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

Andrzej Daszkiewicz; Witold Kraśkiewicz; Tomasz Przebinda

Open Mathematics (2005)

- Volume: 3, Issue: 3, page 430-474
- ISSN: 2391-5455

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topAndrzej Daszkiewicz, Witold Kraśkiewicz, and Tomasz Przebinda. "Dual pairs and Kostant-Sekiguchi correspondence. II. Classification of nilpotent elements." Open Mathematics 3.3 (2005): 430-474. <http://eudml.org/doc/268787>.

@article{AndrzejDaszkiewicz2005,

abstract = {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.},

author = {Andrzej Daszkiewicz, Witold Kraśkiewicz, Tomasz Przebinda},

journal = {Open Mathematics},

keywords = {20G05; 17B75; 22E45},

language = {eng},

number = {3},

pages = {430-474},

title = {Dual pairs and Kostant-Sekiguchi correspondence. II. Classification of nilpotent elements},

url = {http://eudml.org/doc/268787},

volume = {3},

year = {2005},

}

TY - JOUR

AU - Andrzej Daszkiewicz

AU - Witold Kraśkiewicz

AU - Tomasz Przebinda

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

JO - Open Mathematics

PY - 2005

VL - 3

IS - 3

SP - 430

EP - 474

AB - 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.

LA - eng

KW - 20G05; 17B75; 22E45

UR - http://eudml.org/doc/268787

ER -

## References

top- [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 Zbl0932.17008
- [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 Zbl0896.22004
- [4] D. Collingwood and W. McGovern: Nilpotent orbits in complex semisimple Lie algebras, Reinhold, Van Nostrand, New York, 1993. Zbl0972.17008
- [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 Zbl1003.22003
- [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. Zbl0845.22007
- [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 Zbl0131.33302
- [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 Ã 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 Zbl1078.22010
- [14] Nishiyama, Kyo, Zhu and Chen-Bo: “Theta lifting of nilpotent orbits for symmetric pairs”, Trans. Amer. Math. Soc. to appear. Zbl1091.22002
- [15] T. Ohta: “The singularities of the closures of nilpotent orbits in certain symmetric pairs”, Tôhoku Math. J., Vol. 38, (1986), pp. 441–468. Zbl0654.22004
- [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. Zbl0738.22007
- [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 Zbl0788.22018
- [18] J. Sekiguchi: “The nilpotent subvariety of the vector space associated to a symmetric pair”, Publ. RIMS, Vol. 20, (1984), pp. 155–212. Zbl0556.14022
- [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 Zbl0627.22008
- [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 Zbl0960.22009
- [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.