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Polar Symplectic Representations

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Abstract

We study polar representations in the sense of Dadok and Kac which are symplectic. We show that such representations are coisotropic and use this fact to give a classification. We also study their moment maps and prove that they separate closed orbits. Our work can also be seen as a specialization of some of the results of Knop on multiplicity free symplectic representations to the polar case.

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References

  1. Bieliavsky, P., Cahen, M., Gutt, S.: Symmetric symplectic manifolds and deformation quantization. Modern group theoretical methods in physics. In: Bertrand, J., Flato, M., Gazeau, J.-P., Irac-Astaud, M., Sternheimer, D. (eds.) Proceedings of the Conference in Honour of Guy Rideau. Math. Phys. Studies, no. 18, pp. 63–75. Kluwer Academic Publishers (1995)

  2. Bieliavsky, P.: Semisimple symplectic symmetric spaces, Ph.D. thesis, Université Libre de Bruxelles. E-print arXiv:math/0703358 [math.DG] (1995)

  3. Bieliavsky, P.: Semisimple symplectic symmetric spaces. Geom. Dedicata 73(3), 245–273 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Birkes, D.: Orbits of linear algebraic groups. Ann. Math. 93(3), 459–475 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bourbaki, N.: Éléments de mathématique: Groupes et algèbres de Lie, Fascicule XXXIV, Chapitres IV, V, VI, Hermann (1968)

  6. Benson, C., Ratcliff, G.: On multiplicity free actions, Representations of real and p-adic groups, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., vol. 2, pp. 221–304. Singapore Univ. Press, Singapore (2004)

    Google Scholar 

  7. Bremigan, R.: Galois cohomology and real algebraic quotients. J. Algebra 159(2), 275–305 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dadok, J.: Polar coordinates induced by actions of compact Lie groups. Trans. Amer. Math. Soc. 288, 125–137 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dadok, J., Kac, V.: Polar representations. J. Algebra 92, 504–524 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  10. Daszkiewicz, A., Przebinda, T.: On the moment map of a multiplicity free action. Colloq. Math. 71(1), 107–110 (1996)

    MathSciNet  MATH  Google Scholar 

  11. Eschenburg, J., Heintze, E.: On the classification of polar representations. Math. Z. 232, 391–398 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Geatti, L., Gorodski, C.: Polar orthogonal representations of real reductive algebraic groups. J. Algebra 320, 3036–3061 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Huckleberry, A.T., Wurzbacher, T.: Multiplicity-free complex manifolds. Math. Ann. 286, 261–280 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kac, V.: Some remarks on nilpotent orbits. J. Algebra 64(1), 190–213 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  15. Knop, F.: Some remarks on multiplicity free spaces, Representation theories and algebraic geometry (Montreal, PQ, 1997), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 514, pp. 301–317. Kluwer Acad. Publ., Dordrecht (1998). Electronic version http://www.algeo.math.fau.de/fileadmin/algeo/users/knop/papers/montreal.html

  16. Knop, F.: Classification of multiplicity free symplectic representations. J. Algebra 301(2), 531–553 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Knop, F.: Invariant functions on symplectic representations. J. Algebra 313(1), 223–251 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Littelmann, P.: Koreguläre und äquidimensionale Darstellungen. J. Algebra 123(1), 193–222 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  19. Losev, I.V.: Coisotropic representations of reductive groups. (Russian). Tr. Mosk. Mat. Obs. 66, 156–183 (2005). English translation in Trans. Moscow Math. Soc. (2005), 143–168

    MATH  Google Scholar 

  20. Luna, D.: Sur les orbites fermées des groupes algébriques réductifs. Invent. Math. 16, 1–5 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  21. Luna, D.: Slices étales. Memoires Bull. Soc. Math. France 33, 81–105 (1973)

    Article  MATH  Google Scholar 

  22. Milne, J.S.: Algebraic Geometry. http://www.jmilne.org/math/CourseNotes/ag.html (last checked on 2014-6-28) (2012)

  23. Podestà, F., Thorbergsson, G.: Polar and coisotropic actions on Kahler̈ manifolds. Trans. Amer. Math. Soc. 354, 1759–1781 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Popov, V.L., Vinberg, E.B.: Invariant theory, Encyclopaedia of Mathematical Sciences, ch. II, vol. 55, pp. 123–278. Springer-Verlag, Berlin (1994)

    Google Scholar 

  25. Schwarz, G.W.: Lifting smooth homotopies of orbit spaces. I.H.E.S. Publ. Math. 51, 37–135 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  26. Vinberg, E.B.: The Weyl group of a graded Lie algebra. (Russian). Izv. Akad. Nauk SSSR Ser. Mat. 40(3), 488–526, 709 (1976). English translation in Math. USSR-Izv. 10 (1976), 463–495

    MathSciNet  MATH  Google Scholar 

  27. Vinberg, E.B.: Commutative homogeneous spaces and co-isotropic symplectic actions. (Russian). Uspekhi Mat. Nauk 56(1(337)), 3–62 (2001). English translation in Russian. Math. Surveys 56(1) (2001), 1–60

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma 2 Tor Vergata, via della Ricerca Scientifica, 00133, Roma, Italy

    Laura Geatti

  2. Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão, 1010, São Paulo, SP 05508-090, Brazil

    Claudio Gorodski

Authors
  1. Laura Geatti
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  2. Claudio Gorodski
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Correspondence to Claudio Gorodski.

Additional information

Presented by Peter Littelman

The second author has been partially supported by the CNPq grant 303038/2013-6 and the FAPESP project 2011/21362-2.

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Geatti, L., Gorodski, C. Polar Symplectic Representations. Algebr Represent Theor 20, 751–764 (2017). https://doi.org/10.1007/s10468-016-9663-y

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