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On the Moment Map of a Multiplicity Free Action

100%
Daszkiewicz A., Przebinda T.
Colloquium Mathematicum
|
1996
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tom 71
|
nr 1
107-110
EN
The purpose of this note is to show that the Orbit Conjecture of C. Benson, J. Jenkins, R. L. Lipsman and G. Ratcliff [BJLR1] is true. Another proof of that fact has been given by those authors in [BJLR2]. Their proof is based on their earlier results, announced together with the conjecture in [BJLR1]. We follow another path: using a geometric quantization result of Guillemin-Sternberg [G-S] we reduce the conjecture to a similar statement for a projective space, which is a special case of a characterization of projective smooth spherical varieties due to Brion [B2].
2
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The Cauchy Harish-Chandra Integral, for the pair $$\mathfrak{u}_{p,q} ,\mathfrak{u}_1 $$

100%
Daszkiewicz A., Przebinda T.
Open Mathematics
|
2007
|
tom 5
|
nr 4
654-664
EN
For the dual pair considered, the Cauchy Harish-Chandra Integral, as a distribution on the Lie algebra, is the limit of the holomorphic extension of the reciprocal of the determinant. We compute that limit explicitly in terms of the Harish-Chandra orbital integrals.
3
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Dual pairs and Kostant-Sekiguchi correspondence. II. Classification of nilpotent elements

81%
Daszkiewicz A., Kraśkiewicz W., Przebinda T.
Open Mathematics
|
2005
|
tom 3
|
nr 3
430-474
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.
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