Comparison of Unitary Duals of Drinfeld Doubles and Complex Semisimple Lie Groups

Abstract

We determine a substantial part of the unitary representation theory of the Drinfeld double of a q-deformation of a compact Lie group in terms of the complexification of the compact Lie group. Using this, we show that the dual of every q-deformation of a higher rank compact Lie group has central property (T). We also determine the unitary dual of \({SL_q(n,\mathbb{C})}\).

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References

  1. 1

    Andersen H.H., Mazorchuk V.: Category \({\mathcal{O}}\) for quantum groups. J. Eur. Math. Soc. (JEMS) 17(2), 405–431 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2

    Arano, Y.: Unitary spherical representations of Drinfeld doubles. J. Reine Angew. Math. (to appear)

  3. 3

    Baumann P.: Another proof of Joseph and Letzter’s separation of variables theorem for quantum groups. Transform. Groups 5(1), 3–20 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4

    Brannan M.: Approximation properties for free orthogonal and free unitary quantum groups. J. Reine Angew. Math. 672, 223–251 (2012)

    MathSciNet  MATH  Google Scholar 

  5. 5

    De Commer K., Freslon A., Yamashita M.: CCAP for universal discrete quantum groups. Comm. Math. Phys. 331(2), 677–701 (2014)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  6. 6

    Duflo, M.: Reprsentations irrductibles des groupes semi-simples complexes. (French) Analyse harmonique sur les groupes de Lie (Sém., Nancy-Strasbourg, 1973–75), pp. 26–88. Lecture Notes in Math., Vol. 497, Springer, Berlin (1975)

  7. 7

    Etingof P., Kazhdan D.: Quantization of Lie bialgebras. VI. Quantization of generalized Kac-Moody algebras. Transform. Groups 13(3-4), 527–539 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8

    Freslon A.: Examples of weakly amenable discrete quantum groups. J. Funct. Anal. 265(9), 2164–2187 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9

    Ghosh S.K., Jones C.: Annular representation theory for rigid C *-tensor categories. J. Funct. Anal. 270(4), 1537–1584 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10

    Jones C.: Quantum G 2 categories have property (T). Internat. J. Math. 27(2), 23 (2016) (1650015)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11

    Joseph, A.: Quantum groups and their primitive ideals. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 29, pp. x+383. Springer, Berlin (1995). ISBN: 3-540-57057-8

  12. 12

    Joseph A., Letzter G.: Verma module annihilators for quantized enveloping algebras. Ann. Sci. École Norm. Sup. (4) 28(4), 493–526 (1995)

    MathSciNet  MATH  Google Scholar 

  13. 13

    Neshveyev, S., Tuset, L.: Compact quantum groups and their representation categories. Cours Spécialisés [Specialized Courses], 20, pp. vi+169. Société Mathématique de France, Paris, (2013). ISBN: 978-2-85629-777-3

  14. 14

    Neshveyev, S., Yamashita, M.: Drinfeld center and representation theory for monoidal categories. Commun. Math. Phys. (to appear)

  15. 15

    Popa S.: Some properties of the symmetric enveloping algebra of a subfactor, with applications to amenability and property T. Doc. Math. 4, 665–744 (1999)

    MathSciNet  MATH  Google Scholar 

  16. 16

    Popa S., Vaes S.: Representation theory for subfactors, λ-lattices and C *-tensor categories. Commun. Math. Phys. 340(3), 1239–1280 (2015)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  17. 17

    Pusz W.: Irreducible unitary representations of quantum Lorentz group. Commun. Math. Phys. 152(3), 591–626 (1993)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  18. 18

    Voigt C.: The Baum–Connes conjecture for free orthogonal quantum groups. Adv. Math. 227(5), 1873–1913 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19

    Voigt C., Yuncken R.: On the principal series representations of quantized complex semisimple groups (in preparation)

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Correspondence to Yuki Arano.

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Communicated by Y. Kawahigashi

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Arano, Y. Comparison of Unitary Duals of Drinfeld Doubles and Complex Semisimple Lie Groups. Commun. Math. Phys. 351, 1137–1147 (2017). https://doi.org/10.1007/s00220-016-2704-x

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