; this is what turns the covariance of $W_0$ into the orbit picture that defines a Stratonovich–Weyl correspondence.

What would settle it

Take the exact intertwining relation (4.1) with the correct kernels: for $h=(z_0,c_0)$ the left hand side contains $\exp\big(\tfrac{\lambda}{2}\overline{t\cdot z_0}\, z\big)$ and the right hand side contains $\exp\big(-\tfrac{\lambda}{2}\overline{w}\, z_0\big)$; substitute $w=0$ and solve for $b_t(z,w)$. The classification claim holds exactly if the unique solution is $b_t(z,w)=\chi(t)\exp\big(\tfrac{\lambda}{2}\overline{t\cdot w}\, z\big)$; any other solution disproves it.

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Extended reading notes

Core claim

The central discovery is that the map $W_0$ constructed from the Stratonovich–Weyl quantizer $\Omega_0(z)=\rho_\lambda(z,0)R_0\rho_\lambda(z,0)^{-1}$ is a Stratonovich–Weyl correspondence for the triple $(G,\pi,\mathbb{C}^n)$, and that, after composing with the moment map $\psi:\mathbb{C}^n\to \mathfrak{g}^*$, it becomes one for the triple $(G,\pi,O(\xi_0))$ on a genuine coadjoint orbit. The proof combines: (a) a classification of all unitarizable extensions of $\rho_\lambda$ to $G$, each of the form $\sigma(t)f(z)=\chi(t)f(t^{-1}\cdot z)$; (b) covariance identities for both the Berezin symbol $S_\lambda$ and the complex Weyl symbol $W_0$ under the $G$-action $g\cdot z=t\cdot z+z_0$; and (c) the identification $\psi(z)=(-i\,d\chi+\tfrac12\sum_k(1-\lambda|z_k|^2)\alpha_k,\ -\lambda z,\ \lambda)$, a $G$-equivariant bijection onto $O(\xi_0)$. On the way the paper computes the kernels and symbols of $\pi(g)$ and $d\pi(X)$ in closed form.

Load-bearing premise

The classification of extensions in Proposition 4.1 carries everything: the proof assumes every unitary extension of the Heisenberg representation to $G$ is given by a unitary character times the shift $f(t^{-1}\cdot z)$, and the printed derivation of that classification solves an intertwining equation whose displayed form omits complex conjugates, so the classification needs a corrected computation to stand.

Editorial extensions

If this is right

  • For each generic $\pi$, the operator-to-function map $W_0$ gives a concrete quantization of the coadjoint orbit $O(\xi_0)$, with the constant function 1 corresponding to the identity and the trace pairing matched to the Hilbert–Schmidt inner product.
  • The covariance identity means quantizing first and translating on $\mathbb{C}^n$ is the same as conjugating by the representation first; this is exactly what makes the correspondence geometric rather than an arbitrary symbol choice.
  • In the Schrödinger model, the corresponding map $W_1$ is a Stratonovich–Weyl correspondence on $\mathbb{R}^{2n}$, and the identity $W_1(W(f))(x,y)=f(x,\lambda y)$ ties the construction to ordinary Weyl calculus and to the Moyal product.
  • The group law itself yields star-product identities: the relation $\sigma(t+t')=\sigma(t)\sigma(t')$ becomes a Moyal/Gaussian identity for exponentials of quadratic phase functions, reproducing the classical formula for the Moyal product of two Gaussians.
  • Star exponentials of quadratic polynomials are obtained in closed form from $\exp_{*_0}(W_0(d\pi(X)))=W_0(\pi(\exp X))$, giving the $\cos$-$\tan$ formulas of Corollaries 9.3 and 9.4.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial: the same scheme should work when the action of $\mathbb{R}^m$ on $\mathbb{C}^n$ is any unitary action diagonalized by characters, not just the phase action $e^{i\alpha_k(t)}$; the Gaussian-integral technology would then produce analogous closed formulas for a wider class of solvable semi-direct products.
  • Editorial: because $\psi$ exhibits $\mathbb{C}^n$ as the coadjoint orbit $O(\xi_0)$, the construction suggests the complex Weyl correspondence is the quantization map attached to the complex polarization defining the Fock model; the formulas could be recast as a comparison of two polarizations, a reading the paper does not spell out.
  • Editorial: the star-exponential formulas for quadratic Hamiltonians should reproduce unitary evolution generated by the corresponding Weyl operators; checking the $\tan$/$\cos$ expressions against direct time-evolution calculations would give a clean independent test of the whole chain.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies the generalized diamond group G = R^m ⋉ H_n, where R^m acts on C^n by diagonal phase rotations. It constructs generic unitary representations π of G on the Fock space F_λ by extending the Bargmann-Fock representations of the Heisenberg group, and it computes the Berezin and complex Weyl symbols of the representation operators π(g) and dπ(X). The main structural claim is that the complex Weyl correspondence W_0 is a Stratonovich-Weyl correspondence for the triple (G,π,C^n), and that via a moment map it becomes a Stratonovich-Weyl correspondence on the coadjoint orbit O(ξ_0). The paper also develops a Schrödinger model, derives Mehler-type kernel formulas, and applies the results to recover the Moyal product of Gaussians and to compute star exponentials of quadratic polynomials.

Significance. If the central claims hold, the paper gives explicit, covariant quantization maps on coadjoint orbits for a family of solvable groups, with closed formulas for the symbols of representation operators. This is a natural extension of the author's earlier work on the diamond group and the complex Weyl calculus, and the paper is valuable for bringing together the Berezin calculus, the complex Weyl correspondence, and the orbit method in a concrete setting. The final symbol formulas, once the conjugate typos are corrected, are consistent with known identities: W_0(I)=1, the covariance identities in Section 5 are coherent, and the Gaussian star-product computation in Section 9.2 recovers the classical Moyal formula. The paper does not ship machine-checked proofs, but it does provide explicit integral formulas that are independently verifiable. The main weakness is not the mathematical strategy but the number of missing complex conjugates in load-bearing formulas, which currently make the proof of the classification of extensions and the displayed symbol formula invalid as printed.

major comments (3)
  1. [§4, Eq. (4.3)] Equation (4.3) is missing a complex conjugate: the first exponential on the left should be exp(λ/2 z \overline{t·z0}), not exp(λ/2 (t·z0) z). As printed, after setting w=0 and replacing z0 by w, the left-hand side is a product of an anti-holomorphic kernel and a holomorphic exponential, so it is not anti-holomorphic; the claimed dichotomy with the holomorphic right-hand side does not force the kernel to have the stated form. With the conjugate restored, the anti-holomorphic/holomorphic argument works, but the proof must also justify that the resulting factor C(z) is independent of z; this can be done by substituting the solution back into (4.3) or by invoking the one-dimensionality of Hom(ρ_λ, ρ_λ∘t^{-1}) from the Stone-von Neumann theorem. This correction is load-bearing: Proposition 4.1 underpins the covariance results in Section 5 and the Stratonovich-Weyl conclusion in Proposition 7.2.
  2. [§4, multiplication law] The displayed multiplication law for G is printed with the second slot of ω as (t·z', t·z'), but it should be (t·z', \overline{t·z'}). As written, the third component c+c'+1/2 ω((z,\bar z),(t·z',t·z')) is not real-valued in general, so the formula does not define a real Lie group law. The later formulas in the paper implicitly use the corrected semidirect product, so this is a typo, but it appears at the definition of the central object of the paper and must be fixed.
  3. [§6, Proposition 6.3] The displayed formula for W_0(π(g))(z) is missing conjugates in the first factor of the quadratic exponential: it should read exp( λ/2 ( \overline{t^{-1}·z0} + 2\bar z ) (I_n + A(t))^{-1} ( t^{-1}·z0 + 2z ) ). As printed, taking t=0 and z0=0 gives W_0(I)(z) = exp(-λ|z|^2 + λ z^2), which contradicts the Stratonovich-Weyl normalization W_0(I)=1. The same missing conjugate appears in the definition of I(t,z0,z) and of u in the proof. The correction is also needed for consistency with the Section 9.2 formula for W_0(σ(t)), which contains |z_k|^2 rather than z_k^2; this is a load-bearing computation for the later applications.
minor comments (4)
  1. [§8, proof of Proposition 8.1] The kernel of σ'(t) is denoted b_t(x,y) instead of b'_t(x,y) in the proof, and the displayed double integral defining it contains a spurious φ(y) factor in the integrand; both are typos that make the derivation harder to follow.
  2. [§9.3, proof of Corollary 9.3] The formula "α_k(t) = − 2/λ b_k" is ambiguous; it should read α_k(t) = −(2/λ)b_k to be consistent with the subsequent choice of c = c_0/λ + (1/λ^2)∑ b_k.
  3. [§2, after Eq. (2.1)] The notation L^2(F_λ) for the Hilbert space of Hilbert-Schmidt operators on F_λ is unusual and collides with the L^2 notation for function spaces; consider using HS(F_λ) or S_2(F_λ).
  4. [§8, final paragraph before Proposition 8.3] The word "Melher" should be "Mehler" in the sentence introducing the Mehler-type formula.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Stratonovich-Weyl conclusion follows from covariance proved in this paper plus the independent, parameter-free unitarity of the Fock-space Weyl map W0; the missing-conjugate defect in Eq. (4.3) is a repairable correctness typo, not a circular reduction.

full rationale

The derivation chain is self-contained. Proposition 4.1 derives the form of the extensions σ(t) by solving the intertwining equation (4.3) for the kernels b_t; no ansatz is imported from the cited articles [11,14,15], which are used only for background and for the known unitarity/polar-decomposition property of W0 on the Fock space. Covariance of Sλ and W0 (Propositions 5.2 and 5.3) is proved from the kernel formulas and the change of variables w→t^{-1}w; the symbol formulas in Propositions 6.1, 6.3 and 6.4 are direct Gaussian computations. The orbit map ψ in Proposition 7.1 is defined from the computed symbol W0(dπ(X)) and its covariance, and Proposition 7.2 is then a consequence of that proved covariance together with the unitarity of W0, not a restatement of an input. Section 9's recovery of the known Moyal product of Gaussians is an external consistency check, not an input. The only substantive defect found is the missing complex conjugates in the printed equation (4.3), with an analogous missing bar in the multiplication law; as printed this makes the holomorphy/anti-holomorphy argument in Proposition 4.1 false, but it is a repairable typographical error and does not amount to a circular reduction: the conclusion is not assumed in the hypotheses, and the correct classification follows from Stone–von Neumann. No fitted parameter is relabelled as a prediction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper's central claims rely on standard facts in representation theory and Gaussian analysis plus the author's previously published results on W_0. No free parameters are fitted; λ, χ, α_k are inputs defining the group and representation. No new entities are postulated.

assumptions (5)
  • standard math Stone-von Neumann theorem: for each λ>0 there is a unique irreducible unitary representation ρ_λ of H_n with central character e^{iλc}.
    Used in Section 2 to define the Bargmann-Fock realization and in Section 4 to justify extending by ρ_λ.
  • standard math Gaussian integral formula (Lemma 6.2), a variant of Folland's Theorem 3 (ref [22]).
    Used in Sections 6 and 8 to compute W_0(π(g)) and the Mehler kernel b'_t.
  • domain assumption Properties of the complex Weyl correspondence W_0 from the author's prior work: integral formula (2.3), unitarity of W_0, and the SW property for the diamond group (refs [11,12,14,15]).
    Sections 2, 3, and 7 use these published results as black boxes; they are not re-derived here.
  • domain assumption π(exp X) = exp(dπ X) on analytic vectors for the simply connected group G.
    Used in Section 9.3 to compute star exponentials via W_0(π(exp X)) = exp*0(W_0(dπ X)).
  • standard math The Bargmann transform B: L^2(R^n) → F_λ is unitary and intertwines ρ_λ and ρ'_λ.
    Used in Sections 3 and 8 to pass between Fock and Schrödinger models; standard reference [22].

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Pith. "Pith review of Complex Weyl correspondence for a generalized diamond group." pith.science (2026). https://pith.science/paper/YPSOJJJL

@misc{pith2026250908082,
  author       = {Pith},
  title        = {Pith review of: Complex Weyl correspondence for a generalized diamond group},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YPSOJJJL}},
  note         = {Machine review of arXiv:2509.08082}
}
abstract

The generalized diamond group is the semi-direct product $G$ of the abelian group ${\mathbb R}^m$ by the $(2n+1)$-dimensional Heisenberg group $H_n$. We construct the generic representations of $G$ on the Fock space by extending those of $H_n$. Then we study the Berezin correspondence and the complex Weyl correspondence in connection with a generic representation $\pi$ of $G$, proving in particular that these correspondences are covariant with respect to $\pi$. We give also some explicit formulas for the Berezin symbols and the complex Weyl symbols of the representation operators $\pi(g)$ for $g\in G$. These results are applied to recover various formulas involving the Moyal product. Moreover, we relate $\pi$ to a coadjoint orbit of $G$ in the spirit of the Kirillov-Kostant method of orbits. This allows us to establish that the complex Weyl correspondence is a Stratonovich-Weyl correspondence for $\pi$.

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Works this paper leans on

39 extracted references · 38 canonical work pages

  1. [1]

    Abdelmoula, L., Moment sets and unitary dual for the diamond group. Bull. Sci. Math. 134 (2010), no. 4, 379–390

    work page 2010

  2. [2]

    and Upmeier, H., Weyl Calculus for Complex and Real Symmetric Domains, Harmonic analysis on complex homogeneous domains and Lie groups (Rome, 2001)

    Arazy, J. and Upmeier, H., 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 2002

  3. [3]

    Arnal, D. (1988). The∗-Exponential. In: Cahen, M., Flato, M. (eds) Quantum Theories and Geom- etry. Mathematical Physics Studies, vol 10, pp. 23-51. Springer, Dordrecht, 1988

    work page 1988

  4. [4]

    Analytic methods in mathematical physics (Sympos., Indiana Univ., Bloomington, Ind., 1968), pp

    Bargmann, V., Group representations on Hilbert spaces of analytic functions. Analytic methods in mathematical physics (Sympos., Indiana Univ., Bloomington, Ind., 1968), pp. 27–63. Gordon and Breach, New York, 1970

    work page 1968

  5. [5]

    and Maillard, J.-M., Star exponentials of the elements of the inhomogeneous symplectic Lie algebra

    Bayen, F. and Maillard, J.-M., Star exponentials of the elements of the inhomogeneous symplectic Lie algebra. Lett. Math. Phys. 6 (1982), 491–497

    work page 1982

  6. [6]

    A., Quantization

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

    work page 1974

  7. [7]

    A., Quantization in complex symmetric domains

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

    work page 1975

  8. [8]

    and Vergne, M

    Bernat, P., Conze, N., Duflo, M., L´ evy-Nahas, M., Ra¨ ıs, M., Renouard, P. and Vergne, M. Repr´ esentations des groupes de Lie r´ esolubles. Monographies de la Soci´ et´ e Math´ ematique de France, No. 4. Dunod, Paris, 1972

    work page 1972

Show all 39 references
  1. [9]

    Burdet, G., Perrin, M., Weyl quantization and metaplectic representation. Lett. Math. Phys. 2 (1977/78), 93–99

    1977

  2. [10]

    Cahen, B., Berezin Quantization and Holomorphic Representations. Rend. Sem. Mat. Univ. Padova 129 (2013), 277–297. COMPLEX WEYL CORRESPONDENCE... 21

    2013

  3. [11]

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

    2013

  4. [12]

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

    2018

  5. [13]

    Cahen, B., Stratonovich-Weyl correspondence for the generalized Poincar´ e group. J. Lie Theory 28 (2018), 1043–1062

    2018

  6. [14]

    Tsukuba J

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

    2020

  7. [15]

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

    2023

  8. [16]

    Cahen, B., Complex Weyl symbols of the extended metaplectic representation operators. Oper. Matrices 18 (2024), no. 2, 457–477

    2024

  9. [17]

    and Robert, D., Coherent states and applications in mathematical physics

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

    2012

  10. [18]

    Curtright, T. L. and Zachos, C. K., Quantum Mechanics in Phase Space. Preprint, arXiv: 1104.5269v2[physics.hist-ph] (2011)

    2011 arXiv

  11. [19]

    L., Fairlie, D

    Curtright, T. L., Fairlie, D. B. and Zachos, C. K., Quantum mechanics in phase space. An overview with selected papers. World Scientific Series in 20th Century Physics, 34. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005

    2005

  12. [20]

    and Karczmarczyk, M., Quantization of Gaussians

    Derezi´ nski, J. and Karczmarczyk, M., Quantization of Gaussians. Kurasov, Pavel et al. (eds), Anal- ysis as a tool in mathematical physics. Birkh¨ auser. Oper. Theory: Adv. Appl. 276 (2020), 277-304

    2020

  13. [21]

    de Gosson, M. A. Symplectic methods in harmonic analysis and in mathematical physics. Pseudo- Differential Operators. Theory and Applications, 7. Birkh¨ auser-Springer Basel AG, Basel, 2011

    2011

  14. [22]

    Princeton Univ

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

    1989

  15. [23]

    M., Generalized Moyal quantization on homogeneous symplectic spaces

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

    1990

  16. [24]

    M., V` arilly, J

    Gracia-Bond` ıa, J. M., V` arilly, J. C. and Figueroa, H., Elements of noncommutative geometry. Birkh¨ auser Advanced Texts: Basler Lehrb¨ ucher. [Birkh¨ auser Advanced Texts: Basel Textbooks] Birkh¨ auser Boston, Inc., Boston, MA, 2001

    2001

  17. [25]

    M., V` arilly, J

    Gracia-Bond` ıa, J. M., V` arilly, J. C. and Schempp, W., The Moyal representation of quantum me- chanics and special function theory. Acta Appl. Math. 18 (1990), 225-250

    1990

  18. [26]

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

    1985

  19. [27]

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

    1995

  20. [28]

    A., Lectures on the orbit method

    Kirillov, A. A., Lectures on the orbit method. Graduate Studies in Mathematics 64, American Mathematical Society, Providence, RI, 2004

    2004

  21. [29]

    and Vergne, M., On the Segal-Shale-Weil Representations and Harmonic Polynomials

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

    1978

  22. [30]

    Ludwig, J., Dual topology of diamond groups. J. Reine Angew. Math. 467 (1995), 67–87

    1995

  23. [31]

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

    2000

  24. [32]

    de Gruyter Expositions in Mathematics, Vol

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

    2000

  25. [33]

    A., Lectures on Gaussian integral operators and classical groups

    Neretin, Y. A., Lectures on Gaussian integral operators and classical groups. EMS Series of Lectures in Mathematics. European Mathematical Society (EMS), Z¨ urich, 2011

    2011

  26. [34]

    Pravda-Starov, K., Generalized Mehler formula for time-dependent non-selfadjoint quadratic opera- tors and propagation of singularities. Math. Ann. 372 (2018), 1335–1382

    2018

  27. [35]

    L., On distributions in representation space

    Stratonovich, R. L., On distributions in representation space. Soviet Physics. JETP 4 (1957), 891– 898

    1957

  28. [36]

    F., The representations of the oscillator group

    Streater, R. F., The representations of the oscillator group. Comm. Math. Phys. 4 (1967), 217–236

    1967

  29. [37]

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

    1986

  30. [38]

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

    Unterberger, A. and Upmeier, H., Berezin transform and invariant differential operators. Commun. Math. Phys. 164, 3 (1994), 563–597

    1994

  31. [39]

    Voros, A., An Algebra of Pseudo differential operators and the Asymptotics of Quantum Mechanics. J. Funct. Anal. 29 (1978), 104–132. Universit´e de Lorraine, Site de Metz, UFR-MIM, D ´epartement de math ´ematiques, Bˆatiment A, 3 rue Augustin Fresnel, BP 45112, 57073 METZ Cede...

    1978

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