I tried asking this question on stackexchange and received no response.
Given a semisimple Lie group, there is a symplectic structure on the coadjoint orbits arising from the Kirillov 2-form. Can I use this to define a volume form on the adjoint orbits, perhaps pulling it back via a homeomorphism between corresponding adjoint and coadjoint orbits?
I know there is another way to get a volume on adjoint orbits, via 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.
My end goal is this; given $f:\mathfrak{g} \to \mathbb{R}$, and $X \in \mathfrak{g}$ I would like to define some type of integration $\int_{O(X)} f \cdot d\omega$. As mentioned above, there is a way this is usually done by taking a Haar measure on $G$ and getting some normalized quotient measure $\dot{dg}$ over $G/C_G(X)$, giving the so-called orbital integrals $\int_{G/C_G(X)} f(gXg^{-1})\, \dot{dg}$. But I would like to instead have some general volume form on $O(X)$. The Kirillov-Kostant-Souriau symplectic structure on the coadjoint orbits seems to be the way to do this, but I am unsure about the technical details of such an approach.
