Symplectic geometry of semisimple orbits |
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Authors: | Hassan Azada Erik van den Ban Indranil Biswas |
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Institution: | aDepartment of Mathematics and Statistics, King Fahd University, Saudi Arabia;bMathematisch Instituut, Universiteit Utrecht, PO Box 80 010, 3508 TA Utrecht, The Netherlands;cSchool of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India |
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Abstract: | Let G be a complex semisimple group, T G a maximal torus and B a Borel subgroup of G containing T. Let Ω be the Kostant-Kirillov holomorphic symplectic structure on the adjoint orbit O = Ad(G)c G/Z(c), where c Lie(T), and Z(c) is the centralizer of c in G. We prove that the real symplectic form Re Ω (respectively, Im Ω) on O is exact if and only if all the eigenvalues ad(c) are real (respectively, purely imaginary). Furthermore, each of these real symplectic manifolds is symplectomorphic to the cotangent bundle of the partial flag manifold G/Z(cc)B, equipped with the Liouville symplectic form. The latter result is generalized to hyperbolic adjoint orbits in a real semisimple Lie algebra. |
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