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Invariant Differential Operators for Non-compact Lie Groups: The Sp(n, IR) Case

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Lie Theory and Its Applications in Physics

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 36))

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Abstract

In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebras sp(n, I​R), in detail for n = 6. Our choice of these algebras is motivated by the fact that they belong to a narrow class of algebras, which we call “conformal Lie algebras”, which have very similar properties to the conformal algebras of Minkowski space-time. We give the main multiplets and the main reduced multiplets of indecomposable elementary representations for n = 6, including the necessary data for all relevant invariant differential operators. In fact, this gives by reduction also the cases for n < 6, since the main multiplet for fixed n coincides with one reduced case for n + 1.

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Notes

  1. 1.

    For explicit expressions for singular vectors we refer to [22, 23].

  2. 2.

    Generically, the Knapp–Stein operators can be normalized so that indeed \(G_{KS} \circ G_{KS} =\mathrm{ Id}_{\mathcal{C}_{\chi }}\,\). However, this usually fails exactly for the reducible ERs that form the multiplets, cf., e.g., [15–17].

  3. 3.

    For a different use of E 7( − 25), see, e.g., [28].

References

  1. Terning, J.: Modern supersymmetry: Dynamics and duality. In: International Series of Monographs on Physics, vol. 132, Oxford University Press, Oxford (2005)

    Google Scholar 

  2. Dobrev, V.K.: Rev. Math. Phys. 20, 407–449 (2008) [hep-th/0702152]

    Article  MathSciNet  MATH  Google Scholar 

  3. Dobrev, V.K.: J. Phys. A Math. Theor. 42, 285203 (2009) (ar**v:0812.2690 [hep-th])

    Google Scholar 

  4. Faraut, J., Korányi, A.: Analysis on symmetric cones. In: Oxford Mathematical Monographs. Clarendon Press, Oxford (1994)

    Google Scholar 

  5. Gunaydin, M.: Mod. Phys. Lett. A8, 1407–1416 (1993)

    MathSciNet  Google Scholar 

  6. Mack, G., de Riese, M.: J. Math. Phys. 48, 052304 (2007) [hep-th/0410277v2]

    Article  MathSciNet  Google Scholar 

  7. Dobrev, V.K.: Rept. Math. Phys. 25, 159–181 (1988) [first as ICTP Trieste preprint IC/86/393 (1986)]

    Google Scholar 

  8. Langlands, R.P.: On the classification of irreducible representations of real algebraic groups. In: Mathematical Surveys and Monographs, vol. 31. American Mathematical Society (1988) [first as IAS Princeton preprint (1973)]

    Google Scholar 

  9. Zhelobenko, D.P.: Harmonic Analysis on Semisimple Complex Lie Groups. Nauka, Moscow (1974) (in Russian)

    Google Scholar 

  10. Knapp, A.W., Zuckerman, G.J.: Classification theorems for representations of semisimple groups. In: Lecture Notes in Mathematics, vol. 587. Springer, Berlin (1977); Classification of irreducible tempered representations of semisimple groups. Ann. Math. 116, 389–501 (1982)

    Google Scholar 

  11. Harish-Chandra, Discrete series for semisimple Lie groups: II. Ann. Math. 116, 1–111 (1966)

    Google Scholar 

  12. Enright, T., Howe, R., Wallach, W.: A classification of unitary highest weight modules. In: Trombi, P. (ed.) Representations of Reductive Groups, pp. 97–143. Birkhäuser, Boston (1983)

    Chapter  Google Scholar 

  13. Dobrev, V.K.: J. Phys. A Math. Theor. 41, 425206 (2008) [ar**v:0712.4375]

    Article  MathSciNet  Google Scholar 

  14. Grosse, H., Prešnajder, P., Wang, Zh.: J. Math. Phys. 53, 013508 (2012)

    Article  MathSciNet  Google Scholar 

  15. Dobrev, V.K., Mack, G., Petkova, V.B., Petrova, S.G., Todorov, I.T.: Harmonic analysis on the n-dimensional Lorentz group and its applications to conformal quantum field theory. In: Lecture Notes in Physics, vol. 63. Springer, Berlin (1977)

    Google Scholar 

  16. Dobrev, V.K., Mack, G., Petkova, V.B., Petrova, S.G., Todorov, I.T.: Rep. Math. Phys. 9, 219–246 (1976)

    Article  MATH  Google Scholar 

  17. Dobrev, V.K., Petkova, V.B.: Rep. Math. Phys. 13, 233–277 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  18. Knapp, A.W.: Representation Theory of Semisimple Groups (An Overview Based on Examples). Princeton University Press, Princeton (1986)

    MATH  Google Scholar 

  19. Dobrev, V.K.: Lett. Math. Phys. 9, 205–211 (1985); J. Math. Phys. 26, 235–251 (1985)

    Google Scholar 

  20. Bernstein, I.N., Gel’fand, I.M., Gel’fand, S.I.: Structure of representations generated by highest weight vectors. Funkts. Anal. Prilozh. 5(1), 1–9 (1971) [English translation: Funct. Anal. Appl. 5, 1–8 (1971)]

    Google Scholar 

  21. Dixmier, J.: Envelo** Algebras. North Holland, New York (1977)

    Google Scholar 

  22. Dobrev, V.K.: Lett. Math. Phys. 22, 251–266 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  23. Chair, N., Dobrev, V.K., Kanno, H.: Phys. Lett. 283B, 194–202 (1992)

    MathSciNet  Google Scholar 

  24. Satake, I.: Ann. Math. 71, 77–110 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  25. Harish-Chandra, Representations of semisimple Lie groups: IV, V. Am. J. Math. 77, 743–777 (1955); 78, 1–41 (1956)

    Google Scholar 

  26. Bourbaki, N.: Groupes at algèbres de Lie, Chapitres 4, 5 et 6. Hermann, Paris (1968)

    Google Scholar 

  27. Knapp, A.W., Stein, E.M.: Intertwining operators for semisimple groups. Ann. Math. 93, 489–578 (1971); II: Inv. Math. 60, 9–84 (1980)

    Google Scholar 

  28. Cacciatori, S.L., Cerchiai, B.L., Marrani, A.: Magic Coset Decompositions [ar**v:1201.6314], CERN-PH-TH/2012-020

    Google Scholar 

  29. Dobrev, V.K.: Invited lectures at 5th meeting on modern mathematical physics, Belgrade, 6–17 July 2008. In: Dragovich, B., Rakic, Z. (eds.) Proceedings, pp. 95–124. Institute of Physics, Belgrade (2009) (ar**v:0812.2655 [math-ph])

    Google Scholar 

  30. Dobrev, V.K.: Invariant Differential Operators for Non-Compact Lie Groups: the Main SU(n,n) Cases, Plenary talk at SYMPHYS XV (Dubna, 12–16.7.2011). In: Pogosyan, G., et al. (eds.) Proceedings, Physics of Atomic Nuclei.

    Google Scholar 

  31. Witten, E.: Conformal Field Theory in Four and Six Dimensions [ar**v:0712.0157]

    Google Scholar 

  32. Dobrev, V.K., Petkova, V.B.: Phys. Lett. B162, 127–132 (1985); Lett. Math. Phys. 9, 287–298 (1985); Fortsch. Phys. 35, 537–572 (1987)

    Google Scholar 

  33. Dobrev, V.K.: Phys. Lett. B186, 43–51 (1987); J. Phys. A35, 7079–7100 (2002) [hep-th/0201076]; Phys. Part. Nucl. 38, 564–609 (2007) [hep-th/0406154]

    Google Scholar 

  34. Dobrev, V.K., Ganchev, A.Ch.: Mod. Phys. Lett. A3, 127–137 (1988)

    MathSciNet  Google Scholar 

  35. Dobrev, V.K.: J. Math. Phys. 33, 3419–3430 (1992); J. Phys. A26, 1317–1334 (1993) [first as Göttingen University preprint (July 1991)];  J. Phys. A27, 4841–4857 (1994) [Erratum-ibid. A27, 6633–6634 (1994) (hep-th/9405150)];   Phys. Lett. B341, 133–138 (1994) [Erratum-ibid. B346, 427 (1995)]

    Google Scholar 

  36. Dobrev, V.K., Moylan, P.J.: Phys. Lett. B315, 292–298 (1993)

    MathSciNet  Google Scholar 

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Acknowledgements

This work was supported in part by the Bulgarian National Science Fund, grant DO 02-257. The author thanks the Theory Division of CERN for hospitality during the course of this work.

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Authors and Affiliations

  1. Theory Division, Department of Physics, CERN, CH-1211, Geneva 23, Switzerland

    V. K. Dobrev

  2. Institute for Nuclear Research and Nuclear Energy, Tsarigradsko Chaussee 72, 1784, Sofia, Bulgaria

    V. K. Dobrev

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  1. V. K. Dobrev
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Correspondence to V. K. Dobrev .

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  1. Bulgarian Academy of Sciences, 72 Tsarigradsko Chaussee, Sofia, 1784, Sofiya, Bulgaria

    Vladimir Dobrev

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Dobrev, V.K. (2013). Invariant Differential Operators for Non-compact Lie Groups: The Sp(n, IR) Case. In: Dobrev, V. (eds) Lie Theory and Its Applications in Physics. Springer Proceedings in Mathematics & Statistics, vol 36. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54270-4_22

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