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.