Can I bring the Kirilov 2-form on coadjoint orbits to adjoint orbits?

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Given a semisimple lie group, there is a symplectic structure on the coadjoint orbits arising from the Kirilov 2-form. Can I use this to define a volume form on the adjoint orbits, perhaps pulling it back via the homeomorphism between corresponding adjoint and coadjoint orbits?



I know there is another way to get to volume on adjoint orbits via the Haar measure on the lie group and taking the quotient measure identifying the orbit with the quotient of G by the stabilizer, but I would like to do it ignoring this identification entirely.




differential-geometry differential-topology lie-groups lie-algebras




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    Given a semisimple lie group, there is a symplectic structure on the coadjoint orbits arising from the Kirilov 2-form. Can I use this to define a volume form on the adjoint orbits, perhaps pulling it back via the homeomorphism between corresponding adjoint and coadjoint orbits?



    I know there is another way to get to volume on adjoint orbits via the Haar measure on the lie group and taking the quotient measure identifying the orbit with the quotient of G by the stabilizer, but I would like to do it ignoring this identification entirely.




    differential-geometry differential-topology lie-groups lie-algebras




    share|cite|improve this question





















      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      Given a semisimple lie group, there is a symplectic structure on the coadjoint orbits arising from the Kirilov 2-form. Can I use this to define a volume form on the adjoint orbits, perhaps pulling it back via the homeomorphism between corresponding adjoint and coadjoint orbits?



      I know there is another way to get to volume on adjoint orbits via the Haar measure on the lie group and taking the quotient measure identifying the orbit with the quotient of G by the stabilizer, but I would like to do it ignoring this identification entirely.




      differential-geometry differential-topology lie-groups lie-algebras




      share|cite|improve this question











      Given a semisimple lie group, there is a symplectic structure on the coadjoint orbits arising from the Kirilov 2-form. Can I use this to define a volume form on the adjoint orbits, perhaps pulling it back via the homeomorphism between corresponding adjoint and coadjoint orbits?



      I know there is another way to get to volume on adjoint orbits via the Haar measure on the lie group and taking the quotient measure identifying the orbit with the quotient of G by the stabilizer, but I would like to do it ignoring this identification entirely.





      differential-geometry differential-topology lie-groups lie-algebras





      share|cite|improve this question










      share|cite|improve this question




      share|cite|improve this question









      asked Aug 4 at 2:08









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