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2005 | 3 | 3 | 430-474
Tytuł artykułu

Dual pairs and Kostant-Sekiguchi correspondence. II. Classification of nilpotent elements

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We classify the homogeneous nilpotent orbits in certain Lie color algebras and specialize the results to the setting of a real reductive dual pair. For any member of a dual pair, we prove the bijectivity of the two Kostant-Sekiguchi maps by straightforward argument. For a dual pair we determine the correspondence of the real orbits, the correspondence of the complex orbits and explain how these two relations behave under the Kostant-Sekiguchi maps. In particular we prove that for a dual pair in the stable range there is a Kostant-Sekiguchi map such that the conjecture formulated in [6] holds. We also show that the conjecture cannot be true in general.
Wydawca
Czasopismo
Rocznik
Tom
3
Numer
3
Strony
430-474
Opis fizyczny
Daty
wydano
2005-09-01
online
2005-09-01
Twórcy
Bibliografia
  • [1] G. Benkart, Ch.L. Shader and A. Ram: “Tensor product representations for orthosymplectic Lie superalgebras”, J. Pure Appl. Algebra, Vol. 130(1), (1998), pp. 1–48. http://dx.doi.org/10.1016/S0022-4049(97)00084-4
  • [2] N. Burgoyne and R. Cushman: “Conjugacy Classes in Linear Groups”, Journal of Algebra, Vol. 44, (1975), pp. 339–362. http://dx.doi.org/10.1016/0021-8693(77)90186-7
  • [3] D. Barbasch and M. Sepanski: “Closure ordering and the Kostant-Sekiguchi correspondence”, Proc. Amer. Math. Soc., Vol. 126(1), (1998), pp. 311–317. http://dx.doi.org/10.1090/S0002-9939-98-04090-8
  • [4] D. Collingwood and W. McGovern: Nilpotent orbits in complex semisimple Lie algebras, Reinhold, Van Nostrand, New York, 1993.
  • [5] A. Daszkiewicz, W. Kraśkiewicz and T. Przebinda: “Nilpotent Orbits and Complex Dual Pairs”, J. Algebra, Vol. 190, (1997), pp. 518–539. http://dx.doi.org/10.1006/jabr.1996.6910
  • [6] A. Daszkiewicz, W. Kraśkiewicz and T. Przebinda: “Dual Pairs and Kostant-Sekiguchi Correspondence. I.”, J. Algebra, Vol. 250, (2002), pp. 408–426. http://dx.doi.org/10.1006/jabr.2001.9080
  • [7] A. Daszkiewicz and T. Przebinda: “The Oscillator Character Formula, for isometry groups of split forms in deep stable range”, Invent. Math., Vol. 123, (1996), pp. 349–376.
  • [8] D. Ž. Djokovič: Closures of Conjugacy Classes in Classical Real Linear Lie Groups, Lecture Notes in Mathematics, Vol. 848, Springer Verlag, 1980, pp. 63–83. http://dx.doi.org/10.1007/BFb0090557
  • [9] Harish-Chandra: “Invariant distributions on Lie algebras”, Amer. J. Math., Vol. 86, (1964), pp. 271–309. http://dx.doi.org/10.2307/2373165
  • [10] R. Howe: “θ-series and invariant theory”, Proc. Symp. Pure. Math., Vol. 33, (1979), pp. 275–285.
  • [11] R. Howe: A manuscript on dual pairs, preprint.
  • [12] G. Kempken: Eine Darstellung des Köchers à k . Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonner Mathematische Schriften, Vol. 137, Bonn, 1981.
  • [13] Nishiyama, Kyo, Zhu and Chen-Bo: “Theta lifting of unitary lowest weight modules and their associated cycles”, Duke Math. Jour., Vol. 125(3), (2004), pp. 415–465. http://dx.doi.org/10.1215/S0012-7094-04-12531-X
  • [14] Nishiyama, Kyo, Zhu and Chen-Bo: “Theta lifting of nilpotent orbits for symmetric pairs”, Trans. Amer. Math. Soc. to appear.
  • [15] T. Ohta: “The singularities of the closures of nilpotent orbits in certain symmetric pairs”, Tôhoku Math. J., Vol. 38, (1986), pp. 441–468.
  • [16] T. Ohta: “The closures of nilpotent orbits in the classical symmetric pairs and their singularities”, Tôhoku Math. J., Vol. 43, (1991), pp. 161–211.
  • [17] T. Przebinda: “Characters, dual pairs, and unitary representations”, Duke Math. J., Vol. 69(3), (1993), pp. 547–592. http://dx.doi.org/10.1215/S0012-7094-93-06923-2
  • [18] J. Sekiguchi: “The nilpotent subvariety of the vector space associated to a symmetric pair”, Publ. RIMS, Vol. 20, (1984), pp. 155–212.
  • [19] J. Sekiguchi: “Remarks on real nilpotent orbits of a symmetric pair”, J. Math. Soc. Japan, Vol. 39, (1987), pp. 127–138. http://dx.doi.org/10.2969/jmsj/03910127
  • [20] W. Schmid and K. Vilonen: “Characteristic cycles and wave front cycles of representations of reductive Lie groups”, Ann. of Math. 2 Vol. 151(3), (2000), pp. 1071–1118. http://dx.doi.org/10.2307/121129
  • [21] D. Vogan: Representations of reductive Lie groups. A plenary address presented at the International Congress of Mathematicians held in Berkeley, California, August 1986. Introduced by Wilfried Schmid., ICM Series, American Mathematical Society, Providence, RI, 1988.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_BF02475917
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